3.318 \(\int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx\)

Optimal. Leaf size=242 \[ -\frac {3 \log (x) (a d+b c)}{a^4 c^4}-\frac {b^4 (2 b c-5 a d)}{a^3 (a+b x) (b c-a d)^4}-\frac {1}{a^3 c^3 x}-\frac {b^4}{2 a^2 (a+b x)^2 (b c-a d)^3}-\frac {3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}+\frac {3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}+\frac {d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac {d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.33, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ \frac {3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac {3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}-\frac {b^4 (2 b c-5 a d)}{a^3 (a+b x) (b c-a d)^4}-\frac {b^4}{2 a^2 (a+b x)^2 (b c-a d)^3}-\frac {3 \log (x) (a d+b c)}{a^4 c^4}-\frac {1}{a^3 c^3 x}+\frac {d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac {d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^3*(c + d*x)^3),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

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Mathematica [A]  time = 0.44, size = 241, normalized size = 1.00 \[ -\frac {3 \log (x) (a d+b c)}{a^4 c^4}+\frac {b^4 (5 a d-2 b c)}{a^3 (a+b x) (b c-a d)^4}-\frac {1}{a^3 c^3 x}+\frac {b^4}{2 a^2 (a+b x)^2 (a d-b c)^3}-\frac {3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}-\frac {3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (a d-b c)^5}+\frac {d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac {d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^3*(c + d*x)^3),x]

[Out]

