Optimal. Leaf size=60 \[ \frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2}{2 b^2} \]
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Rubi [A] time = 0.0604614, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ \frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (c+d x)}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d}{b^2}+\frac{a (-b c+a d)}{b^2 (a+b x)^2}+\frac{b c-2 a d}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{d x^2}{2 b^2}+\frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.0352775, size = 50, normalized size = 0.83 \[ \frac{\frac{a (b c-a d)}{a+b x^2}+(b c-2 a d) \log \left (a+b x^2\right )+b d x^2}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 74, normalized size = 1.2 \begin{align*}{\frac{d{x}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) ad}{{b}^{3}}}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}-{\frac{{a}^{2}d}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{ac}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46424, size = 80, normalized size = 1.33 \begin{align*} \frac{d x^{2}}{2 \, b^{2}} + \frac{a b c - a^{2} d}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} + \frac{{\left (b c - 2 \, a d\right )} \log \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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Fricas [A] time = 1.53953, size = 165, normalized size = 2.75 \begin{align*} \frac{b^{2} d x^{4} + a b d x^{2} + a b c - a^{2} d +{\left (a b c - 2 \, a^{2} d +{\left (b^{2} c - 2 \, a b d\right )} x^{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 = 0.690072, size = 56, normalized size = 0.93 \begin{align*} - \frac{a^{2} d - a b c}{2 a b^{3} + 2 b^{4} x^{2}} + \frac{d x^{2}}{2 b^{2}} - \frac{\left (2 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13277, size = 122, normalized size = 2.03 \begin{align*} \frac{\frac{{\left (b x^{2} + a\right )} d}{b^{2}} - \frac{{\left (b c - 2 \, a d\right )} \log \left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} + \frac{\frac{a b^{2} c}{b x^{2} + a} - \frac{a^{2} b d}{b x^{2} + a}}{b^{3}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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