Optimal. Leaf size=39 \[ \frac {a^3 x^6}{6}+\frac {3}{4} a^2 b x^4+\frac {3}{2} a b^2 x^2+b^3 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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} \[ \frac {3}{4} a^2 b x^4+\frac {a^3 x^6}{6}+\frac {3}{2} a b^2 x^2+b^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^3 x^5 \, dx &=\int \frac {\left (b+a x^2\right )^3}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^3}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (3 a b^2+\frac {b^3}{x}+3 a^2 b x+a^3 x^2\right ) \, dx,x,x^2\right )\\ &=\frac {3}{2} a b^2 x^2+\frac {3}{4} a^2 b x^4+\frac {a^3 x^6}{6}+b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 39, normalized size = 1.00 \[ \frac {a^3 x^6}{6}+\frac {3}{4} a^2 b x^4+\frac {3}{2} a b^2 x^2+b^3 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 33, normalized size = 0.85 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 0.92 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {1}{2} \, b^{3} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.87 \[ \frac {a^{3} x^{6}}{6}+\frac {3 a^{2} b \,x^{4}}{4}+\frac {3 a \,b^{2} x^{2}}{2}+b^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 36, normalized size = 0.92 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{4} \, a^{2} b x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {1}{2} \, b^{3} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 33, normalized size = 0.85 \[ b^3\,\ln \relax (x)+\frac {a^3\,x^6}{6}+\frac {3\,a\,b^2\,x^2}{2}+\frac {3\,a^2\,b\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 37, normalized size = 0.95 \[ \frac {a^{3} x^{6}}{6} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{2}}{2} + b^{3} \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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