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

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fricas [B]  time = 179.12, size = 1798, normalized size = 7.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^3*b^5*c^8 - 10*a^4*b^4*c^7*d + 20*a^5*b^3*c^6*d^2 - 20*a^6*b^2*c^5*d^3 + 10*a^7*b*c^4*d^4 - 2*a^8*c^
3*d^5 + 6*(a*b^7*c^6*d^2 - 4*a^2*b^6*c^5*d^3 + 5*a^3*b^5*c^4*d^4 - 5*a^4*b^4*c^3*d^5 + 4*a^5*b^3*c^2*d^6 - a^6
*b^2*c*d^7)*x^4 + 3*(4*a*b^7*c^7*d - 13*a^2*b^6*c^6*d^2 + 8*a^3*b^5*c^5*d^3 - 8*a^5*b^3*c^3*d^5 + 13*a^6*b^2*c
^2*d^6 - 4*a^7*b*c*d^7)*x^3 + 2*(3*a*b^7*c^8 - 3*a^2*b^6*c^7*d - 20*a^3*b^5*c^6*d^2 + 36*a^4*b^4*c^5*d^3 - 36*
a^5*b^3*c^4*d^4 + 20*a^6*b^2*c^3*d^5 + 3*a^7*b*c^2*d^6 - 3*a^8*c*d^7)*x^2 + (9*a^2*b^6*c^8 - 32*a^3*b^5*c^7*d
+ 31*a^4*b^4*c^6*d^2 - 31*a^6*b^2*c^4*d^4 + 32*a^7*b*c^3*d^5 - 9*a^8*c^2*d^6)*x - 6*((b^8*c^6*d^2 - 4*a*b^7*c^
5*d^3 + 5*a^2*b^6*c^4*d^4)*x^5 + 2*(b^8*c^7*d - 3*a*b^7*c^6*d^2 + a^2*b^6*c^5*d^3 + 5*a^3*b^5*c^4*d^4)*x^4 + (
b^8*c^8 - 10*a^2*b^6*c^6*d^2 + 16*a^3*b^5*c^5*d^3 + 5*a^4*b^4*c^4*d^4)*x^3 + 2*(a*b^7*c^8 - 3*a^2*b^6*c^7*d +
a^3*b^5*c^6*d^2 + 5*a^4*b^4*c^5*d^3)*x^2 + (a^2*b^6*c^8 - 4*a^3*b^5*c^7*d + 5*a^4*b^4*c^6*d^2)*x)*log(b*x + a)
+ 6*((5*a^4*b^4*c^2*d^6 - 4*a^5*b^3*c*d^7 + a^6*b^2*d^8)*x^5 + 2*(5*a^4*b^4*c^3*d^5 + a^5*b^3*c^2*d^6 - 3*a^6
*b^2*c*d^7 + a^7*b*d^8)*x^4 + (5*a^4*b^4*c^4*d^4 + 16*a^5*b^3*c^3*d^5 - 10*a^6*b^2*c^2*d^6 + a^8*d^8)*x^3 + 2*
(5*a^5*b^3*c^4*d^4 + a^6*b^2*c^3*d^5 - 3*a^7*b*c^2*d^6 + a^8*c*d^7)*x^2 + (5*a^6*b^2*c^4*d^4 - 4*a^7*b*c^3*d^5
+ a^8*c^2*d^6)*x)*log(d*x + c) + 6*((b^8*c^6*d^2 - 4*a*b^7*c^5*d^3 + 5*a^2*b^6*c^4*d^4 - 5*a^4*b^4*c^2*d^6 +
4*a^5*b^3*c*d^7 - a^6*b^2*d^8)*x^5 + 2*(b^8*c^7*d - 3*a*b^7*c^6*d^2 + a^2*b^6*c^5*d^3 + 5*a^3*b^5*c^4*d^4 - 5*
a^4*b^4*c^3*d^5 - a^5*b^3*c^2*d^6 + 3*a^6*b^2*c*d^7 - a^7*b*d^8)*x^4 + (b^8*c^8 - 10*a^2*b^6*c^6*d^2 + 16*a^3*
b^5*c^5*d^3 - 16*a^5*b^3*c^3*d^5 + 10*a^6*b^2*c^2*d^6 - a^8*d^8)*x^3 + 2*(a*b^7*c^8 - 3*a^2*b^6*c^7*d + a^3*b^
5*c^6*d^2 + 5*a^4*b^4*c^5*d^3 - 5*a^5*b^3*c^4*d^4 - a^6*b^2*c^3*d^5 + 3*a^7*b*c^2*d^6 - a^8*c*d^7)*x^2 + (a^2*
b^6*c^8 - 4*a^3*b^5*c^7*d + 5*a^4*b^4*c^6*d^2 - 5*a^6*b^2*c^4*d^4 + 4*a^7*b*c^3*d^5 - a^8*c^2*d^6)*x)*log(x))/
((a^4*b^7*c^9*d^2 - 5*a^5*b^6*c^8*d^3 + 10*a^6*b^5*c^7*d^4 - 10*a^7*b^4*c^6*d^5 + 5*a^8*b^3*c^5*d^6 - a^9*b^2*
c^4*d^7)*x^5 + 2*(a^4*b^7*c^10*d - 4*a^5*b^6*c^9*d^2 + 5*a^6*b^5*c^8*d^3 - 5*a^8*b^3*c^6*d^5 + 4*a^9*b^2*c^5*d
^6 - a^10*b*c^4*d^7)*x^4 + (a^4*b^7*c^11 - a^5*b^6*c^10*d - 9*a^6*b^5*c^9*d^2 + 25*a^7*b^4*c^8*d^3 - 25*a^8*b^
3*c^7*d^4 + 9*a^9*b^2*c^6*d^5 + a^10*b*c^5*d^6 - a^11*c^4*d^7)*x^3 + 2*(a^5*b^6*c^11 - 4*a^6*b^5*c^10*d + 5*a^
7*b^4*c^9*d^2 - 5*a^9*b^2*c^7*d^4 + 4*a^10*b*c^6*d^5 - a^11*c^5*d^6)*x^2 + (a^6*b^5*c^11 - 5*a^7*b^4*c^10*d +
10*a^8*b^3*c^9*d^2 - 10*a^9*b^2*c^8*d^3 + 5*a^10*b*c^7*d^4 - a^11*c^6*d^5)*x)

