Optimal. Leaf size=40 \[ \frac{2}{15} \left (x^3+1\right )^{5/2}-\frac{4}{9} \left (x^3+1\right )^{3/2}+\frac{2 \sqrt{x^3+1}}{3} \]
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Rubi [A] time = 0.0146081, antiderivative size = 40, 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} \[ \frac{2}{15} \left (x^3+1\right )^{5/2}-\frac{4}{9} \left (x^3+1\right )^{3/2}+\frac{2 \sqrt{x^3+1}}{3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1+x}}-2 \sqrt{1+x}+(1+x)^{3/2}\right ) \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}-\frac{4}{9} \left (1+x^3\right )^{3/2}+\frac{2}{15} \left (1+x^3\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0068485, size = 25, normalized size = 0.62 \[ \frac{2}{45} \sqrt{x^3+1} \left (3 x^6-4 x^3+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 3\,{x}^{6}-4\,{x}^{3}+8 \right ) }{45}{\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 = 0.954821, size = 38, normalized size = 0.95 \begin{align*} \frac{2}{15} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - \frac{4}{9} \,{\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 = 1.73187, size = 54, normalized size = 1.35 \begin{align*} \frac{2}{45} \,{\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt{x^{3} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.743006, size = 41, normalized size = 1.02 \begin{align*} \frac{2 x^{6} \sqrt{x^{3} + 1}}{15} - \frac{8 x^{3} \sqrt{x^{3} + 1}}{45} + \frac{16 \sqrt{x^{3} + 1}}{45} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11367, size = 38, normalized size = 0.95 \begin{align*} \frac{2}{15} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - \frac{4}{9} \,{\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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