3.281 \(\int \frac{x^5}{(a+b x)^3 (c+d x)^3} \, dx\)

Optimal. Leaf size=213 \[ \frac{a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}-\frac{a^4 (5 b c-2 a d)}{b^3 (a+b x) (b c-a d)^4}+\frac{c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^5}-\frac{a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}-\frac{c^5}{2 d^3 (c+d x)^2 (b c-a d)^3}+\frac{c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4} \]

[Out]

a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) - (a^4*(5*b*c - 2*a*d))/(b^3*(b*c - a*d)^4
*(a + b*x)) - c^5/(2*d^3*(b*c - a*d)^3*(c + d*x)^2) + (c^4*(2*b*c - 5*a*d))/(d^3
*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[a + b*x]
)/(b^3*(b*c - a*d)^5) + (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d
^3*(b*c - a*d)^5)

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Rubi [A]  time = 0.535407, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}-\frac{a^4 (5 b c-2 a d)}{b^3 (a+b x) (b c-a d)^4}+\frac{c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^5}-\frac{a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}-\frac{c^5}{2 d^3 (c+d x)^2 (b c-a d)^3}+\frac{c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x)^3*(c + d*x)^3),x]

[Out]

a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) - (a^4*(5*b*c - 2*a*d))/(b^3*(b*c - a*d)^4
*(a + b*x)) - c^5/(2*d^3*(b*c - a*d)^3*(c + d*x)^2) + (c^4*(2*b*c - 5*a*d))/(d^3
*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[a + b*x]
)/(b^3*(b*c - a*d)^5) + (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d
^3*(b*c - a*d)^5)

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Rubi in Sympy [A]  time = 135.958, size = 199, normalized size = 0.93 \[ - \frac{a^{5}}{2 b^{3} \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} + \frac{a^{4} \left (2 a d - 5 b c\right )}{b^{3} \left (a + b x\right ) \left (a d - b c\right )^{4}} + \frac{a^{3} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right ) \log{\left (a + b x \right )}}{b^{3} \left (a d - b c\right )^{5}} + \frac{c^{5}}{2 d^{3} \left (c + d x\right )^{2} \left (a d - b c\right )^{3}} - \frac{c^{4} \left (5 a d - 2 b c\right )}{d^{3} \left (c + d x\right ) \left (a d - b c\right )^{4}} - \frac{c^{3} \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{d^{3} \left (a d - b c\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)**3/(d*x+c)**3,x)

[Out]

-a**5/(2*b**3*(a + b*x)**2*(a*d - b*c)**3) + a**4*(2*a*d - 5*b*c)/(b**3*(a + b*x
)*(a*d - b*c)**4) + a**3*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)*log(a + b*x)/(b*
*3*(a*d - b*c)**5) + c**5/(2*d**3*(c + d*x)**2*(a*d - b*c)**3) - c**4*(5*a*d - 2
*b*c)/(d**3*(c + d*x)*(a*d - b*c)**4) - c**3*(10*a**2*d**2 - 5*a*b*c*d + b**2*c*
*2)*log(c + d*x)/(d**3*(a*d - b*c)**5)

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Mathematica [A]  time = 0.633414, size = 213, normalized size = 1. \[ \frac{a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}+\frac{a^4 (2 a d-5 b c)}{b^3 (a+b x) (b c-a d)^4}-\frac{c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (a d-b c)^5}-\frac{a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}+\frac{c^5}{2 d^3 (c+d x)^2 (a d-b c)^3}+\frac{c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x)^3*(c + d*x)^3),x]

[Out]

a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) + (a^4*(-5*b*c + 2*a*d))/(b^3*(b*c - a*d)^
4*(a + b*x)) + c^5/(2*d^3*(-(b*c) + a*d)^3*(c + d*x)^2) + (c^4*(2*b*c - 5*a*d))/
(d^3*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[a +
b*x])/(b^3*(b*c - a*d)^5) - (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a^2*d^2)*Log[c + d*x]
)/(d^3*(-(b*c) + a*d)^5)

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Maple [A]  time = 0.023, size = 315, normalized size = 1.5 \[ -5\,{\frac{{c}^{4}a}{{d}^{2} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+2\,{\frac{b{c}^{5}}{{d}^{3} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{c}^{5}}{2\,{d}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}-10\,{\frac{{c}^{3}\ln \left ( dx+c \right ){a}^{2}}{ \left ( ad-bc \right ) ^{5}d}}+5\,{\frac{{c}^{4}\ln \left ( dx+c \right ) ab}{ \left ( ad-bc \right ) ^{5}{d}^{2}}}-{\frac{{c}^{5}\ln \left ( dx+c \right ){b}^{2}}{ \left ( ad-bc \right ) ^{5}{d}^{3}}}-{\frac{{a}^{5}}{2\,{b}^{3} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}\ln \left ( bx+a \right ){d}^{2}}{ \left ( ad-bc \right ) ^{5}{b}^{3}}}-5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) cd}{ \left ( ad-bc \right ) ^{5}{b}^{2}}}+10\,{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{2}}{ \left ( ad-bc \right ) ^{5}b}}+2\,{\frac{{a}^{5}d}{{b}^{3} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}-5\,{\frac{{a}^{4}c}{{b}^{2} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)^3/(d*x+c)^3,x)