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giac [B]  time = 1.85, size = 630, normalized size = 2.60 \[ \frac {3 \, {\left (b^{7} c^{2} - 4 \, a b^{6} c d + 5 \, a^{2} b^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{5} - 4 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{9} d - 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{7} d^{3} - 10 \, a^{3} b^{2} c^{6} d^{4} + 5 \, a^{4} b c^{5} d^{5} - a^{5} c^{4} d^{6}} - \frac {3 \, {\left (b c + a d\right )} \log \left ({\left | x \right |}\right )}{a^{4} c^{4}} - \frac {2 \, a^{3} b^{4} c^{7} - 8 \, a^{4} b^{3} c^{6} d + 12 \, a^{5} b^{2} c^{5} d^{2} - 8 \, a^{6} b c^{4} d^{3} + 2 \, a^{7} c^{3} d^{4} + 6 \, {\left (a b^{6} c^{5} d^{2} - 3 \, a^{2} b^{5} c^{4} d^{3} + 2 \, a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6}\right )} x^{4} + 3 \, {\left (4 \, a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} - a^{3} b^{4} c^{4} d^{3} - a^{4} b^{3} c^{3} d^{4} - 9 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6}\right )} x^{3} + 2 \, {\left (3 \, a b^{6} c^{7} - 20 \, a^{3} b^{4} c^{5} d^{2} + 16 \, a^{4} b^{3} c^{4} d^{3} - 20 \, a^{5} b^{2} c^{3} d^{4} + 3 \, a^{7} c d^{6}\right )} x^{2} + {\left (9 \, a^{2} b^{5} c^{7} - 23 \, a^{3} b^{4} c^{6} d + 8 \, a^{4} b^{3} c^{5} d^{2} + 8 \, a^{5} b^{2} c^{4} d^{3} - 23 \, a^{6} b c^{3} d^{4} + 9 \, a^{7} c^{2} d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} a^{4} c^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

3*(b^7*c^2 - 4*a*b^6*c*d + 5*a^2*b^5*d^2)*log(abs(b*x + a))/(a^4*b^6*c^5 - 5*a^5*b^5*c^4*d + 10*a^6*b^4*c^3*d^
2 - 10*a^7*b^3*c^2*d^3 + 5*a^8*b^2*c*d^4 - a^9*b*d^5) - 3*(5*b^2*c^2*d^5 - 4*a*b*c*d^6 + a^2*d^7)*log(abs(d*x
+ c))/(b^5*c^9*d - 5*a*b^4*c^8*d^2 + 10*a^2*b^3*c^7*d^3 - 10*a^3*b^2*c^6*d^4 + 5*a^4*b*c^5*d^5 - a^5*c^4*d^6)
- 3*(b*c + a*d)*log(abs(x))/(a^4*c^4) - 1/2*(2*a^3*b^4*c^7 - 8*a^4*b^3*c^6*d + 12*a^5*b^2*c^5*d^2 - 8*a^6*b*c^
4*d^3 + 2*a^7*c^3*d^4 + 6*(a*b^6*c^5*d^2 - 3*a^2*b^5*c^4*d^3 + 2*a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 + a^5*b^2
*c*d^6)*x^4 + 3*(4*a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 - a^3*b^4*c^4*d^3 - a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + 4
*a^6*b*c*d^6)*x^3 + 2*(3*a*b^6*c^7 - 20*a^3*b^4*c^5*d^2 + 16*a^4*b^3*c^4*d^3 - 20*a^5*b^2*c^3*d^4 + 3*a^7*c*d^
6)*x^2 + (9*a^2*b^5*c^7 - 23*a^3*b^4*c^6*d + 8*a^4*b^3*c^5*d^2 + 8*a^5*b^2*c^4*d^3 - 23*a^6*b*c^3*d^4 + 9*a^7*
c^2*d^5)*x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*a^4*c^4*x)

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maple [A]  time = 0.02, size = 349, normalized size = 1.44 \[ \frac {3 a^{2} d^{6} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} c^{4}}-\frac {12 a b \,d^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} c^{3}}-\frac {15 b^{4} d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} a^{2}}+\frac {12 b^{5} c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} a^{3}}-\frac {3 b^{6} c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} a^{4}}+\frac {15 b^{2} d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} c^{2}}-\frac {2 a \,d^{5}}{\left (a d -b c \right )^{4} \left (d x +c \right ) c^{3}}+\frac {5 b^{4} d}{\left (a d -b c \right )^{4} \left (b x +a \right ) a^{2}}-\frac {2 b^{5} c}{\left (a d -b c \right )^{4} \left (b x +a \right ) a^{3}}+\frac {5 b \,d^{4}}{\left (a d -b c \right )^{4} \left (d x +c \right ) c^{2}}+\frac {b^{4}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2} a^{2}}-\frac {d^{4}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2} c^{2}}-\frac {3 d \ln \relax (x )}{a^{3} c^{4}}-\frac {3 b \ln \relax (x )}{a^{4} c^{3}}-\frac {1}{a^{3} c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^3/(d*x+c)^3,x)

