Optimal. Leaf size=38 \[ \frac{1}{5} (x+1)^2 \left (x^2+2 x+2\right )^{3/2}-\frac{2}{15} \left (x^2+2 x+2\right )^{3/2} \]
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Rubi [A] time = 0.0118175, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {692, 629} \[ \frac{1}{5} (x+1)^2 \left (x^2+2 x+2\right )^{3/2}-\frac{2}{15} \left (x^2+2 x+2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 629
Rubi steps
\begin{align*} \int (1+x)^3 \sqrt{2+2 x+x^2} \, dx &=\frac{1}{5} (1+x)^2 \left (2+2 x+x^2\right )^{3/2}-\frac{2}{5} \int (1+x) \sqrt{2+2 x+x^2} \, dx\\ &=-\frac{2}{15} \left (2+2 x+x^2\right )^{3/2}+\frac{1}{5} (1+x)^2 \left (2+2 x+x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0093214, size = 26, normalized size = 0.68 \[ \frac{1}{15} \left (x^2+2 x+2\right )^{3/2} \left (3 x^2+6 x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 23, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{2}+6\,x+1}{15} \left ({x}^{2}+2\,x+2 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51231, size = 55, normalized size = 1.45 \begin{align*} \frac{1}{5} \,{\left (x^{2} + 2 \, x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{2}{5} \,{\left (x^{2} + 2 \, x + 2\right )}^{\frac{3}{2}} x + \frac{1}{15} \,{\left (x^{2} + 2 \, x + 2\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33861, size = 85, normalized size = 2.24 \begin{align*} \frac{1}{15} \,{\left (3 \, x^{4} + 12 \, x^{3} + 19 \, x^{2} + 14 \, x + 2\right )} \sqrt{x^{2} + 2 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.259986, size = 85, normalized size = 2.24 \begin{align*} \frac{x^{4} \sqrt{x^{2} + 2 x + 2}}{5} + \frac{4 x^{3} \sqrt{x^{2} + 2 x + 2}}{5} + \frac{19 x^{2} \sqrt{x^{2} + 2 x + 2}}{15} + \frac{14 x \sqrt{x^{2} + 2 x + 2}}{15} + \frac{2 \sqrt{x^{2} + 2 x + 2}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08866, size = 38, normalized size = 1. \begin{align*} \frac{1}{15} \,{\left ({\left ({\left (3 \,{\left (x + 4\right )} x + 19\right )} x + 14\right )} x + 2\right )} \sqrt{x^{2} + 2 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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