[Out]

-5*c^4/d^2/(a*d-b*c)^4/(d*x+c)*a+2*c^5/d^3/(a*d-b*c)^4/(d*x+c)*b+1/2*c^5/d^3/(a*
d-b*c)^3/(d*x+c)^2-10*c^3/(a*d-b*c)^5/d*ln(d*x+c)*a^2+5*c^4/(a*d-b*c)^5/d^2*ln(d
*x+c)*a*b-c^5/(a*d-b*c)^5/d^3*ln(d*x+c)*b^2-1/2/b^3/(a*d-b*c)^3*a^5/(b*x+a)^2+a^
5/(a*d-b*c)^5/b^3*ln(b*x+a)*d^2-5*a^4/(a*d-b*c)^5/b^2*ln(b*x+a)*c*d+10*a^3/(a*d-
b*c)^5/b*ln(b*x+a)*c^2+2*a^5/b^3/(a*d-b*c)^4/(b*x+a)*d-5*a^4/b^2/(a*d-b*c)^4/(b*
x+a)*c

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Maxima [A]  time = 1.41524, size = 1098, normalized size = 5.15 \[ -\frac{{\left (10 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{8} c^{5} - 5 \, a b^{7} c^{4} d + 10 \, a^{2} b^{6} c^{3} d^{2} - 10 \, a^{3} b^{5} c^{2} d^{3} + 5 \, a^{4} b^{4} c d^{4} - a^{5} b^{3} d^{5}} + \frac{{\left (b^{2} c^{5} - 5 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} d^{3} - 5 \, a b^{4} c^{4} d^{4} + 10 \, a^{2} b^{3} c^{3} d^{5} - 10 \, a^{3} b^{2} c^{2} d^{6} + 5 \, a^{4} b c d^{7} - a^{5} d^{8}} + \frac{3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \,{\left (2 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{4} b^{2} c d^{5} + 2 \, a^{5} b d^{6}\right )} x^{3} +{\left (3 \, b^{6} c^{6} - a b^{5} c^{5} d - 20 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{4} b^{2} c^{2} d^{4} - a^{5} b c d^{5} + 3 \, a^{6} d^{6}\right )} x^{2} + 2 \,{\left (3 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} - 5 \, a^{4} b^{2} c^{3} d^{3} - 7 \, a^{5} b c^{2} d^{4} + 3 \, a^{6} c d^{5}\right )} x}{2 \,{\left (a^{2} b^{7} c^{6} d^{3} - 4 \, a^{3} b^{6} c^{5} d^{4} + 6 \, a^{4} b^{5} c^{4} d^{5} - 4 \, a^{5} b^{4} c^{3} d^{6} + a^{6} b^{3} c^{2} d^{7} +{\left (b^{9} c^{4} d^{5} - 4 \, a b^{8} c^{3} d^{6} + 6 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )} x^{4} + 2 \,{\left (b^{9} c^{5} d^{4} - 3 \, a b^{8} c^{4} d^{5} + 2 \, a^{2} b^{7} c^{3} d^{6} + 2 \, a^{3} b^{6} c^{2} d^{7} - 3 \, a^{4} b^{5} c d^{8} + a^{5} b^{4} d^{9}\right )} x^{3} +{\left (b^{9} c^{6} d^{3} - 9 \, a^{2} b^{7} c^{4} d^{5} + 16 \, a^{3} b^{6} c^{3} d^{6} - 9 \, a^{4} b^{5} c^{2} d^{7} + a^{6} b^{3} d^{9}\right )} x^{2} + 2 \,{\left (a b^{8} c^{6} d^{3} - 3 \, a^{2} b^{7} c^{5} d^{4} + 2 \, a^{3} b^{6} c^{4} d^{5} + 2 \, a^{4} b^{5} c^{3} d^{6} - 3 \, a^{5} b^{4} c^{2} d^{7} + a^{6} b^{3} c d^{8}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)^3*(d*x + c)^3),x, algorithm="maxima")

[Out]

