Optimal. Leaf size=53 \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]
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Rubi [A] time = 0.0344523, antiderivative size = 53, 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} \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{a+\frac{b}{x^2}} \, dx &=\int \frac{x^7}{b+a x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{b+a x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{a^3}-\frac{b x}{a^2}+\frac{x^2}{a}-\frac{b^3}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b^2 x^2}{2 a^3}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a}-\frac{b^3 \log \left (b+a x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0052669, size = 53, normalized size = 1. \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^4}{4 a^2}+\frac{x^6}{6 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 46, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,{a}^{3}}}-{\frac{b{x}^{4}}{4\,{a}^{2}}}+{\frac{{x}^{6}}{6\,a}}-{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992105, size = 62, normalized size = 1.17 \begin{align*} -\frac{b^{3} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac{2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44927, size = 99, normalized size = 1.87 \begin{align*} \frac{2 \, a^{3} x^{6} - 3 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 6 \, b^{3} \log \left (a x^{2} + b\right )}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.302942, size = 44, normalized size = 0.83 \begin{align*} \frac{x^{6}}{6 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{2}}{2 a^{3}} - \frac{b^{3} \log{\left (a x^{2} + b \right )}}{2 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13512, size = 63, normalized size = 1.19 \begin{align*} -\frac{b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac{2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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