Optimal. Leaf size=27 \[ \frac {x^3}{3 a}-\frac {b \log \left (a x^3+b\right )}{3 a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 27, 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} \[ \frac {x^3}{3 a}-\frac {b \log \left (a x^3+b\right )}{3 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {x^2}{a+\frac {b}{x^3}} \, dx &=\int \frac {x^5}{b+a x^3} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{b+a x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx,x,x^3\right )\\ &=\frac {x^3}{3 a}-\frac {b \log \left (b+a x^3\right )}{3 a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ \frac {x^3}{3 a}-\frac {b \log \left (a x^3+b\right )}{3 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 22, normalized size = 0.81 \[ \frac {a x^{3} - b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 24, normalized size = 0.89 \[ \frac {x^{3}}{3 \, a} - \frac {b \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ \frac {x^{3}}{3 a}-\frac {b \ln \left (a \,x^{3}+b \right )}{3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 23, normalized size = 0.85 \[ \frac {x^{3}}{3 \, a} - \frac {b \log \left (a x^{3} + b\right )}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 22, normalized size = 0.81 \[ -\frac {b\,\ln \left (a\,x^3+b\right )-a\,x^3}{3\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 20, normalized size = 0.74 \[ \frac {x^{3}}{3 a} - \frac {b \log {\left (a x^{3} + b \right )}}{3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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