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

Optimal. Leaf size=178 \[ -\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}+\frac {b^4}{a^3 (a+b x) (b c-a d)^2}+\frac {2 (a d+b c)}{a^3 c^3 x}-\frac {1}{2 a^2 c^2 x^2}+\frac {\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3}+\frac {d^4}{c^3 (c+d x) (b c-a d)^2} \]

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Rubi [A]  time = 0.20, antiderivative size = 178, 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} \begin {gather*} \frac {\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}+\frac {b^4}{a^3 (a+b x) (b c-a d)^2}-\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}+\frac {2 (a d+b c)}{a^3 c^3 x}-\frac {1}{2 a^2 c^2 x^2}+\frac {d^4}{c^3 (c+d x) (b c-a d)^2}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.22, size = 176, normalized size = 0.99 \begin {gather*} \frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (a d-b c)^3}+\frac {b^4}{a^3 (a+b x) (b c-a d)^2}+\frac {2 (a d+b c)}{a^3 c^3 x}-\frac {1}{2 a^2 c^2 x^2}+\frac {\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}+\frac {d^4 (3 a d-5 b c) \log (c+d x)}{c^4 (b c-a d)^3}+\frac {d^4}{c^3 (c+d x) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 78.83, size = 750, normalized size = 4.21 \begin {gather*} -\frac {a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} - 2 \, {\left (3 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{2} d^{4} - 3 \, a^{5} b c d^{5}\right )} x^{3} - {\left (6 \, 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} - 6 \, a^{6} c d^{5}\right )} x^{2} - 3 \, {\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x + 2 \, {\left ({\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2}\right )} x^{4} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2}\right )} x^{3} + {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x^{3} + {\left (5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5}\right )} x^{2}\right )} \log \left (d x + c\right ) - 2 \, {\left ({\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{4} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x^{3} + {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d + 5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left ({\left (a^{4} b^{4} c^{7} d - 3 \, a^{5} b^{3} c^{6} d^{2} + 3 \, a^{6} b^{2} c^{5} d^{3} - a^{7} b c^{4} d^{4}\right )} x^{4} + {\left (a^{4} b^{4} c^{8} - 2 \, a^{5} b^{3} c^{7} d + 2 \, a^{7} b c^{5} d^{3} - a^{8} c^{4} d^{4}\right )} x^{3} + {\left (a^{5} b^{3} c^{8} - 3 \, a^{6} b^{2} c^{7} d + 3 \, a^{7} b c^{6} d^{2} - a^{8} c^{5} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.00, size = 459, normalized size = 2.58 \begin {gather*} \frac {b^{9}}{{\left (a^{3} b^{7} c^{2} - 2 \, a^{4} b^{6} c d + a^{5} b^{5} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (5 \, b^{2} c d^{4} - 3 \, a b d^{5}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{7} - 3 \, a b^{3} c^{6} d + 3 \, a^{2} b^{2} c^{5} d^{2} - a^{3} b c^{4} d^{3}} + \frac {{\left (3 \, b^{3} c^{2} + 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b c^{4}} + \frac {5 \, b^{5} c^{5} d - 11 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{3} d^{3} + 7 \, a^{3} b^{2} c^{2} d^{4} - 6 \, a^{4} b c d^{5} + \frac {5 \, b^{7} c^{6} - 22 \, a b^{6} c^{5} d + 28 \, a^{2} b^{5} c^{4} d^{2} - 2 \, a^{3} b^{4} c^{3} d^{3} - 17 \, a^{4} b^{3} c^{2} d^{4} + 12 \, a^{5} b^{2} c d^{5}}{{\left (b x + a\right )} b} - \frac {2 \, {\left (3 \, a b^{8} c^{6} - 10 \, a^{2} b^{7} c^{5} d + 10 \, a^{3} b^{6} c^{4} d^{2} - 5 \, a^{5} b^{4} c^{2} d^{4} + 3 \, a^{6} b^{3} c d^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \, {\left (b c - a d\right )}^{3} a^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} c^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^9/((a^3*b^7*c^2 - 2*a^4*b^6*c*d + a^5*b^5*d^2)*(b*x + a)) - (5*b^2*c*d^4 - 3*a*b*d^5)*log(abs(b*c/(b*x + a)
- a*d/(b*x + a) + d))/(b^4*c^7 - 3*a*b^3*c^6*d + 3*a^2*b^2*c^5*d^2 - a^3*b*c^4*d^3) + (3*b^3*c^2 + 4*a*b^2*c*d
+ 3*a^2*b*d^2)*log(abs(-a/(b*x + a) + 1))/(a^4*b*c^4) + 1/2*(5*b^5*c^5*d - 11*a*b^4*c^4*d^2 + 3*a^2*b^3*c^3*d
^3 + 7*a^3*b^2*c^2*d^4 - 6*a^4*b*c*d^5 + (5*b^7*c^6 - 22*a*b^6*c^5*d + 28*a^2*b^5*c^4*d^2 - 2*a^3*b^4*c^3*d^3
- 17*a^4*b^3*c^2*d^4 + 12*a^5*b^2*c*d^5)/((b*x + a)*b) - 2*(3*a*b^8*c^6 - 10*a^2*b^7*c^5*d + 10*a^3*b^6*c^4*d^
2 - 5*a^5*b^4*c^2*d^4 + 3*a^6*b^3*c*d^5)/((b*x + a)^2*b^2))/((b*c - a*d)^3*a^4*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)*c^4*(a/(b*x + a) - 1)^2)

