Optimal. Leaf size=53 \[ \frac{2}{21} \left (x^3+1\right )^{7/2}-\frac{2}{5} \left (x^3+1\right )^{5/2}+\frac{2}{3} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]
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Rubi [A] time = 0.0178267, antiderivative size = 53, 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} \[ \frac{2}{21} \left (x^3+1\right )^{7/2}-\frac{2}{5} \left (x^3+1\right )^{5/2}+\frac{2}{3} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+3 \sqrt{1+x}-3 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,x^3\right )\\ &=-\frac{2}{3} \sqrt{1+x^3}+\frac{2}{3} \left (1+x^3\right )^{3/2}-\frac{2}{5} \left (1+x^3\right )^{5/2}+\frac{2}{21} \left (1+x^3\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0094386, size = 30, normalized size = 0.57 \[ \frac{2}{105} \sqrt{x^3+1} \left (5 x^9-6 x^6+8 x^3-16\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 38, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 5\,{x}^{9}-6\,{x}^{6}+8\,{x}^{3}-16 \right ) }{105}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953048, size = 50, normalized size = 0.94 \begin{align*} \frac{2}{21} \,{\left (x^{3} + 1\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70776, size = 68, normalized size = 1.28 \begin{align*} \frac{2}{105} \,{\left (5 \, x^{9} - 6 \, x^{6} + 8 \, x^{3} - 16\right )} \sqrt{x^{3} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.78113, size = 56, normalized size = 1.06 \begin{align*} \frac{2 x^{9} \sqrt{x^{3} + 1}}{21} - \frac{4 x^{6} \sqrt{x^{3} + 1}}{35} + \frac{16 x^{3} \sqrt{x^{3} + 1}}{105} - \frac{32 \sqrt{x^{3} + 1}}{105} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14007, size = 50, normalized size = 0.94 \begin{align*} \frac{2}{21} \,{\left (x^{3} + 1\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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