Optimal. Leaf size=52 \[ -\frac{b}{3 a^2 \left (a+b x^3\right )}+\frac{2 b \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b \log (x)}{a^3}-\frac{1}{3 a^2 x^3} \]
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Rubi [A] time = 0.035635, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac{b}{3 a^2 \left (a+b x^3\right )}+\frac{2 b \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b \log (x)}{a^3}-\frac{1}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{3 a^2 x^3}-\frac{b}{3 a^2 \left (a+b x^3\right )}-\frac{2 b \log (x)}{a^3}+\frac{2 b \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0405167, size = 41, normalized size = 0.79 \[ -\frac{a \left (\frac{b}{a+b x^3}+\frac{1}{x^3}\right )-2 b \log \left (a+b x^3\right )+6 b \log (x)}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 47, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}{a}^{2}}}-{\frac{b}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+{\frac{2\,b\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947806, size = 72, normalized size = 1.38 \begin{align*} -\frac{2 \, b x^{3} + a}{3 \,{\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} + \frac{2 \, b \log \left (b x^{3} + a\right )}{3 \, a^{3}} - \frac{2 \, b \log \left (x^{3}\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70176, size = 157, normalized size = 3.02 \begin{align*} -\frac{2 \, a b x^{3} + a^{2} - 2 \,{\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0554, size = 53, normalized size = 1.02 \begin{align*} - \frac{a + 2 b x^{3}}{3 a^{3} x^{3} + 3 a^{2} b x^{6}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{2 b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12927, size = 69, normalized size = 1.33 \begin{align*} \frac{2 \, b \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{2 \, b \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, b x^{3} + a}{3 \,{\left (b x^{6} + a x^{3}\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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