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 = 0.0103061, antiderivative size = 27, normalized size of antiderivative = 1., 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 266
Rule 43
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*}
Mathematica [A] time = 0.0074507, 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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Maple [A] time = 0.003, size = 17, normalized size = 0.6 \begin{align*}{\frac{6\,{x}^{2}-9}{20} \left ({x}^{2}+1 \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945974, size = 26, normalized size = 0.96 \begin{align*} \frac{3}{10} \,{\left (x^{2} + 1\right )}^{\frac{5}{3}} - \frac{3}{4} \,{\left (x^{2} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09868, size = 46, normalized size = 1.7 \begin{align*} \frac{3}{20} \,{\left (2 \, x^{2} - 3\right )}{\left (x^{2} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.798445, size = 26, normalized size = 0.96 \begin{align*} \frac{3 x^{2} \left (x^{2} + 1\right )^{\frac{2}{3}}}{10} - \frac{9 \left (x^{2} + 1\right )^{\frac{2}{3}}}{20} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06786, size = 26, normalized size = 0.96 \begin{align*} \frac{3}{10} \,{\left (x^{2} + 1\right )}^{\frac{5}{3}} - \frac{3}{4} \,{\left (x^{2} + 1\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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