Optimal. Leaf size=27 \[ \frac{1}{5} \left (x^2+1\right )^{5/2}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]
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Rubi [A] time = 0.0101029, 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} \[ \frac{1}{5} \left (x^2+1\right )^{5/2}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt{1+x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \sqrt{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\sqrt{1+x}+(1+x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{3} \left (1+x^2\right )^{3/2}+\frac{1}{5} \left (1+x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0059802, size = 20, normalized size = 0.74 \[ \frac{1}{15} \left (x^2+1\right )^{3/2} \left (3 x^2-2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 17, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{2}-2}{15} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.475, size = 30, normalized size = 1.11 \begin{align*} \frac{1}{5} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{2}{15} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98449, size = 51, normalized size = 1.89 \begin{align*} \frac{1}{15} \,{\left (3 \, x^{4} + x^{2} - 2\right )} \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.663862, size = 37, normalized size = 1.37 \begin{align*} \frac{x^{4} \sqrt{x^{2} + 1}}{5} + \frac{x^{2} \sqrt{x^{2} + 1}}{15} - \frac{2 \sqrt{x^{2} + 1}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0514, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{5} \,{\left (x^{2} + 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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