Optimal. Leaf size=53 \[ -\frac {b^3 \log \left (a x^2+b\right )}{2 a^4}+\frac {b^2 x^2}{2 a^3}-\frac {b x^4}{4 a^2}+\frac {x^6}{6 a} \]
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Rubi [A] time = 0.03, antiderivative size = 53, 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, 266, 43} \[ \frac {b^2 x^2}{2 a^3}-\frac {b^3 \log \left (a x^2+b\right )}{2 a^4}-\frac {b x^4}{4 a^2}+\frac {x^6}{6 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{a+\frac {b}{x^2}} \, dx &=\int \frac {x^7}{b+a x^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{b+a x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2}{a^3}-\frac {b x}{a^2}+\frac {x^2}{a}-\frac {b^3}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {b^2 x^2}{2 a^3}-\frac {b x^4}{4 a^2}+\frac {x^6}{6 a}-\frac {b^3 \log \left (b+a x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 1.00 \[ -\frac {b^3 \log \left (a x^2+b\right )}{2 a^4}+\frac {b^2 x^2}{2 a^3}-\frac {b x^4}{4 a^2}+\frac {x^6}{6 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 45, normalized size = 0.85 \[ \frac {2 \, a^{3} x^{6} - 3 \, a^{2} b x^{4} + 6 \, a b^{2} x^{2} - 6 \, b^{3} \log \left (a x^{2} + b\right )}{12 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 47, normalized size = 0.89 \[ -\frac {b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac {2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.87 \[ \frac {x^{6}}{6 a}-\frac {b \,x^{4}}{4 a^{2}}+\frac {b^{2} x^{2}}{2 a^{3}}-\frac {b^{3} \ln \left (a \,x^{2}+b \right )}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 46, normalized size = 0.87 \[ -\frac {b^{3} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac {2 \, a^{2} x^{6} - 3 \, a b x^{4} + 6 \, b^{2} x^{2}}{12 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 45, normalized size = 0.85 \[ \frac {x^6}{6\,a}-\frac {b\,x^4}{4\,a^2}-\frac {b^3\,\ln \left (a\,x^2+b\right )}{2\,a^4}+\frac {b^2\,x^2}{2\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 44, normalized size = 0.83 \[ \frac {x^{6}}{6 a} - \frac {b x^{4}}{4 a^{2}} + \frac {b^{2} x^{2}}{2 a^{3}} - \frac {b^{3} \log {\left (a x^{2} + b \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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