3.1372 \(\int \frac{1}{(a+b x) (c+d x)^8} \, dx\)

Optimal. Leaf size=202 \[ \frac{b^7 \log (a+b x)}{(b c-a d)^8}-\frac{b^7 \log (c+d x)}{(b c-a d)^8}+\frac{b^6}{(c+d x) (b c-a d)^7}+\frac{b^5}{2 (c+d x)^2 (b c-a d)^6}+\frac{b^4}{3 (c+d x)^3 (b c-a d)^5}+\frac{b^3}{4 (c+d x)^4 (b c-a d)^4}+\frac{b^2}{5 (c+d x)^5 (b c-a d)^3}+\frac{b}{6 (c+d x)^6 (b c-a d)^2}+\frac{1}{7 (c+d x)^7 (b c-a d)} \]

[Out]

1/(7*(b*c - a*d)*(c + d*x)^7) + b/(6*(b*c - a*d)^2*(c + d*x)^6) + b^2/(5*(b*c -
a*d)^3*(c + d*x)^5) + b^3/(4*(b*c - a*d)^4*(c + d*x)^4) + b^4/(3*(b*c - a*d)^5*(
c + d*x)^3) + b^5/(2*(b*c - a*d)^6*(c + d*x)^2) + b^6/((b*c - a*d)^7*(c + d*x))
+ (b^7*Log[a + b*x])/(b*c - a*d)^8 - (b^7*Log[c + d*x])/(b*c - a*d)^8

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Rubi [A]  time = 0.343686, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b^7 \log (a+b x)}{(b c-a d)^8}-\frac{b^7 \log (c+d x)}{(b c-a d)^8}+\frac{b^6}{(c+d x) (b c-a d)^7}+\frac{b^5}{2 (c+d x)^2 (b c-a d)^6}+\frac{b^4}{3 (c+d x)^3 (b c-a d)^5}+\frac{b^3}{4 (c+d x)^4 (b c-a d)^4}+\frac{b^2}{5 (c+d x)^5 (b c-a d)^3}+\frac{b}{6 (c+d x)^6 (b c-a d)^2}+\frac{1}{7 (c+d x)^7 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*(c + d*x)^8),x]

[Out]

1/(7*(b*c - a*d)*(c + d*x)^7) + b/(6*(b*c - a*d)^2*(c + d*x)^6) + b^2/(5*(b*c -
a*d)^3*(c + d*x)^5) + b^3/(4*(b*c - a*d)^4*(c + d*x)^4) + b^4/(3*(b*c - a*d)^5*(
c + d*x)^3) + b^5/(2*(b*c - a*d)^6*(c + d*x)^2) + b^6/((b*c - a*d)^7*(c + d*x))
+ (b^7*Log[a + b*x])/(b*c - a*d)^8 - (b^7*Log[c + d*x])/(b*c - a*d)^8

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x+c)**8,x)

[Out]

Timed out

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Mathematica [A]  time = 0.144475, size = 196, normalized size = 0.97 \[ \frac{420 b^7 (c+d x)^7 \log (a+b x)+420 b^6 (c+d x)^6 (b c-a d)+210 b^5 (c+d x)^5 (b c-a d)^2+140 b^4 (c+d x)^4 (b c-a d)^3+105 b^3 (c+d x)^3 (b c-a d)^4+84 b^2 (c+d x)^2 (b c-a d)^5+70 b (c+d x) (b c-a d)^6+60 (b c-a d)^7-420 b^7 (c+d x)^7 \log (c+d x)}{420 (c+d x)^7 (b c-a d)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)*(c + d*x)^8),x]

[Out]

(60*(b*c - a*d)^7 + 70*b*(b*c - a*d)^6*(c + d*x) + 84*b^2*(b*c - a*d)^5*(c + d*x
)^2 + 105*b^3*(b*c - a*d)^4*(c + d*x)^3 + 140*b^4*(b*c - a*d)^3*(c + d*x)^4 + 21
0*b^5*(b*c - a*d)^2*(c + d*x)^5 + 420*b^6*(b*c - a*d)*(c + d*x)^6 + 420*b^7*(c +
d*x)^7*Log[a + b*x] - 420*b^7*(c + d*x)^7*Log[c + d*x])/(420*(b*c - a*d)^8*(c +
d*x)^7)

