3.1373 \(\int \frac{1}{(a+b x)^2 (c+d x)^8} \, dx\)

Optimal. Leaf size=231 \[ -\frac{b^7}{(a+b x) (b c-a d)^8}-\frac{7 b^6 d}{(c+d x) (b c-a d)^8}-\frac{3 b^5 d}{(c+d x)^2 (b c-a d)^7}-\frac{5 b^4 d}{3 (c+d x)^3 (b c-a d)^6}-\frac{b^3 d}{(c+d x)^4 (b c-a d)^5}-\frac{3 b^2 d}{5 (c+d x)^5 (b c-a d)^4}-\frac{8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac{8 b^7 d \log (c+d x)}{(b c-a d)^9}-\frac{b d}{3 (c+d x)^6 (b c-a d)^3}-\frac{d}{7 (c+d x)^7 (b c-a d)^2} \]

[Out]

-(b^7/((b*c - a*d)^8*(a + b*x))) - d/(7*(b*c - a*d)^2*(c + d*x)^7) - (b*d)/(3*(b*c - a*d)^3*(c + d*x)^6) - (3*
b^2*d)/(5*(b*c - a*d)^4*(c + d*x)^5) - (b^3*d)/((b*c - a*d)^5*(c + d*x)^4) - (5*b^4*d)/(3*(b*c - a*d)^6*(c + d
*x)^3) - (3*b^5*d)/((b*c - a*d)^7*(c + d*x)^2) - (7*b^6*d)/((b*c - a*d)^8*(c + d*x)) - (8*b^7*d*Log[a + b*x])/
(b*c - a*d)^9 + (8*b^7*d*Log[c + d*x])/(b*c - a*d)^9

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Rubi [A]  time = 0.270129, antiderivative size = 231, 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, Rules used = {44} \[ -\frac{b^7}{(a+b x) (b c-a d)^8}-\frac{7 b^6 d}{(c+d x) (b c-a d)^8}-\frac{3 b^5 d}{(c+d x)^2 (b c-a d)^7}-\frac{5 b^4 d}{3 (c+d x)^3 (b c-a d)^6}-\frac{b^3 d}{(c+d x)^4 (b c-a d)^5}-\frac{3 b^2 d}{5 (c+d x)^5 (b c-a d)^4}-\frac{8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac{8 b^7 d \log (c+d x)}{(b c-a d)^9}-\frac{b d}{3 (c+d x)^6 (b c-a d)^3}-\frac{d}{7 (c+d x)^7 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^2*(c + d*x)^8),x]

[Out]

-(b^7/((b*c - a*d)^8*(a + b*x))) - d/(7*(b*c - a*d)^2*(c + d*x)^7) - (b*d)/(3*(b*c - a*d)^3*(c + d*x)^6) - (3*
b^2*d)/(5*(b*c - a*d)^4*(c + d*x)^5) - (b^3*d)/((b*c - a*d)^5*(c + d*x)^4) - (5*b^4*d)/(3*(b*c - a*d)^6*(c + d
*x)^3) - (3*b^5*d)/((b*c - a*d)^7*(c + d*x)^2) - (7*b^6*d)/((b*c - a*d)^8*(c + d*x)) - (8*b^7*d*Log[a + b*x])/
(b*c - a*d)^9 + (8*b^7*d*Log[c + d*x])/(b*c - a*d)^9

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^2 (c+d x)^8} \, dx &=\int \left (\frac{b^8}{(b c-a d)^8 (a+b x)^2}-\frac{8 b^8 d}{(b c-a d)^9 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)^8}+\frac{2 b d^2}{(b c-a d)^3 (c+d x)^7}+\frac{3 b^2 d^2}{(b c-a d)^4 (c+d x)^6}+\frac{4 b^3 d^2}{(b c-a d)^5 (c+d x)^5}+\frac{5 b^4 d^2}{(b c-a d)^6 (c+d x)^4}+\frac{6 b^5 d^2}{(b c-a d)^7 (c+d x)^3}+\frac{7 b^6 d^2}{(b c-a d)^8 (c+d x)^2}+\frac{8 b^7 d^2}{(b c-a d)^9 (c+d x)}\right ) \, dx\\ &=-\frac{b^7}{(b c-a d)^8 (a+b x)}-\frac{d}{7 (b c-a d)^2 (c+d x)^7}-\frac{b d}{3 (b c-a d)^3 (c+d x)^6}-\frac{3 b^2 d}{5 (b c-a d)^4 (c+d x)^5}-\frac{b^3 d}{(b c-a d)^5 (c+d x)^4}-\frac{5 b^4 d}{3 (b c-a d)^6 (c+d x)^3}-\frac{3 b^5 d}{(b c-a d)^7 (c+d x)^2}-\frac{7 b^6 d}{(b c-a d)^8 (c+d x)}-\frac{8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac{8 b^7 d \log (c+d x)}{(b c-a d)^9}\\ \end{align*}

