Optimal. Leaf size=40 \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]
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Rubi [A] time = 0.0256779, antiderivative size = 40, 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} \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{a+\frac{b}{x^2}} \, dx &=\int \frac{x^5}{b+a x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{b+a x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{b}{a^2}+\frac{x}{a}+\frac{b^2}{a^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a}+\frac{b^2 \log \left (b+a x^2\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0050959, size = 40, normalized size = 1. \[ \frac{b^2 \log \left (a x^2+b\right )}{2 a^3}-\frac{b x^2}{2 a^2}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 35, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{2}}{2\,{a}^{2}}}+{\frac{{x}^{4}}{4\,a}}+{\frac{{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02035, size = 46, normalized size = 1.15 \begin{align*} \frac{b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{3}} + \frac{a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40931, size = 73, normalized size = 1.82 \begin{align*} \frac{a^{2} x^{4} - 2 \, a b x^{2} + 2 \, b^{2} \log \left (a x^{2} + b\right )}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.29354, size = 32, normalized size = 0.8 \begin{align*} \frac{x^{4}}{4 a} - \frac{b x^{2}}{2 a^{2}} + \frac{b^{2} \log{\left (a x^{2} + b \right )}}{2 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17355, size = 47, normalized size = 1.18 \begin{align*} \frac{b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} + \frac{a x^{4} - 2 \, b x^{2}}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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