Optimal. Leaf size=96 \[ \frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2}+\frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}-\frac{A}{b^3 x} \]
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Rubi [A] time = 0.118209, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1584, 456, 453, 205} \[ \frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2}+\frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}-\frac{A}{b^3 x} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 456
Rule 453
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{A+B x^2}{x^2 \left (b+c x^2\right )^3} \, dx\\ &=\frac{(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}-\frac{1}{4} \int \frac{-\frac{4 A}{b}-\frac{3 (b B-A c) x^2}{b^2}}{x^2 \left (b+c x^2\right )^2} \, dx\\ &=\frac{(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}+\frac{(3 b B-7 A c) x}{8 b^3 \left (b+c x^2\right )}+\frac{1}{8} \int \frac{\frac{8 A}{b^2}+\frac{(3 b B-7 A c) x^2}{b^3}}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac{A}{b^3 x}+\frac{(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}+\frac{(3 b B-7 A c) x}{8 b^3 \left (b+c x^2\right )}+\frac{(3 (b B-5 A c)) \int \frac{1}{b+c x^2} \, dx}{8 b^3}\\ &=-\frac{A}{b^3 x}+\frac{(b B-A c) x}{4 b^2 \left (b+c x^2\right )^2}+\frac{(3 b B-7 A c) x}{8 b^3 \left (b+c x^2\right )}+\frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0621907, size = 96, normalized size = 1. \[ \frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2}+\frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}-\frac{A}{b^3 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 125, normalized size = 1.3 \begin{align*} -{\frac{A}{{b}^{3}x}}-{\frac{7\,A{x}^{3}{c}^{2}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{3\,Bc{x}^{3}}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,Acx}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{5\,Bx}{8\,b \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,Ac}{8\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{8\,{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02086, size = 686, normalized size = 7.15 \begin{align*} \left [-\frac{16 \, A b^{3} c - 6 \,{\left (B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{4} - 10 \,{\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x^{2} - 3 \,{\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt{-b c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right )}{16 \,{\left (b^{4} c^{3} x^{5} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x\right )}}, -\frac{8 \, A b^{3} c - 3 \,{\left (B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{4} - 5 \,{\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} x^{2} - 3 \,{\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right )}{8 \,{\left (b^{4} c^{3} x^{5} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.900642, size = 194, normalized size = 2.02 \begin{align*} - \frac{3 \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right ) \log{\left (- \frac{3 b^{4} \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right ) \log{\left (\frac{3 b^{4} \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac{- 8 A b^{2} + x^{4} \left (- 15 A c^{2} + 3 B b c\right ) + x^{2} \left (- 25 A b c + 5 B b^{2}\right )}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15514, size = 111, normalized size = 1.16 \begin{align*} \frac{3 \,{\left (B b - 5 \, A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{3}} - \frac{A}{b^{3} x} + \frac{3 \, B b c x^{3} - 7 \, A c^{2} x^{3} + 5 \, B b^{2} x - 9 \, A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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