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 = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 629
Rule 692
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.82, size = 32, normalized size = 0.84 \[ \frac {1}{15} \, {\left (3 \, x^{4} + 12 \, x^{3} + 19 \, x^{2} + 14 \, x + 2\right )} \sqrt {x^{2} + 2 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 25, normalized size = 0.66 \[ \frac {1}{5} \, {\left (x^{2} + 2 \, x + 2\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (x^{2} + 2 \, x + 2\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 23, normalized size = 0.61 \[ \frac {\left (x^{2}+2 x +2\right )^{\frac {3}{2}} \left (3 x^{2}+6 x +1\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.86, size = 41, normalized size = 1.08 \[ \frac {1}{5} \, {\left (x^{2} + 2 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {2}{5} \, {\left (x^{2} + 2 \, x + 2\right )}^{\frac {3}{2}} x + \frac {1}{15} \, {\left (x^{2} + 2 \, x + 2\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 22, normalized size = 0.58 \[ \frac {{\left (x^2+2\,x+2\right )}^{3/2}\,\left (3\,x^2+6\,x+1\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 85, normalized size = 2.24 \[ \frac {x^{4} \sqrt {x^{2} + 2 x + 2}}{5} + \frac {4 x^{3} \sqrt {x^{2} + 2 x + 2}}{5} + \frac {19 x^{2} \sqrt {x^{2} + 2 x + 2}}{15} + \frac {14 x \sqrt {x^{2} + 2 x + 2}}{15} + \frac {2 \sqrt {x^{2} + 2 x + 2}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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