Optimal. Leaf size=27 \[ \frac {3}{10} \left (x^2+1\right )^{5/3}-\frac {3}{4} \left (x^2+1\right )^{2/3} \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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 = {266, 43} \[ \frac {3}{10} \left (x^2+1\right )^{5/3}-\frac {3}{4} \left (x^2+1\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt [3]{1+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [3]{1+x}}+(1+x)^{2/3}\right ) \, dx,x,x^2\right )\\ &=-\frac {3}{4} \left (1+x^2\right )^{2/3}+\frac {3}{10} \left (1+x^2\right )^{5/3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.74 \[ \frac {3}{20} \left (x^2+1\right )^{2/3} \left (2 x^2-3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 16, normalized size = 0.59 \[ \frac {3}{20} \, {\left (2 \, x^{2} - 3\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 19, normalized size = 0.70 \[ \frac {3}{10} \, {\left (x^{2} + 1\right )}^{\frac {5}{3}} - \frac {3}{4} \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.63 \[ \frac {3 \left (x^{2}+1\right )^{\frac {2}{3}} \left (2 x^{2}-3\right )}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 19, normalized size = 0.70 \[ \frac {3}{10} \, {\left (x^{2} + 1\right )}^{\frac {5}{3}} - \frac {3}{4} \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 16, normalized size = 0.59 \[ \frac {3\,{\left (x^2+1\right )}^{2/3}\,\left (2\,x^2-3\right )}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 26, normalized size = 0.96 \[ \frac {3 x^{2} \left (x^{2} + 1\right )^{\frac {2}{3}}}{10} - \frac {9 \left (x^{2} + 1\right )^{\frac {2}{3}}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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