Mathematica [A]  time = 0.232436, size = 213, normalized size = 0.92 \[ -\frac{\frac{105 b^7 (b c-a d)}{a+b x}+\frac{735 b^6 d (b c-a d)}{c+d x}+\frac{315 b^5 d (b c-a d)^2}{(c+d x)^2}+\frac{175 b^4 d (b c-a d)^3}{(c+d x)^3}+\frac{105 b^3 d (b c-a d)^4}{(c+d x)^4}+\frac{63 b^2 d (b c-a d)^5}{(c+d x)^5}+840 b^7 d \log (a+b x)+\frac{35 b d (b c-a d)^6}{(c+d x)^6}-\frac{15 d (a d-b c)^7}{(c+d x)^7}-840 b^7 d \log (c+d x)}{105 (b c-a d)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^2*(c + d*x)^8),x]

[Out]

-((105*b^7*(b*c - a*d))/(a + b*x) - (15*d*(-(b*c) + a*d)^7)/(c + d*x)^7 + (35*b*d*(b*c - a*d)^6)/(c + d*x)^6 +
(63*b^2*d*(b*c - a*d)^5)/(c + d*x)^5 + (105*b^3*d*(b*c - a*d)^4)/(c + d*x)^4 + (175*b^4*d*(b*c - a*d)^3)/(c +
d*x)^3 + (315*b^5*d*(b*c - a*d)^2)/(c + d*x)^2 + (735*b^6*d*(b*c - a*d))/(c + d*x) + 840*b^7*d*Log[a + b*x] -
840*b^7*d*Log[c + d*x])/(105*(b*c - a*d)^9)

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Maple [A]  time = 0.02, size = 223, normalized size = 1. \begin{align*} -{\frac{d}{7\, \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{7}}}-8\,{\frac{d{b}^{7}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{9}}}-7\,{\frac{d{b}^{6}}{ \left ( ad-bc \right ) ^{8} \left ( dx+c \right ) }}+3\,{\frac{d{b}^{5}}{ \left ( ad-bc \right ) ^{7} \left ( dx+c \right ) ^{2}}}-{\frac{5\,d{b}^{4}}{3\, \left ( ad-bc \right ) ^{6} \left ( dx+c \right ) ^{3}}}+{\frac{d{b}^{3}}{ \left ( ad-bc \right ) ^{5} \left ( dx+c \right ) ^{4}}}-{\frac{3\,{b}^{2}d}{5\, \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) ^{5}}}+{\frac{bd}{3\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{6}}}-{\frac{{b}^{7}}{ \left ( ad-bc \right ) ^{8} \left ( bx+a \right ) }}+8\,{\frac{d{b}^{7}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^2/(d*x+c)^8,x)

[Out]

-1/7*d/(a*d-b*c)^2/(d*x+c)^7-8*d/(a*d-b*c)^9*b^7*ln(d*x+c)-7*d/(a*d-b*c)^8*b^6/(d*x+c)+3*d/(a*d-b*c)^7*b^5/(d*
x+c)^2-5/3*d/(a*d-b*c)^6*b^4/(d*x+c)^3+d/(a*d-b*c)^5*b^3/(d*x+c)^4-3/5*d/(a*d-b*c)^4*b^2/(d*x+c)^5+1/3*d/(a*d-
b*c)^3*b/(d*x+c)^6-b^7/(a*d-b*c)^8/(b*x+a)+8*d/(a*d-b*c)^9*b^7*ln(b*x+a)

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Maxima [B]  time = 1.77341, size = 2539, normalized size = 10.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="maxima")

[Out]