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Maple [A]  time = 0.025, size = 192, normalized size = 1. \[ -{\frac{1}{ \left ( 7\,ad-7\,bc \right ) \left ( dx+c \right ) ^{7}}}-{\frac{{b}^{2}}{5\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{5}}}-{\frac{{b}^{4}}{3\, \left ( ad-bc \right ) ^{5} \left ( dx+c \right ) ^{3}}}-{\frac{{b}^{6}}{ \left ( ad-bc \right ) ^{7} \left ( dx+c \right ) }}+{\frac{b}{6\, \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{6}}}+{\frac{{b}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) ^{4}}}+{\frac{{b}^{5}}{2\, \left ( ad-bc \right ) ^{6} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{7}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{8}}}+{\frac{{b}^{7}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)^8,x)

[Out]

-1/7/(a*d-b*c)/(d*x+c)^7-1/5*b^2/(a*d-b*c)^3/(d*x+c)^5-1/3*b^4/(a*d-b*c)^5/(d*x+
c)^3-b^6/(a*d-b*c)^7/(d*x+c)+1/6*b/(a*d-b*c)^2/(d*x+c)^6+1/4*b^3/(a*d-b*c)^4/(d*
x+c)^4+1/2*b^5/(a*d-b*c)^6/(d*x+c)^2-b^7/(a*d-b*c)^8*ln(d*x+c)+b^7/(a*d-b*c)^8*l
n(b*x+a)

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Maxima [A]  time = 1.54782, size = 1914, normalized size = 9.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^8),x, algorithm="maxima")

[Out]

b^7*log(b*x + a)/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*
d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d
^7 + a^8*d^8) - b^7*log(d*x + c)/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 -
56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d
^6 - 8*a^7*b*c*d^7 + a^8*d^8) + 1/420*(420*b^6*d^6*x^6 + 1089*b^6*c^6 - 1851*a*b
^5*c^5*d + 2559*a^2*b^4*c^4*d^2 - 2341*a^3*b^3*c^3*d^3 + 1334*a^4*b^2*c^2*d^4 -
430*a^5*b*c*d^5 + 60*a^6*d^6 + 210*(13*b^6*c*d^5 - a*b^5*d^6)*x^5 + 70*(107*b^6*
c^2*d^4 - 19*a*b^5*c*d^5 + 2*a^2*b^4*d^6)*x^4 + 35*(319*b^6*c^3*d^3 - 101*a*b^5*
c^2*d^4 + 25*a^2*b^4*c*d^5 - 3*a^3*b^3*d^6)*x^3 + 21*(459*b^6*c^4*d^2 - 241*a*b^
5*c^3*d^3 + 109*a^2*b^4*c^2*d^4 - 31*a^3*b^3*c*d^5 + 4*a^4*b^2*d^6)*x^2 + 7*(669
*b^6*c^5*d - 591*a*b^5*c^4*d^2 + 459*a^2*b^4*c^3*d^3 - 241*a^3*b^3*c^2*d^4 + 74*
a^4*b^2*c*d^5 - 10*a^5*b*d^6)*x)/(b^7*c^14 - 7*a*b^6*c^13*d + 21*a^2*b^5*c^12*d^
2 - 35*a^3*b^4*c^11*d^3 + 35*a^4*b^3*c^10*d^4 - 21*a^5*b^2*c^9*d^5 + 7*a^6*b*c^8
*d^6 - a^7*c^7*d^7 + (b^7*c^7*d^7 - 7*a*b^6*c^6*d^8 + 21*a^2*b^5*c^5*d^9 - 35*a^
3*b^4*c^4*d^10 + 35*a^4*b^3*c^3*d^11 - 21*a^5*b^2*c^2*d^12 + 7*a^6*b*c*d^13 - a^
7*d^14)*x^7 + 7*(b^7*c^8*d^6 - 7*a*b^6*c^7*d^7 + 21*a^2*b^5*c^6*d^8 - 35*a^3*b^4
*c^5*d^9 + 35*a^4*b^3*c^4*d^10 - 21*a^5*b^2*c^3*d^11 + 7*a^6*b*c^2*d^12 - a^7*c*
d^13)*x^6 + 21*(b^7*c^9*d^5 - 7*a*b^6*c^8*d^6 + 21*a^2*b^5*c^7*d^7 - 35*a^3*b^4*
c^6*d^8 + 35*a^4*b^3*c^5*d^9 - 21*a^5*b^2*c^4*d^10 + 7*a^6*b*c^3*d^11 - a^7*c^2*
d^12)*x^5 + 35*(b^7*c^10*d^4 - 7*a*b^6*c^9*d^5 + 21*a^2*b^5*c^8*d^6 - 35*a^3*b^4
*c^7*d^7 + 35*a^4*b^3*c^6*d^8 - 21*a^5*b^2*c^5*d^9 + 7*a^6*b*c^4*d^10 - a^7*c^3*
d^11)*x^4 + 35*(b^7*c^11*d^3 - 7*a*b^6*c^10*d^4 + 21*a^2*b^5*c^9*d^5 - 35*a^3*b^
4*c^8*d^6 + 35*a^4*b^3*c^7*d^7 - 21*a^5*b^2*c^6*d^8 + 7*a^6*b*c^5*d^9 - a^7*c^4*
d^10)*x^3 + 21*(b^7*c^12*d^2 - 7*a*b^6*c^11*d^3 + 21*a^2*b^5*c^10*d^4 - 35*a^3*b
^4*c^9*d^5 + 35*a^4*b^3*c^8*d^6 - 21*a^5*b^2*c^7*d^7 + 7*a^6*b*c^6*d^8 - a^7*c^5
*d^9)*x^2 + 7*(b^7*c^13*d - 7*a*b^6*c^12*d^2 + 21*a^2*b^5*c^11*d^3 - 35*a^3*b^4*
c^10*d^4 + 35*a^4*b^3*c^9*d^5 - 21*a^5*b^2*c^8*d^6 + 7*a^6*b*c^7*d^7 - a^7*c^6*d
^8)*x)

