Optimal. Leaf size=41 \[ \frac {(A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {A}{b x} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1593, 453, 208} \begin {gather*} \frac {(A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {A}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 453
Rule 1593
Rubi steps
\begin {align*} \int \frac {A+B x^2}{b x^2-c x^4} \, dx &=\int \frac {A+B x^2}{x^2 \left (b-c x^2\right )} \, dx\\ &=-\frac {A}{b x}+\frac {(b B+A c) \int \frac {1}{b-c x^2} \, dx}{b}\\ &=-\frac {A}{b x}+\frac {(b B+A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.00 \begin {gather*} \frac {(A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {A}{b x} \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}{b x^2-c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 103, normalized size = 2.51 \begin {gather*} \left [\frac {{\left (B b + A c\right )} \sqrt {b c} x \log \left (\frac {c x^{2} + 2 \, \sqrt {b c} x + b}{c x^{2} - b}\right ) - 2 \, A b c}{2 \, b^{2} c x}, -\frac {{\left (B b + A c\right )} \sqrt {-b c} x \arctan \left (\frac {\sqrt {-b c} x}{b}\right ) + A b c}{b^{2} c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 38, normalized size = 0.93 \begin {gather*} -\frac {{\left (B b + A c\right )} \arctan \left (\frac {c x}{\sqrt {-b c}}\right )}{\sqrt {-b c} b} - \frac {A}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.95 \begin {gather*} -\frac {\left (-A c -b B \right ) \arctanh \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}-\frac {A}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 51, normalized size = 1.24 \begin {gather*} -\frac {{\left (B b + A c\right )} \log \left (\frac {c x - \sqrt {b c}}{c x + \sqrt {b c}}\right )}{2 \, \sqrt {b c} b} - \frac {A}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 33, normalized size = 0.80 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c+B\,b\right )}{b^{3/2}\,\sqrt {c}}-\frac {A}{b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 75, normalized size = 1.83 \begin {gather*} - \frac {A}{b x} - \frac {\sqrt {\frac {1}{b^{3} c}} \left (A c + B b\right ) \log {\left (- b^{2} \sqrt {\frac {1}{b^{3} c}} + x \right )}}{2} + \frac {\sqrt {\frac {1}{b^{3} c}} \left (A c + B b\right ) \log {\left (b^{2} \sqrt {\frac {1}{b^{3} c}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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