-8*b^7*d*log(b*x + a)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4
- 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9) + 8*b^7*d*log(d*x
+ c)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^
4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9) - 1/105*(840*b^7*d^7*x^7 + 105*b^7*
c^7 + 1443*a*b^6*c^6*d - 1497*a^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^4*b^3*c^3*d^4 + 463*a^5*b^2*c^2*
d^5 - 125*a^6*b*c*d^6 + 15*a^7*d^7 + 420*(13*b^7*c*d^6 + a*b^6*d^7)*x^6 + 140*(107*b^7*c^2*d^5 + 20*a*b^6*c*d^
6 - a^2*b^5*d^7)*x^5 + 70*(319*b^7*c^3*d^4 + 113*a*b^6*c^2*d^5 - 13*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 14*(137
7*b^7*c^4*d^3 + 872*a*b^6*c^3*d^4 - 178*a^2*b^5*c^2*d^5 + 32*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 14*(669*b^7*
c^5*d^2 + 786*a*b^6*c^4*d^3 - 264*a^2*b^5*c^3*d^4 + 86*a^3*b^4*c^2*d^5 - 19*a^4*b^3*c*d^6 + 2*a^5*b^2*d^7)*x^2
+ 2*(1089*b^7*c^6*d + 2832*a*b^6*c^5*d^2 - 1578*a^2*b^5*c^4*d^3 + 872*a^3*b^4*c^3*d^4 - 353*a^4*b^3*c^2*d^5 +
88*a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x)/(a*b^8*c^15 - 8*a^2*b^7*c^14*d + 28*a^3*b^6*c^13*d^2 - 56*a^4*b^5*c^12*d^
3 + 70*a^5*b^4*c^11*d^4 - 56*a^6*b^3*c^10*d^5 + 28*a^7*b^2*c^9*d^6 - 8*a^8*b*c^8*d^7 + a^9*c^7*d^8 + (b^9*c^8*
d^7 - 8*a*b^8*c^7*d^8 + 28*a^2*b^7*c^6*d^9 - 56*a^3*b^6*c^5*d^10 + 70*a^4*b^5*c^4*d^11 - 56*a^5*b^4*c^3*d^12 +
28*a^6*b^3*c^2*d^13 - 8*a^7*b^2*c*d^14 + a^8*b*d^15)*x^8 + (7*b^9*c^9*d^6 - 55*a*b^8*c^8*d^7 + 188*a^2*b^7*c^
7*d^8 - 364*a^3*b^6*c^6*d^9 + 434*a^4*b^5*c^5*d^10 - 322*a^5*b^4*c^4*d^11 + 140*a^6*b^3*c^3*d^12 - 28*a^7*b^2*
c^2*d^13 - a^8*b*c*d^14 + a^9*d^15)*x^7 + 7*(3*b^9*c^10*d^5 - 23*a*b^8*c^9*d^6 + 76*a^2*b^7*c^8*d^7 - 140*a^3*
b^6*c^7*d^8 + 154*a^4*b^5*c^6*d^9 - 98*a^5*b^4*c^5*d^10 + 28*a^6*b^3*c^4*d^11 + 4*a^7*b^2*c^3*d^12 - 5*a^8*b*c
^2*d^13 + a^9*c*d^14)*x^6 + 7*(5*b^9*c^11*d^4 - 37*a*b^8*c^10*d^5 + 116*a^2*b^7*c^9*d^6 - 196*a^3*b^6*c^8*d^7
+ 182*a^4*b^5*c^7*d^8 - 70*a^5*b^4*c^6*d^9 - 28*a^6*b^3*c^5*d^10 + 44*a^7*b^2*c^4*d^11 - 19*a^8*b*c^3*d^12 + 3
*a^9*c^2*d^13)*x^5 + 35*(b^9*c^12*d^3 - 7*a*b^8*c^11*d^4 + 20*a^2*b^7*c^10*d^5 - 28*a^3*b^6*c^9*d^6 + 14*a^4*b
^5*c^8*d^7 + 14*a^5*b^4*c^7*d^8 - 28*a^6*b^3*c^6*d^9 + 20*a^7*b^2*c^5*d^10 - 7*a^8*b*c^4*d^11 + a^9*c^3*d^12)*
x^4 + 7*(3*b^9*c^13*d^2 - 19*a*b^8*c^12*d^3 + 44*a^2*b^7*c^11*d^4 - 28*a^3*b^6*c^10*d^5 - 70*a^4*b^5*c^9*d^6 +
182*a^5*b^4*c^8*d^7 - 196*a^6*b^3*c^7*d^8 + 116*a^7*b^2*c^6*d^9 - 37*a^8*b*c^5*d^10 + 5*a^9*c^4*d^11)*x^3 + 7
*(b^9*c^14*d - 5*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 + 28*a^3*b^6*c^11*d^4 - 98*a^4*b^5*c^10*d^5 + 154*a^5*b^4
*c^9*d^6 - 140*a^6*b^3*c^8*d^7 + 76*a^7*b^2*c^7*d^8 - 23*a^8*b*c^6*d^9 + 3*a^9*c^5*d^10)*x^2 + (b^9*c^15 - a*b
^8*c^14*d - 28*a^2*b^7*c^13*d^2 + 140*a^3*b^6*c^12*d^3 - 322*a^4*b^5*c^11*d^4 + 434*a^5*b^4*c^10*d^5 - 364*a^6
*b^3*c^9*d^6 + 188*a^7*b^2*c^8*d^7 - 55*a^8*b*c^7*d^8 + 7*a^9*c^6*d^9)*x)