-(10*a^3*b^2*c^2 - 5*a^4*b*c*d + a^5*d^2)*log(b*x + a)/(b^8*c^5 - 5*a*b^7*c^4*d
+ 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5) + (b^
2*c^5 - 5*a*b*c^4*d + 10*a^2*c^3*d^2)*log(d*x + c)/(b^5*c^5*d^3 - 5*a*b^4*c^4*d^
4 + 10*a^2*b^3*c^3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c*d^7 - a^5*d^8) + 1/2*(3*
a^2*b^4*c^6 - 9*a^3*b^3*c^5*d - 9*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + 2*(2*b^6*c^5*d
- 5*a*b^5*c^4*d^2 - 5*a^4*b^2*c*d^5 + 2*a^5*b*d^6)*x^3 + (3*b^6*c^6 - a*b^5*c^5
*d - 20*a^2*b^4*c^4*d^2 - 20*a^4*b^2*c^2*d^4 - a^5*b*c*d^5 + 3*a^6*d^6)*x^2 + 2*
(3*a*b^5*c^6 - 7*a^2*b^4*c^5*d - 5*a^3*b^3*c^4*d^2 - 5*a^4*b^2*c^3*d^3 - 7*a^5*b
*c^2*d^4 + 3*a^6*c*d^5)*x)/(a^2*b^7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c^4*
d^5 - 4*a^5*b^4*c^3*d^6 + a^6*b^3*c^2*d^7 + (b^9*c^4*d^5 - 4*a*b^8*c^3*d^6 + 6*a
^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c^5*d^4 - 3*a*b^8*c
^4*d^5 + 2*a^2*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^9)*
x^3 + (b^9*c^6*d^3 - 9*a^2*b^7*c^4*d^5 + 16*a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2*d^7
+ a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*a^2*b^7*c^5*d^4 + 2*a^3*b^6*c^4*d^5 +
2*a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*x)

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Fricas [A]  time = 0.273908, size = 1710, normalized size = 8.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)^3*(d*x + c)^3),x, algorithm="fricas")

[Out]

1/2*(3*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 9*a^4*b^3*c^5*d^2 - 9*a^5*b^2*c^4*d^3 +
12*a^6*b*c^3*d^4 - 3*a^7*c^2*d^5 + 2*(2*b^7*c^6*d - 7*a*b^6*c^5*d^2 + 5*a^2*b^5*
c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 7*a^5*b^2*c*d^6 - 2*a^6*b*d^7)*x^3 + (3*b^7*c^7 -
4*a*b^6*c^6*d - 19*a^2*b^5*c^5*d^2 + 20*a^3*b^4*c^4*d^3 - 20*a^4*b^3*c^3*d^4 + 1
9*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 - 3*a^7*d^7)*x^2 + 2*(3*a*b^6*c^7 - 10*a^2*b^5
*c^6*d + 2*a^3*b^4*c^5*d^2 - 2*a^5*b^2*c^3*d^4 + 10*a^6*b*c^2*d^5 - 3*a^7*c*d^6)
*x - 2*(10*a^5*b^2*c^4*d^3 - 5*a^6*b*c^3*d^4 + a^7*c^2*d^5 + (10*a^3*b^4*c^2*d^5
- 5*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(10*a^3*b^4*c^3*d^4 + 5*a^4*b^3*c^2*d^
5 - 4*a^5*b^2*c*d^6 + a^6*b*d^7)*x^3 + (10*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4
- 9*a^5*b^2*c^2*d^5 - a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(10*a^4*b^3*c^4*d^3 + 5*a^5
*b^2*c^3*d^4 - 4*a^6*b*c^2*d^5 + a^7*c*d^6)*x)*log(b*x + a) + 2*(a^2*b^5*c^7 - 5
*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^
5*c^3*d^4)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4
*c^3*d^4)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3
+ 10*a^4*b^3*c^3*d^4)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 +
10*a^4*b^3*c^4*d^3)*x)*log(d*x + c))/(a^2*b^8*c^7*d^3 - 5*a^3*b^7*c^6*d^4 + 10*
a^4*b^6*c^5*d^5 - 10*a^5*b^5*c^4*d^6 + 5*a^6*b^4*c^3*d^7 - a^7*b^3*c^2*d^8 + (b^
10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b
^6*c*d^9 - a^5*b^5*d^10)*x^4 + 2*(b^10*c^6*d^4 - 4*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4
*d^6 - 5*a^4*b^6*c^2*d^8 + 4*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^3 + (b^10*c^7*d^3 -
a*b^9*c^6*d^4 - 9*a^2*b^8*c^5*d^5 + 25*a^3*b^7*c^4*d^6 - 25*a^4*b^6*c^3*d^7 + 9
*a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^2 + 2*(a*b^9*c^7*d^3 - 4*a^2*
b^8*c^6*d^4 + 5*a^3*b^7*c^5*d^5 - 5*a^5*b^5*c^3*d^7 + 4*a^6*b^4*c^2*d^8 - a^7*b^
3*c*d^9)*x)

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Sympy [A]  time = 47.7727, size = 1622, normalized size = 7.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)**3/(d*x+c)**3,x)

