Optimal. Leaf size=34 \[ -\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 325, 205} \begin {gather*} -\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{b x^2+c x^4} \, dx &=\int \frac {1}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac {1}{b x}-\frac {c \int \frac {1}{b+c x^2} \, dx}{b}\\ &=-\frac {1}{b x}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}-\frac {1}{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 {1}{b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.60, size = 82, normalized size = 2.41 \begin {gather*} \left [\frac {x \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} - 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) - 2}{2 \, b x}, -\frac {x \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) + 1}{b x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 29, normalized size = 0.85 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b} - \frac {1}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.88 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}-\frac {1}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 29, normalized size = 0.85 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b} - \frac {1}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.27, size = 26, normalized size = 0.76 \begin {gather*} -\frac {1}{b\,x}-\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 65, normalized size = 1.91 \begin {gather*} \frac {\sqrt {- \frac {c}{b^{3}}} \log {\left (- \frac {b^{2} \sqrt {- \frac {c}{b^{3}}}}{c} + x \right )}}{2} - \frac {\sqrt {- \frac {c}{b^{3}}} \log {\left (\frac {b^{2} \sqrt {- \frac {c}{b^{3}}}}{c} + x \right )}}{2} - \frac {1}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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