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Fricas [A]  time = 0.247166, size = 2145, normalized size = 10.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^8),x, algorithm="fricas")

[Out]

1/420*(1089*b^7*c^7 - 2940*a*b^6*c^6*d + 4410*a^2*b^5*c^5*d^2 - 4900*a^3*b^4*c^4
*d^3 + 3675*a^4*b^3*c^3*d^4 - 1764*a^5*b^2*c^2*d^5 + 490*a^6*b*c*d^6 - 60*a^7*d^
7 + 420*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 210*(13*b^7*c^2*d^5 - 14*a*b^6*c*d^6 + a^2
*b^5*d^7)*x^5 + 70*(107*b^7*c^3*d^4 - 126*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 - 2*a
^3*b^4*d^7)*x^4 + 35*(319*b^7*c^4*d^3 - 420*a*b^6*c^3*d^4 + 126*a^2*b^5*c^2*d^5
- 28*a^3*b^4*c*d^6 + 3*a^4*b^3*d^7)*x^3 + 21*(459*b^7*c^5*d^2 - 700*a*b^6*c^4*d^
3 + 350*a^2*b^5*c^3*d^4 - 140*a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 - 4*a^5*b^2*d^7
)*x^2 + 7*(669*b^7*c^6*d - 1260*a*b^6*c^5*d^2 + 1050*a^2*b^5*c^4*d^3 - 700*a^3*b
^4*c^3*d^4 + 315*a^4*b^3*c^2*d^5 - 84*a^5*b^2*c*d^6 + 10*a^6*b*d^7)*x + 420*(b^7
*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^2*d^5*x^5 + 35*b^7*c^3*d^4*x^4 + 35*b^7*c^
4*d^3*x^3 + 21*b^7*c^5*d^2*x^2 + 7*b^7*c^6*d*x + b^7*c^7)*log(b*x + a) - 420*(b^
7*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^2*d^5*x^5 + 35*b^7*c^3*d^4*x^4 + 35*b^7*c
^4*d^3*x^3 + 21*b^7*c^5*d^2*x^2 + 7*b^7*c^6*d*x + b^7*c^7)*log(d*x + c))/(b^8*c^
15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^1
1*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8
+ (b^8*c^8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 56*a^3*b^5*c^5*d^10 + 7
0*a^4*b^4*c^4*d^11 - 56*a^5*b^3*c^3*d^12 + 28*a^6*b^2*c^2*d^13 - 8*a^7*b*c*d^14
+ a^8*d^15)*x^7 + 7*(b^8*c^9*d^6 - 8*a*b^7*c^8*d^7 + 28*a^2*b^6*c^7*d^8 - 56*a^3
*b^5*c^6*d^9 + 70*a^4*b^4*c^5*d^10 - 56*a^5*b^3*c^4*d^11 + 28*a^6*b^2*c^3*d^12 -
8*a^7*b*c^2*d^13 + a^8*c*d^14)*x^6 + 21*(b^8*c^10*d^5 - 8*a*b^7*c^9*d^6 + 28*a^
2*b^6*c^8*d^7 - 56*a^3*b^5*c^7*d^8 + 70*a^4*b^4*c^6*d^9 - 56*a^5*b^3*c^5*d^10 +
28*a^6*b^2*c^4*d^11 - 8*a^7*b*c^3*d^12 + a^8*c^2*d^13)*x^5 + 35*(b^8*c^11*d^4 -
8*a*b^7*c^10*d^5 + 28*a^2*b^6*c^9*d^6 - 56*a^3*b^5*c^8*d^7 + 70*a^4*b^4*c^7*d^8
- 56*a^5*b^3*c^6*d^9 + 28*a^6*b^2*c^5*d^10 - 8*a^7*b*c^4*d^11 + a^8*c^3*d^12)*x^
4 + 35*(b^8*c^12*d^3 - 8*a*b^7*c^11*d^4 + 28*a^2*b^6*c^10*d^5 - 56*a^3*b^5*c^9*d
^6 + 70*a^4*b^4*c^8*d^7 - 56*a^5*b^3*c^7*d^8 + 28*a^6*b^2*c^6*d^9 - 8*a^7*b*c^5*
d^10 + a^8*c^4*d^11)*x^3 + 21*(b^8*c^13*d^2 - 8*a*b^7*c^12*d^3 + 28*a^2*b^6*c^11
*d^4 - 56*a^3*b^5*c^10*d^5 + 70*a^4*b^4*c^9*d^6 - 56*a^5*b^3*c^8*d^7 + 28*a^6*b^
2*c^7*d^8 - 8*a^7*b*c^6*d^9 + a^8*c^5*d^10)*x^2 + 7*(b^8*c^14*d - 8*a*b^7*c^13*d
^2 + 28*a^2*b^6*c^12*d^3 - 56*a^3*b^5*c^11*d^4 + 70*a^4*b^4*c^10*d^5 - 56*a^5*b^
3*c^9*d^6 + 28*a^6*b^2*c^8*d^7 - 8*a^7*b*c^7*d^8 + a^8*c^6*d^9)*x)

