Optimal. Leaf size=42 \[ \frac {(b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {A}{b x} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1593, 453, 205} \begin {gather*} \frac {(b B-A c) \tan ^{-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 205
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) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 1.00 \begin {gather*} \frac {(b B-A c) \tan ^{-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 = 0.40, size = 105, normalized size = 2.50 \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 = 0.16, size = 36, normalized size = 0.86 \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 = 48, normalized size = 1.14 \begin {gather*} -\frac {A c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}+\frac {B \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}}-\frac {A}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 36, normalized size = 0.86 \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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mupad [B] time = 0.09, size = 35, normalized size = 0.83 \begin {gather*} -\frac {A}{b\,x}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{3/2}\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 82, normalized size = 1.95 \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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