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Fricas [B]  time = 2.55013, size = 4788, normalized size = 20.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/105*(105*b^8*c^8 + 1338*a*b^7*c^7*d - 2940*a^2*b^6*c^6*d^2 + 2940*a^3*b^5*c^5*d^3 - 2450*a^4*b^4*c^4*d^4 +
1470*a^5*b^3*c^3*d^5 - 588*a^6*b^2*c^2*d^6 + 140*a^7*b*c*d^7 - 15*a^8*d^8 + 840*(b^8*c*d^7 - a*b^7*d^8)*x^7 +
420*(13*b^8*c^2*d^6 - 12*a*b^7*c*d^7 - a^2*b^6*d^8)*x^6 + 140*(107*b^8*c^3*d^5 - 87*a*b^7*c^2*d^6 - 21*a^2*b^6
*c*d^7 + a^3*b^5*d^8)*x^5 + 70*(319*b^8*c^4*d^4 - 206*a*b^7*c^3*d^5 - 126*a^2*b^6*c^2*d^6 + 14*a^3*b^5*c*d^7 -
a^4*b^4*d^8)*x^4 + 14*(1377*b^8*c^5*d^3 - 505*a*b^7*c^4*d^4 - 1050*a^2*b^6*c^3*d^5 + 210*a^3*b^5*c^2*d^6 - 35
*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 14*(669*b^8*c^6*d^2 + 117*a*b^7*c^5*d^3 - 1050*a^2*b^6*c^4*d^4 + 350*a^3
*b^5*c^3*d^5 - 105*a^4*b^4*c^2*d^6 + 21*a^5*b^3*c*d^7 - 2*a^6*b^2*d^8)*x^2 + 2*(1089*b^8*c^7*d + 1743*a*b^7*c^
6*d^2 - 4410*a^2*b^6*c^5*d^3 + 2450*a^3*b^5*c^4*d^4 - 1225*a^4*b^4*c^3*d^5 + 441*a^5*b^3*c^2*d^6 - 98*a^6*b^2*
c*d^7 + 10*a^7*b*d^8)*x + 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 +
a*b^7*c*d^7)*x^6 + 7*(5*b^8*c^3*d^5 + 3*a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 + a*b^7*c^3*d^5)*x^4 + 7*(3*b^8*c
^5*d^3 + 5*a*b^7*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(b
*x + a) - 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 + a*b^7*c*d^7)*x^6
+ 7*(5*b^8*c^3*d^5 + 3*a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 + a*b^7*c^3*d^5)*x^4 + 7*(3*b^8*c^5*d^3 + 5*a*b^7
*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(d*x + c))/(a*b^9*
c^16 - 9*a^2*b^8*c^15*d + 36*a^3*b^7*c^14*d^2 - 84*a^4*b^6*c^13*d^3 + 126*a^5*b^5*c^12*d^4 - 126*a^6*b^4*c^11*
d^5 + 84*a^7*b^3*c^10*d^6 - 36*a^8*b^2*c^9*d^7 + 9*a^9*b*c^8*d^8 - a^10*c^7*d^9 + (b^10*c^9*d^7 - 9*a*b^9*c^8*
d^8 + 36*a^2*b^8*c^7*d^9 - 84*a^3*b^7*c^6*d^10 + 126*a^4*b^6*c^5*d^11 - 126*a^5*b^5*c^4*d^12 + 84*a^6*b^4*c^3*
d^13 - 36*a^7*b^3*c^2*d^14 + 9*a^8*b^2*c*d^15 - a^9*b*d^16)*x^8 + (7*b^10*c^10*d^6 - 62*a*b^9*c^9*d^7 + 243*a^
2*b^8*c^8*d^8 - 552*a^3*b^7*c^7*d^9 + 798*a^4*b^6*c^6*d^10 - 756*a^5*b^5*c^5*d^11 + 462*a^6*b^4*c^4*d^12 - 168
*a^7*b^3*c^3*d^13 + 27*a^8*b^2*c^2*d^14 + 2*a^9*b*c*d^15 - a^10*d^16)*x^7 + 7*(3*b^10*c^11*d^5 - 26*a*b^9*c^10
*d^6 + 99*a^2*b^8*c^9*d^7 - 216*a^3*b^7*c^8*d^8 + 294*a^4*b^6*c^7*d^9 - 252*a^5*b^5*c^6*d^10 + 126*a^6*b^4*c^5
*d^11 - 24*a^7*b^3*c^4*d^12 - 9*a^8*b^2*c^3*d^13 + 6*a^9*b*c^2*d^14 - a^10*c*d^15)*x^6 + 7*(5*b^10*c^12*d^4 -
42*a*b^9*c^11*d^5 + 153*a^2*b^8*c^10*d^6 - 312*a^3*b^7*c^9*d^7 + 378*a^4*b^6*c^8*d^8 - 252*a^5*b^5*c^7*d^9 + 4
2*a^6*b^4*c^6*d^10 + 72*a^7*b^3*c^5*d^11 - 63*a^8*b^2*c^4*d^12 + 22*a^9*b*c^3*d^13 - 3*a^10*c^2*d^14)*x^5 + 35
*(b^10*c^13*d^3 - 8*a*b^9*c^12*d^4 + 27*a^2*b^8*c^11*d^5 - 48*a^3*b^7*c^10*d^6 + 42*a^4*b^6*c^9*d^7 - 42*a^6*b
^4*c^7*d^9 + 48*a^7*b^3*c^6*d^10 - 27*a^8*b^2*c^5*d^11 + 8*a^9*b*c^4*d^12 - a^10*c^3*d^13)*x^4 + 7*(3*b^10*c^1
4*d^2 - 22*a*b^9*c^13*d^3 + 63*a^2*b^8*c^12*d^4 - 72*a^3*b^7*c^11*d^5 - 42*a^4*b^6*c^10*d^6 + 252*a^5*b^5*c^9*
d^7 - 378*a^6*b^4*c^8*d^8 + 312*a^7*b^3*c^7*d^9 - 153*a^8*b^2*c^6*d^10 + 42*a^9*b*c^5*d^11 - 5*a^10*c^4*d^12)*
x^3 + 7*(b^10*c^15*d - 6*a*b^9*c^14*d^2 + 9*a^2*b^8*c^13*d^3 + 24*a^3*b^7*c^12*d^4 - 126*a^4*b^6*c^11*d^5 + 25
2*a^5*b^5*c^10*d^6 - 294*a^6*b^4*c^9*d^7 + 216*a^7*b^3*c^8*d^8 - 99*a^8*b^2*c^7*d^9 + 26*a^9*b*c^6*d^10 - 3*a^
10*c^5*d^11)*x^2 + (b^10*c^16 - 2*a*b^9*c^15*d - 27*a^2*b^8*c^14*d^2 + 168*a^3*b^7*c^13*d^3 - 462*a^4*b^6*c^12
*d^4 + 756*a^5*b^5*c^11*d^5 - 798*a^6*b^4*c^10*d^6 + 552*a^7*b^3*c^9*d^7 - 243*a^8*b^2*c^8*d^8 + 62*a^9*b*c^7*
d^9 - 7*a^10*c^6*d^10)*x)

