Optimal. Leaf size=55 \[ -\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{7/2}}+\frac {b^2 x}{a^3}-\frac {b x^3}{3 a^2}+\frac {x^5}{5 a} \]
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Rubi [A] time = 0.02, antiderivative size = 55, 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} \[ \frac {b^2 x}{a^3}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{7/2}}-\frac {b x^3}{3 a^2}+\frac {x^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 302
Rubi steps
\begin {align*} \int \frac {x^4}{a+\frac {b}{x^2}} \, dx &=\int \frac {x^6}{b+a x^2} \, dx\\ &=\int \left (\frac {b^2}{a^3}-\frac {b x^2}{a^2}+\frac {x^4}{a}-\frac {b^3}{a^3 \left (b+a x^2\right )}\right ) \, dx\\ &=\frac {b^2 x}{a^3}-\frac {b x^3}{3 a^2}+\frac {x^5}{5 a}-\frac {b^3 \int \frac {1}{b+a x^2} \, dx}{a^3}\\ &=\frac {b^2 x}{a^3}-\frac {b x^3}{3 a^2}+\frac {x^5}{5 a}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 1.00 \[ -\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{7/2}}+\frac {b^2 x}{a^3}-\frac {b x^3}{3 a^2}+\frac {x^5}{5 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 126, normalized size = 2.29 \[ \left [\frac {6 \, a^{2} x^{5} - 10 \, a b x^{3} + 15 \, b^{2} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, a^{3}}, \frac {3 \, a^{2} x^{5} - 5 \, a b x^{3} - 15 \, b^{2} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) + 15 \, b^{2} x}{15 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 55, normalized size = 1.00 \[ -\frac {b^{3} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {3 \, a^{4} x^{5} - 5 \, a^{3} b x^{3} + 15 \, a^{2} b^{2} x}{15 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 49, normalized size = 0.89 \[ \frac {x^{5}}{5 a}-\frac {b \,x^{3}}{3 a^{2}}-\frac {b^{3} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {b^{2} x}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 50, normalized size = 0.91 \[ -\frac {b^{3} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {3 \, a^{2} x^{5} - 5 \, a b x^{3} + 15 \, b^{2} x}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 43, normalized size = 0.78 \[ \frac {x^5}{5\,a}-\frac {b\,x^3}{3\,a^2}+\frac {b^2\,x}{a^3}-\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 95, normalized size = 1.73 \[ \frac {\sqrt {- \frac {b^{5}}{a^{7}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b^{5}}{a^{7}}}}{b^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{a^{7}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b^{5}}{a^{7}}}}{b^{2}} + x \right )}}{2} + \frac {x^{5}}{5 a} - \frac {b x^{3}}{3 a^{2}} + \frac {b^{2} x}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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