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maple [A]  time = 0.02, size = 223, normalized size = 1.25 \begin {gather*} -\frac {3 a \,d^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{4}}-\frac {5 b^{4} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{3}}+\frac {3 b^{5} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{4}}+\frac {5 b \,d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{3}}+\frac {b^{4}}{\left (a d -b c \right )^{2} \left (b x +a \right ) a^{3}}+\frac {d^{4}}{\left (a d -b c \right )^{2} \left (d x +c \right ) c^{3}}+\frac {3 d^{2} \ln \relax (x )}{a^{2} c^{4}}+\frac {4 b d \ln \relax (x )}{a^{3} c^{3}}+\frac {3 b^{2} \ln \relax (x )}{a^{4} c^{2}}+\frac {2 d}{a^{2} c^{3} x}+\frac {2 b}{a^{3} c^{2} x}-\frac {1}{2 a^{2} c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.22, size = 472, normalized size = 2.65 \begin {gather*} -\frac {{\left (3 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}} - \frac {{\left (5 \, b c d^{4} - 3 \, a d^{5}\right )} \log \left (d x + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}} - \frac {a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - 2 \, {\left (3 \, b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} - 2 \, a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x^{3} - {\left (6 \, b^{4} c^{4} - a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + 6 \, a^{4} d^{4}\right )} x^{2} - 3 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x}{2 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{4} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x^{2}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \relax (x)}{a^{4} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.06, size = 367, normalized size = 2.06 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (3\,b^5\,c-5\,a\,b^4\,d\right )}{a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d-a^4\,b^3\,c^3}-\frac {\frac {1}{2\,a\,c}-\frac {3\,x\,\left (a\,d+b\,c\right )}{2\,a^2\,c^2}+\frac {x^2\,\left (-6\,a^4\,d^4+a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d-6\,b^4\,c^4\right )}{2\,a^3\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {x^3\,\left (a\,d+b\,c\right )\,\left (3\,a^2\,b\,d^3-5\,a\,b^2\,c\,d^2+3\,b^3\,c^2\,d\right )}{a^3\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^4+\left (a\,d+b\,c\right )\,x^3+a\,c\,x^2}+\frac {\ln \left (c+d\,x\right )\,\left (3\,a\,d^5-5\,b\,c\,d^4\right )}{-a^3\,c^4\,d^3+3\,a^2\,b\,c^5\,d^2-3\,a\,b^2\,c^6\,d+b^3\,c^7}+\frac {\ln \relax (x)\,\left (3\,a^2\,d^2+4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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