[Out]

a**3*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)*log(x + (a**9*d**8*(a**2*d**2 - 5*a*
b*c*d + 10*b**2*c**2)/(b*(a*d - b*c)**5) - 6*a**8*c*d**7*(a**2*d**2 - 5*a*b*c*d
+ 10*b**2*c**2)/(a*d - b*c)**5 + 15*a**7*b*c**2*d**6*(a**2*d**2 - 5*a*b*c*d + 10
*b**2*c**2)/(a*d - b*c)**5 - 20*a**6*b**2*c**3*d**5*(a**2*d**2 - 5*a*b*c*d + 10*
b**2*c**2)/(a*d - b*c)**5 + 15*a**5*b**3*c**4*d**4*(a**2*d**2 - 5*a*b*c*d + 10*b
**2*c**2)/(a*d - b*c)**5 + a**5*c*d**4 - 6*a**4*b**4*c**5*d**3*(a**2*d**2 - 5*a*
b*c*d + 10*b**2*c**2)/(a*d - b*c)**5 - 5*a**4*b*c**2*d**3 + a**3*b**5*c**6*d**2*
(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)/(a*d - b*c)**5 + 20*a**3*b**2*c**3*d**2 -
5*a**2*b**3*c**4*d + a*b**4*c**5)/(a**5*d**5 - 5*a**4*b*c*d**4 + 10*a**3*b**2*c
**2*d**3 + 10*a**2*b**3*c**3*d**2 - 5*a*b**4*c**4*d + b**5*c**5))/(b**3*(a*d - b
*c)**5) - c**3*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)*log(x + (-a**6*b**2*c**3*d
**5*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 6*a**5*b**3*c**4*d**
4*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + a**5*c*d**4 - 15*a**4*
b**4*c**5*d**3*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 5*a**4*b*
c**2*d**3 + 20*a**3*b**5*c**6*d**2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)/(a*d -
b*c)**5 + 20*a**3*b**2*c**3*d**2 - 15*a**2*b**6*c**7*d*(10*a**2*d**2 - 5*a*b*c*
d + b**2*c**2)/(a*d - b*c)**5 - 5*a**2*b**3*c**4*d + 6*a*b**7*c**8*(10*a**2*d**2
- 5*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + a*b**4*c**5 - b**8*c**9*(10*a**2*d**2
- 5*a*b*c*d + b**2*c**2)/(d*(a*d - b*c)**5))/(a**5*d**5 - 5*a**4*b*c*d**4 + 10*
a**3*b**2*c**2*d**3 + 10*a**2*b**3*c**3*d**2 - 5*a*b**4*c**4*d + b**5*c**5))/(d*
*3*(a*d - b*c)**5) + (3*a**6*c**2*d**4 - 9*a**5*b*c**3*d**3 - 9*a**3*b**3*c**5*d
+ 3*a**2*b**4*c**6 + x**3*(4*a**5*b*d**6 - 10*a**4*b**2*c*d**5 - 10*a*b**5*c**4
*d**2 + 4*b**6*c**5*d) + x**2*(3*a**6*d**6 - a**5*b*c*d**5 - 20*a**4*b**2*c**2*d
**4 - 20*a**2*b**4*c**4*d**2 - a*b**5*c**5*d + 3*b**6*c**6) + x*(6*a**6*c*d**5 -
14*a**5*b*c**2*d**4 - 10*a**4*b**2*c**3*d**3 - 10*a**3*b**3*c**4*d**2 - 14*a**2
*b**4*c**5*d + 6*a*b**5*c**6))/(2*a**6*b**3*c**2*d**7 - 8*a**5*b**4*c**3*d**6 +
12*a**4*b**5*c**4*d**5 - 8*a**3*b**6*c**5*d**4 + 2*a**2*b**7*c**6*d**3 + x**4*(2
*a**4*b**5*d**9 - 8*a**3*b**6*c*d**8 + 12*a**2*b**7*c**2*d**7 - 8*a*b**8*c**3*d*
*6 + 2*b**9*c**4*d**5) + x**3*(4*a**5*b**4*d**9 - 12*a**4*b**5*c*d**8 + 8*a**3*b
**6*c**2*d**7 + 8*a**2*b**7*c**3*d**6 - 12*a*b**8*c**4*d**5 + 4*b**9*c**5*d**4)
+ x**2*(2*a**6*b**3*d**9 - 18*a**4*b**5*c**2*d**7 + 32*a**3*b**6*c**3*d**6 - 18*
a**2*b**7*c**4*d**5 + 2*b**9*c**6*d**3) + x*(4*a**6*b**3*c*d**8 - 12*a**5*b**4*c
**2*d**7 + 8*a**4*b**5*c**3*d**6 + 8*a**3*b**6*c**4*d**5 - 12*a**2*b**7*c**5*d**
4 + 4*a*b**8*c**6*d**3))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)^3*(d*x + c)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError