Optimal. Leaf size=38 \[ -\frac {\log \left (a x^3+b\right )}{3 b^2}+\frac {1}{3 b \left (a x^3+b\right )}+\frac {\log (x)}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 44} \[ -\frac {\log \left (a x^3+b\right )}{3 b^2}+\frac {1}{3 b \left (a x^3+b\right )}+\frac {\log (x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^3}\right )^2 x^7} \, dx &=\int \frac {1}{x \left (b+a x^3\right )^2} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{b^2 x}-\frac {a}{b (b+a x)^2}-\frac {a}{b^2 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{3 b \left (b+a x^3\right )}+\frac {\log (x)}{b^2}-\frac {\log \left (b+a x^3\right )}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.87 \[ \frac {\frac {b}{a x^3+b}-\log \left (a x^3+b\right )+3 \log (x)}{3 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 47, normalized size = 1.24 \[ -\frac {{\left (a x^{3} + b\right )} \log \left (a x^{3} + b\right ) - 3 \, {\left (a x^{3} + b\right )} \log \relax (x) - b}{3 \, {\left (a b^{2} x^{3} + b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 1.18 \[ -\frac {\log \left ({\left | a x^{3} + b \right |}\right )}{3 \, b^{2}} + \frac {\log \left ({\left | x \right |}\right )}{b^{2}} + \frac {a x^{3} + 2 \, b}{3 \, {\left (a x^{3} + b\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.92 \[ \frac {1}{3 \left (a \,x^{3}+b \right ) b}+\frac {\ln \relax (x )}{b^{2}}-\frac {\ln \left (a \,x^{3}+b \right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 37, normalized size = 0.97 \[ \frac {1}{3 \, {\left (a b x^{3} + b^{2}\right )}} - \frac {\log \left (a x^{3} + b\right )}{3 \, b^{2}} + \frac {\log \left (x^{3}\right )}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 34, normalized size = 0.89 \[ \frac {\ln \relax (x)}{b^2}+\frac {1}{3\,b\,\left (a\,x^3+b\right )}-\frac {\ln \left (a\,x^3+b\right )}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 34, normalized size = 0.89 \[ \frac {1}{3 a b x^{3} + 3 b^{2}} + \frac {\log {\relax (x )}}{b^{2}} - \frac {\log {\left (x^{3} + \frac {b}{a} \right )}}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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