Optimal. Leaf size=24 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {c}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1584, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^2}{b x^2+c x^4} \, dx &=\int \frac {1}{b+c x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {c}} \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 {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 = 0.62, size = 67, normalized size = 2.79 \begin {gather*} \left [-\frac {\sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{2 \, b c}, \frac {\sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{b c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 15, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.67 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 15, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 16, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{\sqrt {b}\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 53, normalized size = 2.21 \begin {gather*} - \frac {\sqrt {- \frac {1}{b c}} \log {\left (- b \sqrt {- \frac {1}{b c}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{b c}} \log {\left (b \sqrt {- \frac {1}{b c}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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