Optimal. Leaf size=27 \[ \frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.0176019, antiderivative size = 27, 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} \[ \frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0042308, size = 27, normalized size = 1. \[ \frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 24, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,b}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.65654, size = 31, normalized size = 1.15 \begin{align*} \frac{x^{2}}{2 \, b} - \frac{a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29075, size = 49, normalized size = 1.81 \begin{align*} \frac{b x^{2} - a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.280649, size = 20, normalized size = 0.74 \begin{align*} - \frac{a \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.65943, size = 32, normalized size = 1.19 \begin{align*} \frac{x^{2}}{2 \, b} - \frac{a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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