Optimal. Leaf size=61 \[ -\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}}-\frac {b B-A c}{b^2 x}-\frac {A}{3 b x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 453, 325, 205} \[ -\frac {b B-A c}{b^2 x}-\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}}-\frac {A}{3 b x^3} \]
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^2 \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^4 \left (b+c x^2\right )} \, dx\\ &=-\frac {A}{3 b x^3}-\frac {(-3 b B+3 A c) \int \frac {1}{x^2 \left (b+c x^2\right )} \, dx}{3 b}\\ &=-\frac {A}{3 b x^3}-\frac {b B-A c}{b^2 x}-\frac {(c (b B-A c)) \int \frac {1}{b+c x^2} \, dx}{b^2}\\ &=-\frac {A}{3 b x^3}-\frac {b B-A c}{b^2 x}-\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 60, normalized size = 0.98 \[ -\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}}+\frac {A c-b B}{b^2 x}-\frac {A}{3 b x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 135, normalized size = 2.21 \[ \left [-\frac {3 \, {\left (B b - A c\right )} x^{3} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) + 6 \, {\left (B b - A c\right )} x^{2} + 2 \, A b}{6 \, b^{2} x^{3}}, -\frac {3 \, {\left (B b - A c\right )} x^{3} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) + 3 \, {\left (B b - A c\right )} x^{2} + A b}{3 \, b^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 57, normalized size = 0.93 \[ -\frac {{\left (B b c - A c^{2}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} - \frac {3 \, B b x^{2} - 3 \, A c x^{2} + A b}{3 \, b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 72, normalized size = 1.18 \[ \frac {A \,c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{2}}-\frac {B c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}+\frac {A c}{b^{2} x}-\frac {B}{b x}-\frac {A}{3 b \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.82, size = 56, normalized size = 0.92 \[ -\frac {{\left (B b c - A c^{2}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} - \frac {3 \, {\left (B b - A c\right )} x^{2} + A b}{3 \, b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 53, normalized size = 0.87 \[ \frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{5/2}}-\frac {\frac {A}{3\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{b^2}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 129, normalized size = 2.11 \[ \frac {\sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right ) \log {\left (- \frac {b^{3} \sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right )}{- A c^{2} + B b c} + x \right )}}{2} - \frac {\sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right ) \log {\left (\frac {b^{3} \sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right )}{- A c^{2} + B b c} + x \right )}}{2} + \frac {- A b + x^{2} \left (3 A c - 3 B b\right )}{3 b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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