Optimal. Leaf size=34 \[ \frac{2}{7} (x+1)^{7/2}-\frac{4}{5} (x+1)^{5/2}+\frac{2}{3} (x+1)^{3/2} \]
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Rubi [A] time = 0.0051121, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2}{7} (x+1)^{7/2}-\frac{4}{5} (x+1)^{5/2}+\frac{2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^2 \sqrt{1+x} \, dx &=\int \left (\sqrt{1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx\\ &=\frac{2}{3} (1+x)^{3/2}-\frac{4}{5} (1+x)^{5/2}+\frac{2}{7} (1+x)^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0056155, size = 21, normalized size = 0.62 \[ \frac{2}{105} (x+1)^{3/2} \left (15 x^2-12 x+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 0.5 \begin{align*}{\frac{30\,{x}^{2}-24\,x+16}{105} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93806, size = 30, normalized size = 0.88 \begin{align*} \frac{2}{7} \,{\left (x + 1\right )}^{\frac{7}{2}} - \frac{4}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.417573, size = 62, normalized size = 1.82 \begin{align*} \frac{2}{105} \,{\left (15 \, x^{3} + 3 \, x^{2} - 4 \, x + 8\right )} \sqrt{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19745, size = 48, normalized size = 1.41 \begin{align*} \frac{2 x^{3} \sqrt{x + 1}}{7} + \frac{2 x^{2} \sqrt{x + 1}}{35} - \frac{8 x \sqrt{x + 1}}{105} + \frac{16 \sqrt{x + 1}}{105} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06609, size = 30, normalized size = 0.88 \begin{align*} \frac{2}{7} \,{\left (x + 1\right )}^{\frac{7}{2}} - \frac{4}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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