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Sympy [A]  time = 19.67, size = 1776, normalized size = 8.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)/(d*x+c)**8,x)

[Out]

-b**7*log(x + (-a**9*b**7*d**9/(a*d - b*c)**8 + 9*a**8*b**8*c*d**8/(a*d - b*c)**
8 - 36*a**7*b**9*c**2*d**7/(a*d - b*c)**8 + 84*a**6*b**10*c**3*d**6/(a*d - b*c)*
*8 - 126*a**5*b**11*c**4*d**5/(a*d - b*c)**8 + 126*a**4*b**12*c**5*d**4/(a*d - b
*c)**8 - 84*a**3*b**13*c**6*d**3/(a*d - b*c)**8 + 36*a**2*b**14*c**7*d**2/(a*d -
b*c)**8 - 9*a*b**15*c**8*d/(a*d - b*c)**8 + a*b**7*d + b**16*c**9/(a*d - b*c)**
8 + b**8*c)/(2*b**8*d))/(a*d - b*c)**8 + b**7*log(x + (a**9*b**7*d**9/(a*d - b*c
)**8 - 9*a**8*b**8*c*d**8/(a*d - b*c)**8 + 36*a**7*b**9*c**2*d**7/(a*d - b*c)**8
- 84*a**6*b**10*c**3*d**6/(a*d - b*c)**8 + 126*a**5*b**11*c**4*d**5/(a*d - b*c)
**8 - 126*a**4*b**12*c**5*d**4/(a*d - b*c)**8 + 84*a**3*b**13*c**6*d**3/(a*d - b
*c)**8 - 36*a**2*b**14*c**7*d**2/(a*d - b*c)**8 + 9*a*b**15*c**8*d/(a*d - b*c)**
8 + a*b**7*d - b**16*c**9/(a*d - b*c)**8 + b**8*c)/(2*b**8*d))/(a*d - b*c)**8 -
(60*a**6*d**6 - 430*a**5*b*c*d**5 + 1334*a**4*b**2*c**2*d**4 - 2341*a**3*b**3*c*
*3*d**3 + 2559*a**2*b**4*c**4*d**2 - 1851*a*b**5*c**5*d + 1089*b**6*c**6 + 420*b
**6*d**6*x**6 + x**5*(-210*a*b**5*d**6 + 2730*b**6*c*d**5) + x**4*(140*a**2*b**4
*d**6 - 1330*a*b**5*c*d**5 + 7490*b**6*c**2*d**4) + x**3*(-105*a**3*b**3*d**6 +
875*a**2*b**4*c*d**5 - 3535*a*b**5*c**2*d**4 + 11165*b**6*c**3*d**3) + x**2*(84*
a**4*b**2*d**6 - 651*a**3*b**3*c*d**5 + 2289*a**2*b**4*c**2*d**4 - 5061*a*b**5*c
**3*d**3 + 9639*b**6*c**4*d**2) + x*(-70*a**5*b*d**6 + 518*a**4*b**2*c*d**5 - 16
87*a**3*b**3*c**2*d**4 + 3213*a**2*b**4*c**3*d**3 - 4137*a*b**5*c**4*d**2 + 4683
