Optimal. Leaf size=133 \[ \frac {b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{11/2}}-\frac {b^3 x (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac {b^2 x (4 b B-3 A c)}{c^5}+\frac {b x^3 (3 b B-2 A c)}{3 c^4}-\frac {x^5 (2 b B-A c)}{5 c^3}+\frac {B x^7}{7 c^2} \]
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Rubi [A] time = 0.17, antiderivative size = 133, normalized size of antiderivative = 1.00, 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 205
Rule 455
Rule 1584
Rule 1810
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*}
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Mathematica [A] time = 0.11, size = 134, normalized size = 1.01 \[ \frac {b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{11/2}}-\frac {b^2 x (4 b B-3 A c)}{c^5}+\frac {x \left (A b^3 c-b^4 B\right )}{2 c^5 \left (b+c x^2\right )}+\frac {b x^3 (3 b B-2 A c)}{3 c^4}+\frac {x^5 (A c-2 b B)}{5 c^3}+\frac {B x^7}{7 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 350, normalized size = 2.63 \[ \left [\frac {60 \, B c^{4} x^{9} - 12 \, {\left (9 \, B b c^{3} - 7 \, A c^{4}\right )} x^{7} + 28 \, {\left (9 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{5} - 140 \, {\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3} - 105 \, {\left (9 \, B b^{4} - 7 \, A b^{3} c + {\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) - 210 \, {\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} x}{420 \, {\left (c^{6} x^{2} + b c^{5}\right )}}, \frac {30 \, B c^{4} x^{9} - 6 \, {\left (9 \, B b c^{3} - 7 \, A c^{4}\right )} x^{7} + 14 \, {\left (9 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{5} - 70 \, {\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3} + 105 \, {\left (9 \, B b^{4} - 7 \, A b^{3} c + {\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 105 \, {\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} x}{210 \, {\left (c^{6} x^{2} + b c^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 139, normalized size = 1.05 \[ \frac {{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{5}} - \frac {B b^{4} x - A b^{3} c x}{2 \, {\left (c x^{2} + b\right )} c^{5}} + \frac {15 \, B c^{12} x^{7} - 42 \, B b c^{11} x^{5} + 21 \, A c^{12} x^{5} + 105 \, B b^{2} c^{10} x^{3} - 70 \, A b c^{11} x^{3} - 420 \, B b^{3} c^{9} x + 315 \, A b^{2} c^{10} x}{105 \, c^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 155, normalized size = 1.17 \[ \frac {B \,x^{7}}{7 c^{2}}+\frac {A \,x^{5}}{5 c^{2}}-\frac {2 B b \,x^{5}}{5 c^{3}}-\frac {2 A b \,x^{3}}{3 c^{3}}+\frac {B \,b^{2} x^{3}}{c^{4}}+\frac {A \,b^{3} x}{2 \left (c \,x^{2}+b \right ) c^{4}}-\frac {7 A \,b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c^{4}}-\frac {B \,b^{4} x}{2 \left (c \,x^{2}+b \right ) c^{5}}+\frac {9 B \,b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c^{5}}+\frac {3 A \,b^{2} x}{c^{4}}-\frac {4 B \,b^{3} x}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 136, normalized size = 1.02 \[ -\frac {{\left (B b^{4} - A b^{3} c\right )} x}{2 \, {\left (c^{6} x^{2} + b c^{5}\right )}} + \frac {{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} c^{5}} + \frac {15 \, B c^{3} x^{7} - 21 \, {\left (2 \, B b c^{2} - A c^{3}\right )} x^{5} + 35 \, {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} x^{3} - 105 \, {\left (4 \, B b^{3} - 3 \, A b^{2} c\right )} x}{105 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 203, normalized size = 1.53 \[ x\,\left (\frac {2\,b\,\left (\frac {2\,b\,\left (\frac {A}{c^2}-\frac {2\,B\,b}{c^3}\right )}{c}+\frac {B\,b^2}{c^4}\right )}{c}-\frac {b^2\,\left (\frac {A}{c^2}-\frac {2\,B\,b}{c^3}\right )}{c^2}\right )+x^5\,\left (\frac {A}{5\,c^2}-\frac {2\,B\,b}{5\,c^3}\right )-x^3\,\left (\frac {2\,b\,\left (\frac {A}{c^2}-\frac {2\,B\,b}{c^3}\right )}{3\,c}+\frac {B\,b^2}{3\,c^4}\right )+\frac {B\,x^7}{7\,c^2}-\frac {x\,\left (\frac {B\,b^4}{2}-\frac {A\,b^3\,c}{2}\right )}{c^6\,x^2+b\,c^5}+\frac {b^{5/2}\,\mathrm {atan}\left (\frac {b^{5/2}\,\sqrt {c}\,x\,\left (7\,A\,c-9\,B\,b\right )}{9\,B\,b^4-7\,A\,b^3\,c}\right )\,\left (7\,A\,c-9\,B\,b\right )}{2\,c^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.01, size = 238, normalized size = 1.79 \[ \frac {B x^{7}}{7 c^{2}} + x^{5} \left (\frac {A}{5 c^{2}} - \frac {2 B b}{5 c^{3}}\right ) + x^{3} \left (- \frac {2 A b}{3 c^{3}} + \frac {B b^{2}}{c^{4}}\right ) + x \left (\frac {3 A b^{2}}{c^{4}} - \frac {4 B b^{3}}{c^{5}}\right ) + \frac {x \left (A b^{3} c - B b^{4}\right )}{2 b c^{5} + 2 c^{6} x^{2}} - \frac {\sqrt {- \frac {b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right ) \log {\left (- \frac {c^{5} \sqrt {- \frac {b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right )}{- 7 A b^{2} c + 9 B b^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right ) \log {\left (\frac {c^{5} \sqrt {- \frac {b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right )}{- 7 A b^{2} c + 9 B b^{3}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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