Optimal. Leaf size=42 \[ \frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a x}{b^2}+\frac {x^3}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {302, 205} \[ \frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a x}{b^2}+\frac {x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rubi steps
\begin {align*} \int \frac {x^4}{a+b x^2} \, dx &=\int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {a x}{b^2}+\frac {x^3}{3 b}+\frac {a^2 \int \frac {1}{a+b x^2} \, dx}{b^2}\\ &=-\frac {a x}{b^2}+\frac {x^3}{3 b}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ \frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a x}{b^2}+\frac {x^3}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 99, normalized size = 2.36 \[ \left [\frac {2 \, b x^{3} + 3 \, a \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 6 \, a x}{6 \, b^{2}}, \frac {b x^{3} + 3 \, a \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 3 \, a x}{3 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 40, normalized size = 0.95 \[ \frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} x^{3} - 3 \, a b x}{3 \, b^{3}} \]
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 \[ \frac {x^{3}}{3 b}+\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {a x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.80, size = 37, normalized size = 0.88 \[ \frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b x^{3} - 3 \, a x}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 32, normalized size = 0.76 \[ \frac {x^3}{3\,b}+\frac {a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a\,x}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 80, normalized size = 1.90 \[ - \frac {a x}{b^{2}} - \frac {\sqrt {- \frac {a^{3}}{b^{5}}} \log {\left (x - \frac {b^{2} \sqrt {- \frac {a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac {\sqrt {- \frac {a^{3}}{b^{5}}} \log {\left (x + \frac {b^{2} \sqrt {- \frac {a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac {x^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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