Optimal. Leaf size=87 \[ \frac {b^2 \log \left (a+b x^3\right )}{3 a^2 (b c-a d)}-\frac {\log (x) (a d+b c)}{a^2 c^2}-\frac {d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac {1}{3 a c x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 72} \begin {gather*} \frac {b^2 \log \left (a+b x^3\right )}{3 a^2 (b c-a d)}-\frac {\log (x) (a d+b c)}{a^2 c^2}-\frac {d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac {1}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x) (c+d x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{a c x^2}+\frac {-b c-a d}{a^2 c^2 x}-\frac {b^3}{a^2 (-b c+a d) (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{3 a c x^3}-\frac {(b c+a d) \log (x)}{a^2 c^2}+\frac {b^2 \log \left (a+b x^3\right )}{3 a^2 (b c-a d)}-\frac {d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 88, normalized size = 1.01 \begin {gather*} -\frac {b^2 \log \left (a+b x^3\right )}{3 a^2 (a d-b c)}+\frac {\log (x) (-a d-b c)}{a^2 c^2}-\frac {d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac {1}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 4.79, size = 99, normalized size = 1.14 \begin {gather*} \frac {b^{2} c^{2} x^{3} \log \left (b x^{3} + a\right ) - a^{2} d^{2} x^{3} \log \left (d x^{3} + c\right ) - 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3} \log \relax (x) - a b c^{2} + a^{2} c d}{3 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 111, normalized size = 1.28 \begin {gather*} \frac {b^{3} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, {\left (a^{2} b^{2} c - a^{3} b d\right )}} - \frac {d^{3} \log \left ({\left | d x^{3} + c \right |}\right )}{3 \, {\left (b c^{3} d - a c^{2} d^{2}\right )}} - \frac {{\left (b c + a d\right )} \log \left ({\left | x \right |}\right )}{a^{2} c^{2}} + \frac {b c x^{3} + a d x^{3} - a c}{3 \, a^{2} c^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^{2} \ln \left (b \,x^{3}+a \right )}{3 \left (a d -b c \right ) a^{2}}+\frac {d^{2} \ln \left (d \,x^{3}+c \right )}{3 \left (a d -b c \right ) c^{2}}-\frac {d \ln \relax (x )}{a \,c^{2}}-\frac {b \ln \relax (x )}{a^{2} c}-\frac {1}{3 a c \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 87, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (b x^{3} + a\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \log \left (d x^{3} + c\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {{\left (b c + a d\right )} \log \left (x^{3}\right )}{3 \, a^{2} c^{2}} - \frac {1}{3 \, a c x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.22, size = 87, normalized size = 1.00 \begin {gather*} -\frac {b^2\,\ln \left (b\,x^3+a\right )}{3\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {d^2\,\ln \left (d\,x^3+c\right )}{3\,\left (b\,c^3-a\,c^2\,d\right )}-\frac {1}{3\,a\,c\,x^3}-\frac {\ln \relax (x)\,\left (a\,d+b\,c\right )}{a^2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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