Optimal. Leaf size=44 \[ -\frac{a^2}{2 b^3 \left (a+b x^2\right )}-\frac{a \log \left (a+b x^2\right )}{b^3}+\frac{x^2}{2 b^2} \]
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Rubi [A] time = 0.031901, antiderivative size = 44, 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} \[ -\frac{a^2}{2 b^3 \left (a+b x^2\right )}-\frac{a \log \left (a+b x^2\right )}{b^3}+\frac{x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 b^2}-\frac{a^2}{2 b^3 \left (a+b x^2\right )}-\frac{a \log \left (a+b x^2\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0224456, size = 38, normalized size = 0.86 \[ \frac{-\frac{a^2}{a+b x^2}-2 a \log \left (a+b x^2\right )+b x^2}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 41, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{2}}}-{\frac{{a}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{a\ln \left ( b{x}^{2}+a \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41121, size = 58, normalized size = 1.32 \begin{align*} -\frac{a^{2}}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} + \frac{x^{2}}{2 \, b^{2}} - \frac{a \log \left (b x^{2} + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22029, size = 113, normalized size = 2.57 \begin{align*} \frac{b^{2} x^{4} + a b x^{2} - a^{2} - 2 \,{\left (a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.368169, size = 39, normalized size = 0.89 \begin{align*} - \frac{a^{2}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{a \log{\left (a + b x^{2} \right )}}{b^{3}} + \frac{x^{2}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.29296, size = 66, normalized size = 1.5 \begin{align*} \frac{x^{2}}{2 \, b^{2}} - \frac{a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{3}} + \frac{2 \, a b x^{2} + a^{2}}{2 \,{\left (b x^{2} + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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