*b**6*c**5*d))/(420*a**7*c**7*d**7 - 2940*a**6*b*c**8*d**6 + 8820*a**5*b**2*c**9
*d**5 - 14700*a**4*b**3*c**10*d**4 + 14700*a**3*b**4*c**11*d**3 - 8820*a**2*b**5
*c**12*d**2 + 2940*a*b**6*c**13*d - 420*b**7*c**14 + x**7*(420*a**7*d**14 - 2940
*a**6*b*c*d**13 + 8820*a**5*b**2*c**2*d**12 - 14700*a**4*b**3*c**3*d**11 + 14700
*a**3*b**4*c**4*d**10 - 8820*a**2*b**5*c**5*d**9 + 2940*a*b**6*c**6*d**8 - 420*b
**7*c**7*d**7) + x**6*(2940*a**7*c*d**13 - 20580*a**6*b*c**2*d**12 + 61740*a**5*
b**2*c**3*d**11 - 102900*a**4*b**3*c**4*d**10 + 102900*a**3*b**4*c**5*d**9 - 617
40*a**2*b**5*c**6*d**8 + 20580*a*b**6*c**7*d**7 - 2940*b**7*c**8*d**6) + x**5*(8
820*a**7*c**2*d**12 - 61740*a**6*b*c**3*d**11 + 185220*a**5*b**2*c**4*d**10 - 30
8700*a**4*b**3*c**5*d**9 + 308700*a**3*b**4*c**6*d**8 - 185220*a**2*b**5*c**7*d*
*7 + 61740*a*b**6*c**8*d**6 - 8820*b**7*c**9*d**5) + x**4*(14700*a**7*c**3*d**11
- 102900*a**6*b*c**4*d**10 + 308700*a**5*b**2*c**5*d**9 - 514500*a**4*b**3*c**6
*d**8 + 514500*a**3*b**4*c**7*d**7 - 308700*a**2*b**5*c**8*d**6 + 102900*a*b**6*
c**9*d**5 - 14700*b**7*c**10*d**4) + x**3*(14700*a**7*c**4*d**10 - 102900*a**6*b
*c**5*d**9 + 308700*a**5*b**2*c**6*d**8 - 514500*a**4*b**3*c**7*d**7 + 514500*a*
*3*b**4*c**8*d**6 - 308700*a**2*b**5*c**9*d**5 + 102900*a*b**6*c**10*d**4 - 1470
0*b**7*c**11*d**3) + x**2*(8820*a**7*c**5*d**9 - 61740*a**6*b*c**6*d**8 + 185220
*a**5*b**2*c**7*d**7 - 308700*a**4*b**3*c**8*d**6 + 308700*a**3*b**4*c**9*d**5 -
185220*a**2*b**5*c**10*d**4 + 61740*a*b**6*c**11*d**3 - 8820*b**7*c**12*d**2) +
x*(2940*a**7*c**6*d**8 - 20580*a**6*b*c**7*d**7 + 61740*a**5*b**2*c**8*d**6 - 1
02900*a**4*b**3*c**9*d**5 + 102900*a**3*b**4*c**10*d**4 - 61740*a**2*b**5*c**11*
d**3 + 20580*a*b**6*c**12*d**2 - 2940*b**7*c**13*d))