[Out]

-1/a^3/c^3/x-3/a^3/c^4*ln(x)*d-3/a^4/c^3*ln(x)*b-1/2*d^4/c^2/(a*d-b*c)^3/(d*x+c)^2-2*d^5/c^3/(a*d-b*c)^4/(d*x+
c)*a+5*d^4/c^2/(a*d-b*c)^4/(d*x+c)*b+3*d^6/c^4/(a*d-b*c)^5*ln(d*x+c)*a^2-12*d^5/c^3/(a*d-b*c)^5*ln(d*x+c)*a*b+
15*d^4/c^2/(a*d-b*c)^5*ln(d*x+c)*b^2+1/2*b^4/a^2/(a*d-b*c)^3/(b*x+a)^2+5*b^4/a^2/(a*d-b*c)^4/(b*x+a)*d-2*b^5/a
^3/(a*d-b*c)^4/(b*x+a)*c-15*b^4/a^2/(a*d-b*c)^5*ln(b*x+a)*d^2+12*b^5/a^3/(a*d-b*c)^5*ln(b*x+a)*c*d-3*b^6/a^4/(
a*d-b*c)^5*ln(b*x+a)*c^2

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maxima [B]  time = 1.34, size = 936, normalized size = 3.87 \[ \frac {3 \, {\left (b^{6} c^{2} - 4 \, a b^{5} c d + 5 \, a^{2} b^{4} d^{2}\right )} \log \left (b x + a\right )}{a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{4} - 4 \, a b c d^{5} + a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{5} c^{9} - 5 \, a b^{4} c^{8} d + 10 \, a^{2} b^{3} c^{7} d^{2} - 10 \, a^{3} b^{2} c^{6} d^{3} + 5 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}} - \frac {2 \, a^{2} b^{4} c^{6} - 8 \, a^{3} b^{3} c^{5} d + 12 \, a^{4} b^{2} c^{4} d^{2} - 8 \, a^{5} b c^{3} d^{3} + 2 \, a^{6} c^{2} d^{4} + 6 \, {\left (b^{6} c^{4} d^{2} - 3 \, a b^{5} c^{3} d^{3} + 2 \, a^{2} b^{4} c^{2} d^{4} - 3 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 3 \, {\left (4 \, b^{6} c^{5} d - 9 \, a b^{5} c^{4} d^{2} - a^{2} b^{4} c^{3} d^{3} - a^{3} b^{3} c^{2} d^{4} - 9 \, a^{4} b^{2} c d^{5} + 4 \, a^{5} b d^{6}\right )} x^{3} + 2 \, {\left (3 \, b^{6} c^{6} - 20 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 20 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{6} d^{6}\right )} x^{2} + {\left (9 \, a b^{5} c^{6} - 23 \, a^{2} b^{4} c^{5} d + 8 \, a^{3} b^{3} c^{4} d^{2} + 8 \, a^{4} b^{2} c^{3} d^{3} - 23 \, a^{5} b c^{2} d^{4} + 9 \, a^{6} c d^{5}\right )} x}{2 \, {\left ({\left (a^{3} b^{6} c^{7} d^{2} - 4 \, a^{4} b^{5} c^{6} d^{3} + 6 \, a^{5} b^{4} c^{5} d^{4} - 4 \, a^{6} b^{3} c^{4} d^{5} + a^{7} b^{2} c^{3} d^{6}\right )} x^{5} + 2 \, {\left (a^{3} b^{6} c^{8} d - 3 \, a^{4} b^{5} c^{7} d^{2} + 2 \, a^{5} b^{4} c^{6} d^{3} + 2 \, a^{6} b^{3} c^{5} d^{4} - 3 \, a^{7} b^{2} c^{4} d^{5} + a^{8} b c^{3} d^{6}\right )} x^{4} + {\left (a^{3} b^{6} c^{9} - 9 \, a^{5} b^{4} c^{7} d^{2} + 16 \, a^{6} b^{3} c^{6} d^{3} - 9 \, a^{7} b^{2} c^{5} d^{4} + a^{9} c^{3} d^{6}\right )} x^{3} + 2 \, {\left (a^{4} b^{5} c^{9} - 3 \, a^{5} b^{4} c^{8} d + 2 \, a^{6} b^{3} c^{7} d^{2} + 2 \, a^{7} b^{2} c^{6} d^{3} - 3 \, a^{8} b c^{5} d^{4} + a^{9} c^{4} d^{5}\right )} x^{2} + {\left (a^{5} b^{4} c^{9} - 4 \, a^{6} b^{3} c^{8} d + 6 \, a^{7} b^{2} c^{7} d^{2} - 4 \, a^{8} b c^{6} d^{3} + a^{9} c^{5} d^{4}\right )} x\right )}} - \frac {3 \, {\left (b c + a d\right )} \log \relax (x)}{a^{4} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

