Optimal. Leaf size=111 \[ \frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}}+\frac {c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac {c (2 b B-3 A c)}{b^4 x}-\frac {b B-2 A c}{3 b^3 x^3}-\frac {A}{5 b^2 x^5} \]
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Rubi [A] time = 0.20, antiderivative size = 111, 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, 456, 1802, 205} \begin {gather*} \frac {c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}}-\frac {b B-2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}-\frac {A}{5 b^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 456
Rule 1584
Rule 1802
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{x^6 \left (b+c x^2\right )^2} \, dx\\ &=\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}-\frac {1}{2} c^2 \int \frac {-\frac {2 A}{b c^2}-\frac {2 (b B-A c) x^2}{b^2 c^2}+\frac {2 (b B-A c) x^4}{b^3 c}-\frac {(b B-A c) x^6}{b^4}}{x^6 \left (b+c x^2\right )} \, dx\\ &=\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}-\frac {1}{2} c^2 \int \left (-\frac {2 A}{b^2 c^2 x^6}-\frac {2 (b B-2 A c)}{b^3 c^2 x^4}+\frac {2 (2 b B-3 A c)}{b^4 c x^2}+\frac {-5 b B+7 A c}{b^4 \left (b+c x^2\right )}\right ) \, dx\\ &=-\frac {A}{5 b^2 x^5}-\frac {b B-2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}+\frac {\left (c^2 (5 b B-7 A c)\right ) \int \frac {1}{b+c x^2} \, dx}{2 b^4}\\ &=-\frac {A}{5 b^2 x^5}-\frac {b B-2 A c}{3 b^3 x^3}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c) x}{2 b^4 \left (b+c x^2\right )}+\frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 112, normalized size = 1.01 \begin {gather*} \frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{9/2}}+\frac {c^2 x (b B-A c)}{2 b^4 \left (b+c x^2\right )}+\frac {c (2 b B-3 A c)}{b^4 x}+\frac {2 A c-b B}{3 b^3 x^3}-\frac {A}{5 b^2 x^5} \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 {A+B x^2}{x^2 \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 = 0.42, size = 308, normalized size = 2.77 \begin {gather*} \left [\frac {30 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 20 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 12 \, A b^{3} - 4 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} - 15 \, {\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} - 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right )}{60 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}, \frac {15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 6 \, A b^{3} - 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2} + 15 \, {\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{7} + {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{5}\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right )}{30 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 112, normalized size = 1.01 \begin {gather*} \frac {{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{4}} + \frac {B b c^{2} x - A c^{3} x}{2 \, {\left (c x^{2} + b\right )} b^{4}} + \frac {30 \, B b c x^{4} - 45 \, A c^{2} x^{4} - 5 \, B b^{2} x^{2} + 10 \, A b c x^{2} - 3 \, A b^{2}}{15 \, b^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 136, normalized size = 1.23 \begin {gather*} -\frac {A \,c^{3} x}{2 \left (c \,x^{2}+b \right ) b^{4}}-\frac {7 A \,c^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, b^{4}}+\frac {B \,c^{2} x}{2 \left (c \,x^{2}+b \right ) b^{3}}+\frac {5 B \,c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}\, b^{3}}-\frac {3 A \,c^{2}}{b^{4} x}+\frac {2 B c}{b^{3} x}+\frac {2 A c}{3 b^{3} x^{3}}-\frac {B}{3 b^{2} x^{3}}-\frac {A}{5 b^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 119, normalized size = 1.07 \begin {gather*} \frac {15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{6} + 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{4} - 6 \, A b^{3} - 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2}}{30 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}} + \frac {{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 104, normalized size = 0.94 \begin {gather*} -\frac {\frac {A}{5\,b}-\frac {x^2\,\left (7\,A\,c-5\,B\,b\right )}{15\,b^2}+\frac {c^2\,x^6\,\left (7\,A\,c-5\,B\,b\right )}{2\,b^4}+\frac {c\,x^4\,\left (7\,A\,c-5\,B\,b\right )}{3\,b^3}}{c\,x^7+b\,x^5}-\frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (7\,A\,c-5\,B\,b\right )}{2\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.70, size = 218, normalized size = 1.96 \begin {gather*} - \frac {\sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right ) \log {\left (- \frac {b^{5} \sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right )}{- 7 A c^{3} + 5 B b c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right ) \log {\left (\frac {b^{5} \sqrt {- \frac {c^{3}}{b^{9}}} \left (- 7 A c + 5 B b\right )}{- 7 A c^{3} + 5 B b c^{2}} + x \right )}}{4} + \frac {- 6 A b^{3} + x^{6} \left (- 105 A c^{3} + 75 B b c^{2}\right ) + x^{4} \left (- 70 A b c^{2} + 50 B b^{2} c\right ) + x^{2} \left (14 A b^{2} c - 10 B b^{3}\right )}{30 b^{5} x^{5} + 30 b^{4} c x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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