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Sympy [B]  time = 22.2064, size = 2334, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**2/(d*x+c)**8,x)

[Out]

-8*b**7*d*log(x + (-8*a**10*b**7*d**11/(a*d - b*c)**9 + 80*a**9*b**8*c*d**10/(a*d - b*c)**9 - 360*a**8*b**9*c*
*2*d**9/(a*d - b*c)**9 + 960*a**7*b**10*c**3*d**8/(a*d - b*c)**9 - 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 +
2016*a**5*b**12*c**5*d**6/(a*d - b*c)**9 - 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 + 960*a**3*b**14*c**7*d**4
/(a*d - b*c)**9 - 360*a**2*b**15*c**8*d**3/(a*d - b*c)**9 + 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**
2 - 8*b**17*c**10*d/(a*d - b*c)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)**9 + 8*b**7*d*log(x + (8*a**10*b*
*7*d**11/(a*d - b*c)**9 - 80*a**9*b**8*c*d**10/(a*d - b*c)**9 + 360*a**8*b**9*c**2*d**9/(a*d - b*c)**9 - 960*a
**7*b**10*c**3*d**8/(a*d - b*c)**9 + 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 - 2016*a**5*b**12*c**5*d**6/(a*d
- b*c)**9 + 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 - 960*a**3*b**14*c**7*d**4/(a*d - b*c)**9 + 360*a**2*b**
15*c**8*d**3/(a*d - b*c)**9 - 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**2 + 8*b**17*c**10*d/(a*d - b*c
)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)**9 - (15*a**7*d**7 - 125*a**6*b*c*d**6 + 463*a**5*b**2*c**2*d**
5 - 1007*a**4*b**3*c**3*d**4 + 1443*a**3*b**4*c**4*d**3 - 1497*a**2*b**5*c**5*d**2 + 1443*a*b**6*c**6*d + 105*
b**7*c**7 + 840*b**7*d**7*x**7 + x**6*(420*a*b**6*d**7 + 5460*b**7*c*d**6) + x**5*(-140*a**2*b**5*d**7 + 2800*
a*b**6*c*d**6 + 14980*b**7*c**2*d**5) + x**4*(70*a**3*b**4*d**7 - 910*a**2*b**5*c*d**6 + 7910*a*b**6*c**2*d**5
+ 22330*b**7*c**3*d**4) + x**3*(-42*a**4*b**3*d**7 + 448*a**3*b**4*c*d**6 - 2492*a**2*b**5*c**2*d**5 + 12208*
a*b**6*c**3*d**4 + 19278*b**7*c**4*d**3) + x**2*(28*a**5*b**2*d**7 - 266*a**4*b**3*c*d**6 + 1204*a**3*b**4*c**
2*d**5 - 3696*a**2*b**5*c**3*d**4 + 11004*a*b**6*c**4*d**3 + 9366*b**7*c**5*d**2) + x*(-20*a**6*b*d**7 + 176*a
**5*b**2*c*d**6 - 706*a**4*b**3*c**2*d**5 + 1744*a**3*b**4*c**3*d**4 - 3156*a**2*b**5*c**4*d**3 + 5664*a*b**6*
c**5*d**2 + 2178*b**7*c**6*d))/(105*a**9*c**7*d**8 - 840*a**8*b*c**8*d**7 + 2940*a**7*b**2*c**9*d**6 - 5880*a*
