Optimal. Leaf size=29 \[ \frac{c x^2}{2 b}-\frac{a c \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.0213888, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {21, 266, 43} \[ \frac{c x^2}{2 b}-\frac{a c \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx &=c \int \frac{x^3}{a+b x^2} \, dx\\ &=\frac{1}{2} c \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} c \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{c x^2}{2 b}-\frac{a c \log \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0044315, size = 29, normalized size = 1. \[ c \left (\frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.9 \begin{align*}{\frac{c{x}^{2}}{2\,b}}-{\frac{ac\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 = 1.00552, size = 34, normalized size = 1.17 \begin{align*} \frac{c x^{2}}{2 \, b} - \frac{a c \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 = 1.19585, size = 54, normalized size = 1.86 \begin{align*} \frac{b c x^{2} - a c \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 = 0.290732, size = 22, normalized size = 0.76 \begin{align*} c \left (- \frac{a \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x^{2}}{2 b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.65468, size = 63, normalized size = 2.17 \begin{align*} \frac{\frac{a c \log \left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b} + \frac{{\left (b x^{2} + a\right )} c}{b}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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