Optimal. Leaf size=50 \[ -\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {3}{2} \sinh ^{-1}\left (\frac {1+x}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {654, 626, 633,
221} \begin {gather*} \frac {1}{3} \left (x^2+2 x+4\right )^{3/2}-\frac {1}{2} (x+1) \sqrt {x^2+2 x+4}-\frac {3}{2} \sinh ^{-1}\left (\frac {x+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 626
Rule 633
Rule 654
Rubi steps
\begin {align*} \int x \sqrt {4+2 x+x^2} \, dx &=\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\int \sqrt {4+2 x+x^2} \, dx\\ &=-\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {3}{2} \int \frac {1}{\sqrt {4+2 x+x^2}} \, dx\\ &=-\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {1}{4} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{12}}} \, dx,x,2+2 x\right )\\ &=-\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {3}{2} \sinh ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 0.94 \begin {gather*} \frac {1}{6} \sqrt {4+2 x+x^2} \left (5+x+2 x^2\right )+\frac {3}{2} \log \left (-1-x+\sqrt {4+2 x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 42, normalized size = 0.84
method | result | size |
risch | \(\frac {\left (2 x^{2}+x +5\right ) \sqrt {x^{2}+2 x +4}}{6}-\frac {3 \arcsinh \left (\frac {\left (x +1\right ) \sqrt {3}}{3}\right )}{2}\) | \(33\) |
trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{6} x +\frac {5}{6}\right ) \sqrt {x^{2}+2 x +4}+\frac {3 \ln \left (\sqrt {x^{2}+2 x +4}-1-x \right )}{2}\) | \(41\) |
default | \(\frac {\left (x^{2}+2 x +4\right )^{\frac {3}{2}}}{3}-\frac {\left (2+2 x \right ) \sqrt {x^{2}+2 x +4}}{4}-\frac {3 \arcsinh \left (\frac {\left (x +1\right ) \sqrt {3}}{3}\right )}{2}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 49, normalized size = 0.98 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 4} x - \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 4} - \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.29, size = 39, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{2} + x + 5\right )} \sqrt {x^{2} + 2 \, x + 4} + \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 2 \, x + 4} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x^{2} + 2 x + 4}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 40, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left ({\left (2 \, x + 1\right )} x + 5\right )} \sqrt {x^{2} + 2 \, x + 4} + \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 2 \, x + 4} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 39, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x^2+2\,x+4}\,\left (8\,x^2+4\,x+20\right )}{24}-\frac {3\,\ln \left (x+\sqrt {x^2+2\,x+4}+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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