*6*b**3*c**10*d**5 + 7350*a**5*b**4*c**11*d**4 - 5880*a**4*b**5*c**12*d**3 + 2940*a**3*b**6*c**13*d**2 - 840*a
**2*b**7*c**14*d + 105*a*b**8*c**15 + x**8*(105*a**8*b*d**15 - 840*a**7*b**2*c*d**14 + 2940*a**6*b**3*c**2*d**
13 - 5880*a**5*b**4*c**3*d**12 + 7350*a**4*b**5*c**4*d**11 - 5880*a**3*b**6*c**5*d**10 + 2940*a**2*b**7*c**6*d
**9 - 840*a*b**8*c**7*d**8 + 105*b**9*c**8*d**7) + x**7*(105*a**9*d**15 - 105*a**8*b*c*d**14 - 2940*a**7*b**2*
c**2*d**13 + 14700*a**6*b**3*c**3*d**12 - 33810*a**5*b**4*c**4*d**11 + 45570*a**4*b**5*c**5*d**10 - 38220*a**3
*b**6*c**6*d**9 + 19740*a**2*b**7*c**7*d**8 - 5775*a*b**8*c**8*d**7 + 735*b**9*c**9*d**6) + x**6*(735*a**9*c*d
**14 - 3675*a**8*b*c**2*d**13 + 2940*a**7*b**2*c**3*d**12 + 20580*a**6*b**3*c**4*d**11 - 72030*a**5*b**4*c**5*
d**10 + 113190*a**4*b**5*c**6*d**9 - 102900*a**3*b**6*c**7*d**8 + 55860*a**2*b**7*c**8*d**7 - 16905*a*b**8*c**
9*d**6 + 2205*b**9*c**10*d**5) + x**5*(2205*a**9*c**2*d**13 - 13965*a**8*b*c**3*d**12 + 32340*a**7*b**2*c**4*d
**11 - 20580*a**6*b**3*c**5*d**10 - 51450*a**5*b**4*c**6*d**9 + 133770*a**4*b**5*c**7*d**8 - 144060*a**3*b**6*
c**8*d**7 + 85260*a**2*b**7*c**9*d**6 - 27195*a*b**8*c**10*d**5 + 3675*b**9*c**11*d**4) + x**4*(3675*a**9*c**3
*d**12 - 25725*a**8*b*c**4*d**11 + 73500*a**7*b**2*c**5*d**10 - 102900*a**6*b**3*c**6*d**9 + 51450*a**5*b**4*c
**7*d**8 + 51450*a**4*b**5*c**8*d**7 - 102900*a**3*b**6*c**9*d**6 + 73500*a**2*b**7*c**10*d**5 - 25725*a*b**8*
c**11*d**4 + 3675*b**9*c**12*d**3) + x**3*(3675*a**9*c**4*d**11 - 27195*a**8*b*c**5*d**10 + 85260*a**7*b**2*c*
*6*d**9 - 144060*a**6*b**3*c**7*d**8 + 133770*a**5*b**4*c**8*d**7 - 51450*a**4*b**5*c**9*d**6 - 20580*a**3*b**
6*c**10*d**5 + 32340*a**2*b**7*c**11*d**4 - 13965*a*b**8*c**12*d**3 + 2205*b**9*c**13*d**2) + x**2*(2205*a**9*
c**5*d**10 - 16905*a**8*b*c**6*d**9 + 55860*a**7*b**2*c**7*d**8 - 102900*a**6*b**3*c**8*d**7 + 113190*a**5*b**
4*c**9*d**6 - 72030*a**4*b**5*c**10*d**5 + 20580*a**3*b**6*c**11*d**4 + 2940*a**2*b**7*c**12*d**3 - 3675*a*b**
8*c**13*d**2 + 735*b**9*c**14*d) + x*(735*a**9*c**6*d**9 - 5775*a**8*b*c**7*d**8 + 19740*a**7*b**2*c**8*d**7 -
38220*a**6*b**3*c**9*d**6 + 45570*a**5*b**4*c**10*d**5 - 33810*a**4*b**5*c**11*d**4 + 14700*a**3*b**6*c**12*d
**3 - 2940*a**2*b**7*c**13*d**2 - 105*a*b**8*c**14*d + 105*b**9*c**15))

