Optimal. Leaf size=78 \[ \frac {c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{7/2}}+\frac {c (b B-A c)}{b^3 x}-\frac {b B-A c}{3 b^2 x^3}-\frac {A}{5 b x^5} \]
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Rubi [A] time = 0.07, antiderivative size = 78, 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, 453, 325, 205} \[ \frac {c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{7/2}}-\frac {b B-A c}{3 b^2 x^3}+\frac {c (b B-A c)}{b^3 x}-\frac {A}{5 b x^5} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 453
Rule 1584
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^4 \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^6 \left (b+c x^2\right )} \, dx\\ &=-\frac {A}{5 b x^5}-\frac {(-5 b B+5 A c) \int \frac {1}{x^4 \left (b+c x^2\right )} \, dx}{5 b}\\ &=-\frac {A}{5 b x^5}-\frac {b B-A c}{3 b^2 x^3}-\frac {(c (b B-A c)) \int \frac {1}{x^2 \left (b+c x^2\right )} \, dx}{b^2}\\ &=-\frac {A}{5 b x^5}-\frac {b B-A c}{3 b^2 x^3}+\frac {c (b B-A c)}{b^3 x}+\frac {\left (c^2 (b B-A c)\right ) \int \frac {1}{b+c x^2} \, dx}{b^3}\\ &=-\frac {A}{5 b x^5}-\frac {b B-A c}{3 b^2 x^3}+\frac {c (b B-A c)}{b^3 x}+\frac {c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 78, normalized size = 1.00 \[ \frac {c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{7/2}}+\frac {c (b B-A c)}{b^3 x}+\frac {A c-b B}{3 b^2 x^3}-\frac {A}{5 b x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 184, normalized size = 2.36 \[ \left [-\frac {15 \, {\left (B b c - A c^{2}\right )} x^{5} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} - 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) - 30 \, {\left (B b c - A c^{2}\right )} x^{4} + 6 \, A b^{2} + 10 \, {\left (B b^{2} - A b c\right )} x^{2}}{30 \, b^{3} x^{5}}, \frac {15 \, {\left (B b c - A c^{2}\right )} x^{5} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) + 15 \, {\left (B b c - A c^{2}\right )} x^{4} - 3 \, A b^{2} - 5 \, {\left (B b^{2} - A b c\right )} x^{2}}{15 \, b^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 81, normalized size = 1.04 \[ \frac {{\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} + \frac {15 \, B b c x^{4} - 15 \, A c^{2} x^{4} - 5 \, B b^{2} x^{2} + 5 \, A b c x^{2} - 3 \, A b^{2}}{15 \, b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 96, normalized size = 1.23 \[ -\frac {A \,c^{3} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{3}}+\frac {B \,c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{2}}-\frac {A \,c^{2}}{b^{3} x}+\frac {B c}{b^{2} x}+\frac {A c}{3 b^{2} x^{3}}-\frac {B}{3 b \,x^{3}}-\frac {A}{5 b \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 79, normalized size = 1.01 \[ \frac {{\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} + \frac {15 \, {\left (B b c - A c^{2}\right )} x^{4} - 3 \, A b^{2} - 5 \, {\left (B b^{2} - A b c\right )} x^{2}}{15 \, b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 70, normalized size = 0.90 \[ -\frac {\frac {A}{5\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{3\,b^2}+\frac {c\,x^4\,\left (A\,c-B\,b\right )}{b^3}}{x^5}-\frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.58, size = 163, normalized size = 2.09 \[ - \frac {\sqrt {- \frac {c^{3}}{b^{7}}} \left (- A c + B b\right ) \log {\left (- \frac {b^{4} \sqrt {- \frac {c^{3}}{b^{7}}} \left (- A c + B b\right )}{- A c^{3} + B b c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {c^{3}}{b^{7}}} \left (- A c + B b\right ) \log {\left (\frac {b^{4} \sqrt {- \frac {c^{3}}{b^{7}}} \left (- A c + B b\right )}{- A c^{3} + B b c^{2}} + x \right )}}{2} + \frac {- 3 A b^{2} + x^{4} \left (- 15 A c^{2} + 15 B b c\right ) + x^{2} \left (5 A b c - 5 B b^{2}\right )}{15 b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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