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GIAC/XCAS [A]  time = 0.22798, size = 949, normalized size = 4.7 \[ \frac{b^{8}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{9} c^{8} - 8 \, a b^{8} c^{7} d + 28 \, a^{2} b^{7} c^{6} d^{2} - 56 \, a^{3} b^{6} c^{5} d^{3} + 70 \, a^{4} b^{5} c^{4} d^{4} - 56 \, a^{5} b^{4} c^{3} d^{5} + 28 \, a^{6} b^{3} c^{2} d^{6} - 8 \, a^{7} b^{2} c d^{7} + a^{8} b d^{8}} - \frac{b^{7} d{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{8} c^{8} d - 8 \, a b^{7} c^{7} d^{2} + 28 \, a^{2} b^{6} c^{6} d^{3} - 56 \, a^{3} b^{5} c^{5} d^{4} + 70 \, a^{4} b^{4} c^{4} d^{5} - 56 \, a^{5} b^{3} c^{3} d^{6} + 28 \, a^{6} b^{2} c^{2} d^{7} - 8 \, a^{7} b c d^{8} + a^{8} d^{9}} + \frac{1089 \, b^{7} c^{7} - 2940 \, a b^{6} c^{6} d + 4410 \, a^{2} b^{5} c^{5} d^{2} - 4900 \, a^{3} b^{4} c^{4} d^{3} + 3675 \, a^{4} b^{3} c^{3} d^{4} - 1764 \, a^{5} b^{2} c^{2} d^{5} + 490 \, a^{6} b c d^{6} - 60 \, a^{7} d^{7} + 420 \,{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 210 \,{\left (13 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \,{\left (107 \, b^{7} c^{3} d^{4} - 126 \, a b^{6} c^{2} d^{5} + 21 \, a^{2} b^{5} c d^{6} - 2 \, a^{3} b^{4} d^{7}\right )} x^{4} + 35 \,{\left (319 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 126 \, a^{2} b^{5} c^{2} d^{5} - 28 \, a^{3} b^{4} c d^{6} + 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 21 \,{\left (459 \, b^{7} c^{5} d^{2} - 700 \, a b^{6} c^{4} d^{3} + 350 \, a^{2} b^{5} c^{3} d^{4} - 140 \, a^{3} b^{4} c^{2} d^{5} + 35 \, a^{4} b^{3} c d^{6} - 4 \, a^{5} b^{2} d^{7}\right )} x^{2} + 7 \,{\left (669 \, b^{7} c^{6} d - 1260 \, a b^{6} c^{5} d^{2} + 1050 \, a^{2} b^{5} c^{4} d^{3} - 700 \, a^{3} b^{4} c^{3} d^{4} + 315 \, a^{4} b^{3} c^{2} d^{5} - 84 \, a^{5} b^{2} c d^{6} + 10 \, a^{6} b d^{7}\right )} x}{420 \,{\left (b c - a d\right )}^{8}{\left (d x + c\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^8),x, algorithm="giac")

[Out]

b^8*ln(abs(b*x + a))/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*
c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b
^2*c*d^7 + a^8*b*d^8) - b^7*d*ln(abs(d*x + c))/(b^8*c^8*d - 8*a*b^7*c^7*d^2 + 28
*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6
+ 28*a^6*b^2*c^2*d^7 - 8*a^7*b*c*d^8 + a^8*d^9) + 1/420*(1089*b^7*c^7 - 2940*a*b
^6*c^6*d + 4410*a^2*b^5*c^5*d^2 - 4900*a^3*b^4*c^4*d^3 + 3675*a^4*b^3*c^3*d^4 -
1764*a^5*b^2*c^2*d^5 + 490*a^6*b*c*d^6 - 60*a^7*d^7 + 420*(b^7*c*d^6 - a*b^6*d^7
)*x^6 + 210*(13*b^7*c^2*d^5 - 14*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 70*(107*b^7*c^
3*d^4 - 126*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 35*(319*b^7*
c^4*d^3 - 420*a*b^6*c^3*d^4 + 126*a^2*b^5*c^2*d^5 - 28*a^3*b^4*c*d^6 + 3*a^4*b^3
*d^7)*x^3 + 21*(459*b^7*c^5*d^2 - 700*a*b^6*c^4*d^3 + 350*a^2*b^5*c^3*d^4 - 140*
a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 - 4*a^5*b^2*d^7)*x^2 + 7*(669*b^7*c^6*d - 126
0*a*b^6*c^5*d^2 + 1050*a^2*b^5*c^4*d^3 - 700*a^3*b^4*c^3*d^4 + 315*a^4*b^3*c^2*d
^5 - 84*a^5*b^2*c*d^6 + 10*a^6*b*d^7)*x)/((b*c - a*d)^8*(d*x + c)^7)