Optimal. Leaf size=95 \[ -\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}}+\frac {x (9 b B-5 A c)}{8 c^3 \left (b+c x^2\right )}-\frac {b x (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac {B x}{c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1584, 455, 1157, 388, 205} \[ \frac {x (9 b B-5 A c)}{8 c^3 \left (b+c x^2\right )}-\frac {b x (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}}+\frac {B x}{c^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 455
Rule 1157
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{10} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^4 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}-\frac {\int \frac {-b (b B-A c)+4 c (b B-A c) x^2-4 B c^2 x^4}{\left (b+c x^2\right )^2} \, dx}{4 c^3}\\ &=-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}+\frac {\int \frac {-b (7 b B-3 A c)+8 b B c x^2}{b+c x^2} \, dx}{8 b c^3}\\ &=\frac {B x}{c^3}-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}-\frac {(3 (5 b B-A c)) \int \frac {1}{b+c x^2} \, dx}{8 c^3}\\ &=\frac {B x}{c^3}-\frac {b (b B-A c) x}{4 c^3 \left (b+c x^2\right )^2}+\frac {(9 b B-5 A c) x}{8 c^3 \left (b+c x^2\right )}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 92, normalized size = 0.97 \[ \frac {x \left (b \left (25 B c x^2-3 A c\right )+c^2 x^2 \left (8 B x^2-5 A\right )+15 b^2 B\right )}{8 c^3 \left (b+c x^2\right )^2}-\frac {3 (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 \sqrt {b} c^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 328, normalized size = 3.45 \[ \left [\frac {16 \, B b c^{3} x^{5} + 10 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x^{3} + 3 \, {\left ({\left (5 \, B b c^{2} - A c^{3}\right )} x^{4} + 5 \, B b^{3} - A b^{2} c + 2 \, {\left (5 \, B b^{2} c - A b c^{2}\right )} x^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right ) + 6 \, {\left (5 \, B b^{3} c - A b^{2} c^{2}\right )} x}{16 \, {\left (b c^{6} x^{4} + 2 \, b^{2} c^{5} x^{2} + b^{3} c^{4}\right )}}, \frac {8 \, B b c^{3} x^{5} + 5 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 3 \, {\left ({\left (5 \, B b c^{2} - A c^{3}\right )} x^{4} + 5 \, B b^{3} - A b^{2} c + 2 \, {\left (5 \, B b^{2} c - A b c^{2}\right )} x^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right ) + 3 \, {\left (5 \, B b^{3} c - A b^{2} c^{2}\right )} x}{8 \, {\left (b c^{6} x^{4} + 2 \, b^{2} c^{5} x^{2} + b^{3} c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 80, normalized size = 0.84 \[ \frac {B x}{c^{3}} - \frac {3 \, {\left (5 \, B b - A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{3}} + \frac {9 \, B b c x^{3} - 5 \, A c^{2} x^{3} + 7 \, B b^{2} x - 3 \, A b c x}{8 \, {\left (c x^{2} + b\right )}^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 122, normalized size = 1.28 \[ -\frac {5 A \,x^{3}}{8 \left (c \,x^{2}+b \right )^{2} c}+\frac {9 B b \,x^{3}}{8 \left (c \,x^{2}+b \right )^{2} c^{2}}-\frac {3 A b x}{8 \left (c \,x^{2}+b \right )^{2} c^{2}}+\frac {7 B \,b^{2} x}{8 \left (c \,x^{2}+b \right )^{2} c^{3}}+\frac {3 A \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, c^{2}}-\frac {15 B b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}\, c^{3}}+\frac {B x}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 94, normalized size = 0.99 \[ \frac {{\left (9 \, B b c - 5 \, A c^{2}\right )} x^{3} + {\left (7 \, B b^{2} - 3 \, A b c\right )} x}{8 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac {B x}{c^{3}} - \frac {3 \, {\left (5 \, B b - A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 92, normalized size = 0.97 \[ \frac {B\,x}{c^3}-\frac {x^3\,\left (\frac {5\,A\,c^2}{8}-\frac {9\,B\,b\,c}{8}\right )-x\,\left (\frac {7\,B\,b^2}{8}-\frac {3\,A\,b\,c}{8}\right )}{b^2\,c^3+2\,b\,c^4\,x^2+c^5\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-5\,B\,b\right )}{8\,\sqrt {b}\,c^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.11, size = 194, normalized size = 2.04 \[ \frac {B x}{c^{3}} + \frac {3 \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right ) \log {\left (- \frac {3 b c^{3} \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right )}{- 3 A c + 15 B b} + x \right )}}{16} - \frac {3 \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right ) \log {\left (\frac {3 b c^{3} \sqrt {- \frac {1}{b c^{7}}} \left (- A c + 5 B b\right )}{- 3 A c + 15 B b} + x \right )}}{16} + \frac {x^{3} \left (- 5 A c^{2} + 9 B b c\right ) + x \left (- 3 A b c + 7 B b^{2}\right )}{8 b^{2} c^{3} + 16 b c^{4} x^{2} + 8 c^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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