An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

210 The integral of explicitly irrational functions. Bk. 1l. ch. IT1.
fF(x, /R) d x, where i = a + bx + c2 + dx3, (e 0),
can be reduced to algebraic and logarithmic functions and to three
fundamental integrals:
rdx rxdx f dx
J V ' J / ' (x - Q) VR
When arranged according to its rational and irrational parts, we
have (compare ~ 119):
F(xV ) = G X) + H(x) /R =( ) + (X)
(p and p denoting rational functions; the integration of Sp leads to
algebraic and logarithmic expressions; the second part gives us:,?(X} f /c) 1 An(Xn ) Bn _1f
Ip*)B f(x) _ A ln y___ 1
xn R vn V R (C-) VR
The integral fx dx can be reduced by a recurring formula: From
d(xn-2 V_) _ (n-2)xn-3(a+bx+cx+dx3) dx+ XS-(b+2cx+3dX2 dx,
VP 2 y/A
we find by integration:
x._-2j - anr x-3 dx s)Jx d2 (7
Ie x- /2R a(n - 2)J X + &( 3) J 3 2Z
\ ~xn-s x ~ n-\ dx 7X' dx
+ c(" - J -~r + C(n- ( 2)J.
Now putting in succession for n the values 2, 3, 4..., we get:
xI2x 2 b idx _ 2c xdx
R- - d -dJ YP 3dJ VP]
cd 2= - 2a / dX 3b 6- dc 4c -'c/tS
rx.dx _ _^_ /v 7 ^a_ r~d^, ~ __ f^l^, _ al jxtd
J V R 5d, 5XV -5dJ VR 5dJ /V 5dJ V
In general for n > 2:
v= n-i
fx s d /R 2vxn + IJdx + Jxdx
means of the integral; for these relations Euler (in the Nov. Corn. Petrop., 1761,
Vols. VI, VII; see also his Inst. calc. integral., Vol. I, Sect. 2, Cap. VI, ~~ 606-649)
discovered the general ground in the theorem of addition: "par nne combinaison
qu'on peut regarder comme fort heureuse, quoique ces hasards n'arrivent qu'A,
ceux qui savent les faire naitre", as Legendre says in the Introduction to his great
work. Euler perceived that by these integrals new functions are introduced into
analysis, so that the group of transcendental functions (logarithmic and circular, and
their inverses: the exponential and trigonometric) becomes legitimately enlarged.
Legendre (1752-1833) established a theory of these new transcendents by his:
Traitd des fonctions elliptiques et des integrales Euleriennes, 1825-26.