Optimal. Leaf size=40 \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac{a x^2}{2 b^2}+\frac{x^4}{4 b} \]
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Rubi [A] time = 0.0267045, antiderivative size = 40, 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, 43} \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac{a x^2}{2 b^2}+\frac{x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a x^2}{2 b^2}+\frac{x^4}{4 b}+\frac{a^2 \log \left (a+b x^2\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.0052148, size = 40, normalized size = 1. \[ \frac{a^2 \log \left (a+b x^2\right )}{2 b^3}-\frac{a x^2}{2 b^2}+\frac{x^4}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 35, normalized size = 0.9 \begin{align*} -{\frac{a{x}^{2}}{2\,{b}^{2}}}+{\frac{{x}^{4}}{4\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.88058, size = 46, normalized size = 1.15 \begin{align*} \frac{a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23125, size = 73, normalized size = 1.82 \begin{align*} \frac{b^{2} x^{4} - 2 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.300961, size = 32, normalized size = 0.8 \begin{align*} \frac{a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} - \frac{a x^{2}}{2 b^{2}} + \frac{x^{4}}{4 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.52663, size = 47, normalized size = 1.18 \begin{align*} \frac{a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} + \frac{b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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