Optimal. Leaf size=74 \[ -\frac{b^4}{4 a^5 \left (a x^2+b\right )^2}+\frac{2 b^3}{a^5 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3} \]
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Rubi [A] time = 0.0558214, antiderivative size = 74, 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, 43} \[ -\frac{b^4}{4 a^5 \left (a x^2+b\right )^2}+\frac{2 b^3}{a^5 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^3} \, dx &=\int \frac{x^9}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{3 b}{a^4}+\frac{x}{a^3}+\frac{b^4}{a^4 (b+a x)^3}-\frac{4 b^3}{a^4 (b+a x)^2}+\frac{6 b^2}{a^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 b x^2}{2 a^4}+\frac{x^4}{4 a^3}-\frac{b^4}{4 a^5 \left (b+a x^2\right )^2}+\frac{2 b^3}{a^5 \left (b+a x^2\right )}+\frac{3 b^2 \log \left (b+a x^2\right )}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0420365, size = 58, normalized size = 0.78 \[ \frac{a^2 x^4+\frac{b^3 \left (8 a x^2+7 b\right )}{\left (a x^2+b\right )^2}+12 b^2 \log \left (a x^2+b\right )-6 a b x^2}{4 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 69, normalized size = 0.9 \begin{align*} -{\frac{3\,b{x}^{2}}{2\,{a}^{4}}}+{\frac{{x}^{4}}{4\,{a}^{3}}}-{\frac{{b}^{4}}{4\,{a}^{5} \left ( a{x}^{2}+b \right ) ^{2}}}+2\,{\frac{{b}^{3}}{{a}^{5} \left ( a{x}^{2}+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31239, size = 104, normalized size = 1.41 \begin{align*} \frac{8 \, a b^{3} x^{2} + 7 \, b^{4}}{4 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}} + \frac{3 \, b^{2} \log \left (a x^{2} + b\right )}{a^{5}} + \frac{a x^{4} - 6 \, b x^{2}}{4 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38963, size = 211, normalized size = 2.85 \begin{align*} \frac{a^{4} x^{8} - 4 \, a^{3} b x^{6} - 11 \, a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + 7 \, b^{4} + 12 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.728944, size = 78, normalized size = 1.05 \begin{align*} \frac{8 a b^{3} x^{2} + 7 b^{4}}{4 a^{7} x^{4} + 8 a^{6} b x^{2} + 4 a^{5} b^{2}} + \frac{x^{4}}{4 a^{3}} - \frac{3 b x^{2}}{2 a^{4}} + \frac{3 b^{2} \log{\left (a x^{2} + b \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15127, size = 93, normalized size = 1.26 \begin{align*} \frac{3 \, b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac{a^{3} x^{4} - 6 \, a^{2} b x^{2}}{4 \, a^{6}} + \frac{8 \, a b^{3} x^{2} + 7 \, b^{4}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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