Optimal. Leaf size=42 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b x}{a^2}+\frac {x^3}{3 a} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 302, 205} \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b x}{a^2}+\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 302
Rubi steps
\begin {align*} \int \frac {x^2}{a+\frac {b}{x^2}} \, dx &=\int \frac {x^4}{b+a x^2} \, dx\\ &=\int \left (-\frac {b}{a^2}+\frac {x^2}{a}+\frac {b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx\\ &=-\frac {b x}{a^2}+\frac {x^3}{3 a}+\frac {b^2 \int \frac {1}{b+a x^2} \, dx}{a^2}\\ &=-\frac {b x}{a^2}+\frac {x^3}{3 a}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b x}{a^2}+\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 99, normalized size = 2.36 \[ \left [\frac {2 \, a x^{3} + 3 \, b \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) - 6 \, b x}{6 \, a^{2}}, \frac {a x^{3} + 3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) - 3 \, b x}{3 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 40, normalized size = 0.95 \[ \frac {b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a^{2} x^{3} - 3 \, a b x}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.90 \[ \frac {x^{3}}{3 a}+\frac {b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {b x}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.85, size = 37, normalized size = 0.88 \[ \frac {b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a x^{3} - 3 \, b x}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.76 \[ \frac {x^3}{3\,a}+\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b\,x}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 80, normalized size = 1.90 \[ - \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac {x^{3}}{3 a} - \frac {b x}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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