3*(b^6*c^2 - 4*a*b^5*c*d + 5*a^2*b^4*d^2)*log(b*x + a)/(a^4*b^5*c^5 - 5*a^5*b^4*c^4*d + 10*a^6*b^3*c^3*d^2 - 1
0*a^7*b^2*c^2*d^3 + 5*a^8*b*c*d^4 - a^9*d^5) - 3*(5*b^2*c^2*d^4 - 4*a*b*c*d^5 + a^2*d^6)*log(d*x + c)/(b^5*c^9
- 5*a*b^4*c^8*d + 10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 + 5*a^4*b*c^5*d^4 - a^5*c^4*d^5) - 1/2*(2*a^2*b^4*c
^6 - 8*a^3*b^3*c^5*d + 12*a^4*b^2*c^4*d^2 - 8*a^5*b*c^3*d^3 + 2*a^6*c^2*d^4 + 6*(b^6*c^4*d^2 - 3*a*b^5*c^3*d^3
+ 2*a^2*b^4*c^2*d^4 - 3*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 3*(4*b^6*c^5*d - 9*a*b^5*c^4*d^2 - a^2*b^4*c^3*d^3
- a^3*b^3*c^2*d^4 - 9*a^4*b^2*c*d^5 + 4*a^5*b*d^6)*x^3 + 2*(3*b^6*c^6 - 20*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d
^3 - 20*a^4*b^2*c^2*d^4 + 3*a^6*d^6)*x^2 + (9*a*b^5*c^6 - 23*a^2*b^4*c^5*d + 8*a^3*b^3*c^4*d^2 + 8*a^4*b^2*c^3
*d^3 - 23*a^5*b*c^2*d^4 + 9*a^6*c*d^5)*x)/((a^3*b^6*c^7*d^2 - 4*a^4*b^5*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^
3*c^4*d^5 + a^7*b^2*c^3*d^6)*x^5 + 2*(a^3*b^6*c^8*d - 3*a^4*b^5*c^7*d^2 + 2*a^5*b^4*c^6*d^3 + 2*a^6*b^3*c^5*d^
4 - 3*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6)*x^4 + (a^3*b^6*c^9 - 9*a^5*b^4*c^7*d^2 + 16*a^6*b^3*c^6*d^3 - 9*a^7*b^2
*c^5*d^4 + a^9*c^3*d^6)*x^3 + 2*(a^4*b^5*c^9 - 3*a^5*b^4*c^8*d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8
*b*c^5*d^4 + a^9*c^4*d^5)*x^2 + (a^5*b^4*c^9 - 4*a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6*d^3 + a^9*c^5
*d^4)*x) - 3*(b*c + a*d)*log(x)/(a^4*c^4)

