Optimal. Leaf size=54 \[ -\frac{x^2 (b B-A c)}{2 c^2}+\frac{b (b B-A c) \log \left (b+c x^2\right )}{2 c^3}+\frac{B x^4}{4 c} \]
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Rubi [A] time = 0.0687733, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 446, 77} \[ -\frac{x^2 (b B-A c)}{2 c^2}+\frac{b (b B-A c) \log \left (b+c x^2\right )}{2 c^3}+\frac{B x^4}{4 c} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^5 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{x^3 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (A+B x)}{b+c x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{-b B+A c}{c^2}+\frac{B x}{c}+\frac{b (b B-A c)}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{(b B-A c) x^2}{2 c^2}+\frac{B x^4}{4 c}+\frac{b (b B-A c) \log \left (b+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0187509, size = 47, normalized size = 0.87 \[ \frac{c x^2 \left (2 A c-2 b B+B c x^2\right )+2 b (b B-A c) \log \left (b+c x^2\right )}{4 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 62, normalized size = 1.2 \begin{align*}{\frac{B{x}^{4}}{4\,c}}+{\frac{A{x}^{2}}{2\,c}}-{\frac{B{x}^{2}b}{2\,{c}^{2}}}-{\frac{b\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{2}}}+{\frac{{b}^{2}\ln \left ( c{x}^{2}+b \right ) B}{2\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.02292, size = 68, normalized size = 1.26 \begin{align*} \frac{B c x^{4} - 2 \,{\left (B b - A c\right )} x^{2}}{4 \, c^{2}} + \frac{{\left (B b^{2} - A b c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.496878, size = 108, normalized size = 2. \begin{align*} \frac{B c^{2} x^{4} - 2 \,{\left (B b c - A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} - A b c\right )} \log \left (c x^{2} + b\right )}{4 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.423355, size = 44, normalized size = 0.81 \begin{align*} \frac{B x^{4}}{4 c} + \frac{b \left (- A c + B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{3}} - \frac{x^{2} \left (- A c + B b\right )}{2 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25056, size = 70, normalized size = 1.3 \begin{align*} \frac{B c x^{4} - 2 \, B b x^{2} + 2 \, A c x^{2}}{4 \, c^{2}} + \frac{{\left (B b^{2} - A b c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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