Optimal. Leaf size=38 \[ -\frac {2}{15} \left (2+2 x+x^2\right )^{3/2}+\frac {1}{5} (1+x)^2 \left (2+2 x+x^2\right )^{3/2} \]
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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 = {706, 643}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 706
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 = 0.11, size = 36, normalized size = 0.95 \begin {gather*} \frac {1}{15} \sqrt {2+2 x+x^2} \left (2+14 x+19 x^2+12 x^3+3 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 42, normalized size = 1.11
method | result | size |
gosper | \(\frac {\left (x^{2}+2 x +2\right )^{\frac {3}{2}} \left (3 x^{2}+6 x +1\right )}{15}\) | \(23\) |
trager | \(\left (\frac {1}{5} x^{4}+\frac {4}{5} x^{3}+\frac {19}{15} x^{2}+\frac {14}{15} x +\frac {2}{15}\right ) \sqrt {x^{2}+2 x +2}\) | \(32\) |
risch | \(\frac {\left (3 x^{4}+12 x^{3}+19 x^{2}+14 x +2\right ) \sqrt {x^{2}+2 x +2}}{15}\) | \(33\) |
default | \(\frac {x^{2} \left (x^{2}+2 x +2\right )^{\frac {3}{2}}}{5}+\frac {2 x \left (x^{2}+2 x +2\right )^{\frac {3}{2}}}{5}+\frac {\left (x^{2}+2 x +2\right )^{\frac {3}{2}}}{15}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 41, normalized size = 1.08 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.41, size = 32, normalized size = 0.84 \begin {gather*} \frac {1}{15} \, {\left (3 \, x^{4} + 12 \, x^{3} + 19 \, x^{2} + 14 \, x + 2\right )} \sqrt {x^{2} + 2 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (32) = 64\).
time = 0.09, size = 85, normalized size = 2.24 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.95, size = 25, normalized size = 0.66 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \frac {{\left (x^2+2\,x+2\right )}^{3/2}\,\left (3\,x^2+6\,x+1\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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