Optimal. Leaf size=133 \[ -\frac{b^3 x (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}-\frac{x^5 (2 b B-A c)}{5 c^3}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}+\frac{B x^7}{7 c^2} \]
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Rubi [A] time = 0.166286, antiderivative size = 133, 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} \[ -\frac{b^3 x (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}-\frac{x^5 (2 b B-A c)}{5 c^3}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}+\frac{B x^7}{7 c^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 455
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{12} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^8 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{b^3 (b B-A c) x}{2 c^5 \left (b+c x^2\right )}-\frac{\int \frac{-b^3 (b B-A c)+2 b^2 c (b B-A c) x^2-2 b c^2 (b B-A c) x^4+2 c^3 (b B-A c) x^6-2 B c^4 x^8}{b+c x^2} \, dx}{2 c^5}\\ &=-\frac{b^3 (b B-A c) x}{2 c^5 \left (b+c x^2\right )}-\frac{\int \left (2 b^2 (4 b B-3 A c)-2 b c (3 b B-2 A c) x^2+2 c^2 (2 b B-A c) x^4-2 B c^3 x^6+\frac{-9 b^4 B+7 A b^3 c}{b+c x^2}\right ) \, dx}{2 c^5}\\ &=-\frac{b^2 (4 b B-3 A c) x}{c^5}+\frac{b (3 b B-2 A c) x^3}{3 c^4}-\frac{(2 b B-A c) x^5}{5 c^3}+\frac{B x^7}{7 c^2}-\frac{b^3 (b B-A c) x}{2 c^5 \left (b+c x^2\right )}+\frac{\left (b^3 (9 b B-7 A c)\right ) \int \frac{1}{b+c x^2} \, dx}{2 c^5}\\ &=-\frac{b^2 (4 b B-3 A c) x}{c^5}+\frac{b (3 b B-2 A c) x^3}{3 c^4}-\frac{(2 b B-A c) x^5}{5 c^3}+\frac{B x^7}{7 c^2}-\frac{b^3 (b B-A c) x}{2 c^5 \left (b+c x^2\right )}+\frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.103565, size = 134, normalized size = 1.01 \[ \frac{x \left (A b^3 c-b^4 B\right )}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}+\frac{x^5 (A c-2 b B)}{5 c^3}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}+\frac{B x^7}{7 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 155, normalized size = 1.2 \begin{align*}{\frac{B{x}^{7}}{7\,{c}^{2}}}+{\frac{A{x}^{5}}{5\,{c}^{2}}}-{\frac{2\,B{x}^{5}b}{5\,{c}^{3}}}-{\frac{2\,Ab{x}^{3}}{3\,{c}^{3}}}+{\frac{B{x}^{3}{b}^{2}}{{c}^{4}}}+3\,{\frac{A{b}^{2}x}{{c}^{4}}}-4\,{\frac{B{b}^{3}x}{{c}^{5}}}+{\frac{A{b}^{3}x}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{{b}^{4}xB}{2\,{c}^{5} \left ( c{x}^{2}+b \right ) }}-{\frac{7\,A{b}^{3}}{2\,{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{9\,B{b}^{4}}{2\,{c}^{5}}\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 = 0.888588, size = 755, normalized size = 5.68 \begin{align*} \left [\frac{60 \, B c^{4} x^{9} - 12 \,{\left (9 \, B b c^{3} - 7 \, A c^{4}\right )} x^{7} + 28 \,{\left (9 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{5} - 140 \,{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3} - 105 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c +{\left (9 \, B b^{3} c - 7 \, A b^{2} 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 ) - 210 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} x}{420 \,{\left (c^{6} x^{2} + b c^{5}\right )}}, \frac{30 \, B c^{4} x^{9} - 6 \,{\left (9 \, B b c^{3} - 7 \, A c^{4}\right )} x^{7} + 14 \,{\left (9 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{5} - 70 \,{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3} + 105 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c +{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) - 105 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} x}{210 \,{\left (c^{6} x^{2} + b c^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.926563, size = 233, normalized size = 1.75 \begin{align*} \frac{B x^{7}}{7 c^{2}} - \frac{x \left (- A b^{3} c + B b^{4}\right )}{2 b c^{5} + 2 c^{6} x^{2}} - \frac{\sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right ) \log{\left (- \frac{c^{5} \sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right )}{- 7 A b^{2} c + 9 B b^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right ) \log{\left (\frac{c^{5} \sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right )}{- 7 A b^{2} c + 9 B b^{3}} + x \right )}}{4} - \frac{x^{5} \left (- A c + 2 B b\right )}{5 c^{3}} + \frac{x^{3} \left (- 2 A b c + 3 B b^{2}\right )}{3 c^{4}} - \frac{x \left (- 3 A b^{2} c + 4 B b^{3}\right )}{c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33625, size = 188, normalized size = 1.41 \begin{align*} \frac{{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{5}} - \frac{B b^{4} x - A b^{3} c x}{2 \,{\left (c x^{2} + b\right )} c^{5}} + \frac{15 \, B c^{12} x^{7} - 42 \, B b c^{11} x^{5} + 21 \, A c^{12} x^{5} + 105 \, B b^{2} c^{10} x^{3} - 70 \, A b c^{11} x^{3} - 420 \, B b^{3} c^{9} x + 315 \, A b^{2} c^{10} x}{105 \, c^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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