Optimal. Leaf size=35 \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]
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Rubi [A] time = 0.0231146, antiderivative size = 35, normalized size of antiderivative = 1., 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} \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^7} \, dx &=\int \frac{1}{x^4 \left (b+a x^3\right )} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+a x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b x^2}-\frac{a}{b^2 x}+\frac{a^2}{b^2 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{3 b x^3}-\frac{a \log (x)}{b^2}+\frac{a \log \left (b+a x^3\right )}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0069067, size = 35, normalized size = 1. \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b{x}^{3}}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\ln \left ( a{x}^{3}+b \right ) }{3\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00295, size = 45, normalized size = 1.29 \begin{align*} \frac{a \log \left (a x^{3} + b\right )}{3 \, b^{2}} - \frac{a \log \left (x^{3}\right )}{3 \, b^{2}} - \frac{1}{3 \, b x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44066, size = 80, normalized size = 2.29 \begin{align*} \frac{a x^{3} \log \left (a x^{3} + b\right ) - 3 \, a x^{3} \log \left (x\right ) - b}{3 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.655607, size = 31, normalized size = 0.89 \begin{align*} - \frac{a \log{\left (x \right )}}{b^{2}} + \frac{a \log{\left (x^{3} + \frac{b}{a} \right )}}{3 b^{2}} - \frac{1}{3 b x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19547, size = 57, normalized size = 1.63 \begin{align*} \frac{a \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, b^{2}} - \frac{a \log \left ({\left | x \right |}\right )}{b^{2}} + \frac{a x^{3} - b}{3 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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