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mupad [B]  time = 1.64, size = 823, normalized size = 3.40 \[ -\frac {\frac {1}{a\,c}+\frac {3\,x^4\,\left (a^4\,b^2\,d^6-3\,a^3\,b^3\,c\,d^5+2\,a^2\,b^4\,c^2\,d^4-3\,a\,b^5\,c^3\,d^3+b^6\,c^4\,d^2\right )}{a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x^2\,\left (3\,a^6\,d^6-20\,a^4\,b^2\,c^2\,d^4+16\,a^3\,b^3\,c^3\,d^3-20\,a^2\,b^4\,c^4\,d^2+3\,b^6\,c^6\right )}{a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {3\,x^3\,\left (-4\,a^5\,b\,d^6+9\,a^4\,b^2\,c\,d^5+a^3\,b^3\,c^2\,d^4+a^2\,b^4\,c^3\,d^3+9\,a\,b^5\,c^4\,d^2-4\,b^6\,c^5\,d\right )}{2\,a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x\,\left (9\,a^5\,d^5-23\,a^4\,b\,c\,d^4+8\,a^3\,b^2\,c^2\,d^3+8\,a^2\,b^3\,c^3\,d^2-23\,a\,b^4\,c^4\,d+9\,b^5\,c^5\right )}{2\,a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^2\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^4\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2\,x+b^2\,d^2\,x^5}-\frac {\ln \left (a+b\,x\right )\,\left (15\,a^2\,b^4\,d^2-12\,a\,b^5\,c\,d+3\,b^6\,c^2\right )}{a^9\,d^5-5\,a^8\,b\,c\,d^4+10\,a^7\,b^2\,c^2\,d^3-10\,a^6\,b^3\,c^3\,d^2+5\,a^5\,b^4\,c^4\,d-a^4\,b^5\,c^5}-\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,d^6-12\,a\,b\,c\,d^5+15\,b^2\,c^2\,d^4\right )}{-a^5\,c^4\,d^5+5\,a^4\,b\,c^5\,d^4-10\,a^3\,b^2\,c^6\,d^3+10\,a^2\,b^3\,c^7\,d^2-5\,a\,b^4\,c^8\,d+b^5\,c^9}-\frac {3\,\ln \relax (x)\,\left (a\,d+b\,c\right )}{a^4\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x)^3*(c + d*x)^3),x)

[Out]

- (1/(a*c) + (3*x^4*(a^4*b^2*d^6 + b^6*c^4*d^2 - 3*a*b^5*c^3*d^3 - 3*a^3*b^3*c*d^5 + 2*a^2*b^4*c^2*d^4))/(a^3*
c^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (x^2*(3*a^6*d^6 + 3*b^6*c^6 - 2
0*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 20*a^4*b^2*c^2*d^4))/(a^3*c^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2
- 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (3*x^3*(9*a*b^5*c^4*d^2 - 4*b^6*c^5*d - 4*a^5*b*d^6 + 9*a^4*b^2*c*d^5 + a^
2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4))/(2*a^3*c^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*
c*d^3)) + (x*(9*a^5*d^5 + 9*b^5*c^5 + 8*a^2*b^3*c^3*d^2 + 8*a^3*b^2*c^2*d^3 - 23*a*b^4*c^4*d - 23*a^4*b*c*d^4)
)/(2*a^2*c^2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))/(x^3*(a^2*d^2 + b^2*c^2
+ 4*a*b*c*d) + x^2*(2*a*b*c^2 + 2*a^2*c*d) + x^4*(2*a*b*d^2 + 2*b^2*c*d) + a^2*c^2*x + b^2*d^2*x^5) - (log(a
+ b*x)*(3*b^6*c^2 + 15*a^2*b^4*d^2 - 12*a*b^5*c*d))/(a^9*d^5 - a^4*b^5*c^5 + 5*a^5*b^4*c^4*d - 10*a^6*b^3*c^3*
d^2 + 10*a^7*b^2*c^2*d^3 - 5*a^8*b*c*d^4) - (log(c + d*x)*(3*a^2*d^6 + 15*b^2*c^2*d^4 - 12*a*b*c*d^5))/(b^5*c^
9 - a^5*c^4*d^5 + 5*a^4*b*c^5*d^4 + 10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 - 5*a*b^4*c^8*d) - (3*log(x)*(a*d
+ b*c))/(a^4*c^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

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