Optimal. Leaf size=110 \[ -\frac {b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 c^{9/2}}+\frac {b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac {b x (3 b B-2 A c)}{c^4}-\frac {x^3 (2 b B-A c)}{3 c^3}+\frac {B x^5}{5 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 110, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 455
Rule 1584
Rule 1810
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*}
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Mathematica [A] time = 0.09, size = 111, normalized size = 1.01 \begin {gather*} -\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 \left (A b^2 c-b^3 B\right )}{2 c^4 \left (b+c x^2\right )}+\frac {b x (3 b B-2 A c)}{c^4}+\frac {x^3 (A c-2 b B)}{3 c^3}+\frac {B x^5}{5 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{10} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 298, normalized size = 2.71 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 115, normalized size = 1.05 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 132, normalized size = 1.20 \begin {gather*} \frac {B \,x^{5}}{5 c^{2}}+\frac {A \,x^{3}}{3 c^{2}}-\frac {2 B b \,x^{3}}{3 c^{3}}-\frac {A \,b^{2} x}{2 \left (c \,x^{2}+b \right ) c^{3}}+\frac {5 A \,b^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c^{3}}+\frac {B \,b^{3} x}{2 \left (c \,x^{2}+b \right ) c^{4}}-\frac {7 B \,b^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, c^{4}}-\frac {2 A b x}{c^{3}}+\frac {3 B \,b^{2} x}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 112, normalized size = 1.02 \begin {gather*} \frac {{\left (B b^{3} - A b^{2} c\right )} x}{2 \, {\left (c^{5} x^{2} + b c^{4}\right )}} - \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 {3 \, B c^{2} x^{5} - 5 \, {\left (2 \, B b c - A c^{2}\right )} x^{3} + 15 \, {\left (3 \, B b^{2} - 2 \, A b c\right )} x}{15 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 141, normalized size = 1.28 \begin {gather*} x^3\,\left (\frac {A}{3\,c^2}-\frac {2\,B\,b}{3\,c^3}\right )-x\,\left (\frac {2\,b\,\left (\frac {A}{c^2}-\frac {2\,B\,b}{c^3}\right )}{c}+\frac {B\,b^2}{c^4}\right )+\frac {B\,x^5}{5\,c^2}+\frac {x\,\left (\frac {B\,b^3}{2}-\frac {A\,b^2\,c}{2}\right )}{c^5\,x^2+b\,c^4}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {b^{3/2}\,\sqrt {c}\,x\,\left (5\,A\,c-7\,B\,b\right )}{7\,B\,b^3-5\,A\,b^2\,c}\right )\,\left (5\,A\,c-7\,B\,b\right )}{2\,c^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.76, size = 211, normalized size = 1.92 \begin {gather*} \frac {B x^{5}}{5 c^{2}} + x^{3} \left (\frac {A}{3 c^{2}} - \frac {2 B b}{3 c^{3}}\right ) + x \left (- \frac {2 A b}{c^{3}} + \frac {3 B b^{2}}{c^{4}}\right ) + \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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