Optimal. Leaf size=110 \[ \frac{b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{b x (3 b B-2 A c)}{c^4}+\frac{B x^5}{5 c^2} \]
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Rubi [A] time = 0.118471, antiderivative size = 110, 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, 455, 1810, 205} \[ \frac{b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{b x (3 b B-2 A c)}{c^4}+\frac{B x^5}{5 c^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 455
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{10} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^6 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac{b^2 (b B-A c) x}{2 c^4 \left (b+c x^2\right )}-\frac{\int \frac{b^2 (b B-A c)-2 b c (b B-A c) x^2+2 c^2 (b B-A c) x^4-2 B c^3 x^6}{b+c x^2} \, dx}{2 c^4}\\ &=\frac{b^2 (b B-A c) x}{2 c^4 \left (b+c x^2\right )}-\frac{\int \left (-2 b (3 b B-2 A c)+2 c (2 b B-A c) x^2-2 B c^2 x^4+\frac{7 b^3 B-5 A b^2 c}{b+c x^2}\right ) \, dx}{2 c^4}\\ &=\frac{b (3 b B-2 A c) x}{c^4}-\frac{(2 b B-A c) x^3}{3 c^3}+\frac{B x^5}{5 c^2}+\frac{b^2 (b B-A c) x}{2 c^4 \left (b+c x^2\right )}-\frac{\left (b^2 (7 b B-5 A c)\right ) \int \frac{1}{b+c x^2} \, dx}{2 c^4}\\ &=\frac{b (3 b B-2 A c) x}{c^4}-\frac{(2 b B-A c) x^3}{3 c^3}+\frac{B x^5}{5 c^2}+\frac{b^2 (b B-A c) x}{2 c^4 \left (b+c x^2\right )}-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.08294, size = 111, normalized size = 1.01 \[ -\frac{x \left (A b^2 c-b^3 B\right )}{2 c^4 \left (b+c x^2\right )}-\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}+\frac{x^3 (A c-2 b B)}{3 c^3}+\frac{b x (3 b B-2 A c)}{c^4}+\frac{B x^5}{5 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 132, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,{c}^{2}}}+{\frac{A{x}^{3}}{3\,{c}^{2}}}-{\frac{2\,B{x}^{3}b}{3\,{c}^{3}}}-2\,{\frac{Abx}{{c}^{3}}}+3\,{\frac{B{b}^{2}x}{{c}^{4}}}-{\frac{A{b}^{2}x}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{B{b}^{3}x}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}+{\frac{5\,A{b}^{2}}{2\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{7\,B{b}^{3}}{2\,{c}^{4}}\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 = 0.832423, size = 637, normalized size = 5.79 \begin{align*} \left [\frac{12 \, B c^{3} x^{7} - 4 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 20 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} - 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 30 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} x}{60 \,{\left (c^{5} x^{2} + b c^{4}\right )}}, \frac{6 \, B c^{3} x^{7} - 2 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 10 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} - 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) + 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} x}{30 \,{\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.862162, size = 206, normalized size = 1.87 \begin{align*} \frac{B x^{5}}{5 c^{2}} + \frac{x \left (- A b^{2} c + B b^{3}\right )}{2 b c^{4} + 2 c^{5} x^{2}} + \frac{\sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right )}{- 5 A b c + 7 B b^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right )}{- 5 A b c + 7 B b^{2}} + x \right )}}{4} - \frac{x^{3} \left (- A c + 2 B b\right )}{3 c^{3}} + \frac{x \left (- 2 A b c + 3 B b^{2}\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20538, size = 155, normalized size = 1.41 \begin{align*} -\frac{{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{4}} + \frac{B b^{3} x - A b^{2} c x}{2 \,{\left (c x^{2} + b\right )} c^{4}} + \frac{3 \, B c^{8} x^{5} - 10 \, B b c^{7} x^{3} + 5 \, A c^{8} x^{3} + 45 \, B b^{2} c^{6} x - 30 \, A b c^{7} x}{15 \, c^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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