Optimal. Leaf size=31 \[ \frac {x}{b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {321, 205} \[ \frac {x}{b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rubi steps
\begin {align*} \int \frac {x^2}{a+b x^2} \, dx &=\frac {x}{b}-\frac {a \int \frac {1}{a+b x^2} \, dx}{b}\\ &=\frac {x}{b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {x}{b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 82, normalized size = 2.65 \[ \left [\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, x}{2 \, b}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - x}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 26, normalized size = 0.84 \[ -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.87 \[ -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 26, normalized size = 0.84 \[ -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 0.74 \[ \frac {x}{b}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 56, normalized size = 1.81 \[ \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (- b \sqrt {- \frac {a}{b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (b \sqrt {- \frac {a}{b^{3}}} + x \right )}}{2} + \frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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