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Giac [B]  time = 1.13522, size = 964, normalized size = 4.17 \begin{align*} -\frac{b^{15}}{{\left (b^{16} c^{8} - 8 \, a b^{15} c^{7} d + 28 \, a^{2} b^{14} c^{6} d^{2} - 56 \, a^{3} b^{13} c^{5} d^{3} + 70 \, a^{4} b^{12} c^{4} d^{4} - 56 \, a^{5} b^{11} c^{3} d^{5} + 28 \, a^{6} b^{10} c^{2} d^{6} - 8 \, a^{7} b^{9} c d^{7} + a^{8} b^{8} d^{8}\right )}{\left (b x + a\right )}} + \frac{8 \, b^{8} d \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{10} c^{9} - 9 \, a b^{9} c^{8} d + 36 \, a^{2} b^{8} c^{7} d^{2} - 84 \, a^{3} b^{7} c^{6} d^{3} + 126 \, a^{4} b^{6} c^{5} d^{4} - 126 \, a^{5} b^{5} c^{4} d^{5} + 84 \, a^{6} b^{4} c^{3} d^{6} - 36 \, a^{7} b^{3} c^{2} d^{7} + 9 \, a^{8} b^{2} c d^{8} - a^{9} b d^{9}} + \frac{1443 \, b^{7} d^{8} + \frac{9366 \,{\left (b^{9} c d^{7} - a b^{8} d^{8}\right )}}{{\left (b x + a\right )} b} + \frac{25578 \,{\left (b^{11} c^{2} d^{6} - 2 \, a b^{10} c d^{7} + a^{2} b^{9} d^{8}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{37730 \,{\left (b^{13} c^{3} d^{5} - 3 \, a b^{12} c^{2} d^{6} + 3 \, a^{2} b^{11} c d^{7} - a^{3} b^{10} d^{8}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{31850 \,{\left (b^{15} c^{4} d^{4} - 4 \, a b^{14} c^{3} d^{5} + 6 \, a^{2} b^{13} c^{2} d^{6} - 4 \, a^{3} b^{12} c d^{7} + a^{4} b^{11} d^{8}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac{14700 \,{\left (b^{17} c^{5} d^{3} - 5 \, a b^{16} c^{4} d^{4} + 10 \, a^{2} b^{15} c^{3} d^{5} - 10 \, a^{3} b^{14} c^{2} d^{6} + 5 \, a^{4} b^{13} c d^{7} - a^{5} b^{12} d^{8}\right )}}{{\left (b x + a\right )}^{5} b^{5}} + \frac{2940 \,{\left (b^{19} c^{6} d^{2} - 6 \, a b^{18} c^{5} d^{3} + 15 \, a^{2} b^{17} c^{4} d^{4} - 20 \, a^{3} b^{16} c^{3} d^{5} + 15 \, a^{4} b^{15} c^{2} d^{6} - 6 \, a^{5} b^{14} c d^{7} + a^{6} b^{13} d^{8}\right )}}{{\left (b x + a\right )}^{6} b^{6}}}{105 \,{\left (b c - a d\right )}^{9}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="giac")

