Optimal. Leaf size=75 \[ -\frac{b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}-\frac{x^4 (b B-A c)}{4 c^2}+\frac{b x^2 (b B-A c)}{2 c^3}+\frac{B x^6}{6 c} \]
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Rubi [A] time = 0.0938306, antiderivative size = 75, 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} \[ -\frac{b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}-\frac{x^4 (b B-A c)}{4 c^2}+\frac{b x^2 (b B-A c)}{2 c^3}+\frac{B x^6}{6 c} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{x^5 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (A+B x)}{b+c x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b (b B-A c)}{c^3}+\frac{(-b B+A c) x}{c^2}+\frac{B x^2}{c}-\frac{b^2 (b B-A c)}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b (b B-A c) x^2}{2 c^3}-\frac{(b B-A c) x^4}{4 c^2}+\frac{B x^6}{6 c}-\frac{b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.0289057, size = 71, normalized size = 0.95 \[ \frac{c x^2 \left (-3 b c \left (2 A+B x^2\right )+c^2 x^2 \left (3 A+2 B x^2\right )+6 b^2 B\right )+6 b^2 (A c-b B) \log \left (b+c x^2\right )}{12 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 86, normalized size = 1.2 \begin{align*}{\frac{B{x}^{6}}{6\,c}}+{\frac{A{x}^{4}}{4\,c}}-{\frac{B{x}^{4}b}{4\,{c}^{2}}}-{\frac{Ab{x}^{2}}{2\,{c}^{2}}}+{\frac{B{x}^{2}{b}^{2}}{2\,{c}^{3}}}+{\frac{{b}^{2}\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{3}}}-{\frac{{b}^{3}\ln \left ( c{x}^{2}+b \right ) B}{2\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50688, size = 100, normalized size = 1.33 \begin{align*} \frac{2 \, B c^{2} x^{6} - 3 \,{\left (B b c - A c^{2}\right )} x^{4} + 6 \,{\left (B b^{2} - A b c\right )} x^{2}}{12 \, c^{3}} - \frac{{\left (B b^{3} - A b^{2} c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489839, size = 155, normalized size = 2.07 \begin{align*} \frac{2 \, B c^{3} x^{6} - 3 \,{\left (B b c^{2} - A c^{3}\right )} x^{4} + 6 \,{\left (B b^{2} c - A b c^{2}\right )} x^{2} - 6 \,{\left (B b^{3} - A b^{2} c\right )} \log \left (c x^{2} + b\right )}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.445902, size = 65, normalized size = 0.87 \begin{align*} \frac{B x^{6}}{6 c} - \frac{b^{2} \left (- A c + B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{4}} - \frac{x^{4} \left (- A c + B b\right )}{4 c^{2}} + \frac{x^{2} \left (- A b c + B b^{2}\right )}{2 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24536, size = 104, normalized size = 1.39 \begin{align*} \frac{2 \, B c^{2} x^{6} - 3 \, B b c x^{4} + 3 \, A c^{2} x^{4} + 6 \, B b^{2} x^{2} - 6 \, A b c x^{2}}{12 \, c^{3}} - \frac{{\left (B b^{3} - A b^{2} c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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