[Out]

-b^15/((b^16*c^8 - 8*a*b^15*c^7*d + 28*a^2*b^14*c^6*d^2 - 56*a^3*b^13*c^5*d^3 + 70*a^4*b^12*c^4*d^4 - 56*a^5*b
^11*c^3*d^5 + 28*a^6*b^10*c^2*d^6 - 8*a^7*b^9*c*d^7 + a^8*b^8*d^8)*(b*x + a)) + 8*b^8*d*log(abs(b*c/(b*x + a)
- a*d/(b*x + a) + d))/(b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^
4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9) + 1/105*(1443
*b^7*d^8 + 9366*(b^9*c*d^7 - a*b^8*d^8)/((b*x + a)*b) + 25578*(b^11*c^2*d^6 - 2*a*b^10*c*d^7 + a^2*b^9*d^8)/((
b*x + a)^2*b^2) + 37730*(b^13*c^3*d^5 - 3*a*b^12*c^2*d^6 + 3*a^2*b^11*c*d^7 - a^3*b^10*d^8)/((b*x + a)^3*b^3)
+ 31850*(b^15*c^4*d^4 - 4*a*b^14*c^3*d^5 + 6*a^2*b^13*c^2*d^6 - 4*a^3*b^12*c*d^7 + a^4*b^11*d^8)/((b*x + a)^4*
b^4) + 14700*(b^17*c^5*d^3 - 5*a*b^16*c^4*d^4 + 10*a^2*b^15*c^3*d^5 - 10*a^3*b^14*c^2*d^6 + 5*a^4*b^13*c*d^7 -
a^5*b^12*d^8)/((b*x + a)^5*b^5) + 2940*(b^19*c^6*d^2 - 6*a*b^18*c^5*d^3 + 15*a^2*b^17*c^4*d^4 - 20*a^3*b^16*c
^3*d^5 + 15*a^4*b^15*c^2*d^6 - 6*a^5*b^14*c*d^7 + a^6*b^13*d^8)/((b*x + a)^6*b^6))/((b*c - a*d)^9*(b*c/(b*x +
a) - a*d/(b*x + a) + d)^7)