Optimal. Leaf size=50 \[ \frac{1}{3} \left (x^2+2 x+4\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{x^2+2 x+4}-\frac{3}{2} \sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0145362, antiderivative size = 50, normalized size of antiderivative = 1., 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 = {640, 612, 619, 215} \[ \frac{1}{3} \left (x^2+2 x+4\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{x^2+2 x+4}-\frac{3}{2} \sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 619
Rule 215
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} \operatorname{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*}
Mathematica [A] time = 0.0182636, size = 38, normalized size = 0.76 \[ \frac{1}{6} \left (\sqrt{x^2+2 x+4} \left (2 x^2+x+5\right )-9 \sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{3} \left ({x}^{2}+2\,x+4 \right ) ^{{\frac{3}{2}}}}-{\frac{2\,x+2}{4}\sqrt{{x}^{2}+2\,x+4}}-{\frac{3}{2}{\it Arcsinh} \left ({\frac{ \left ( 1+x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42293, size = 66, normalized size = 1.32 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94938, size = 109, normalized size = 2.18 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{x^{2} + 2 x + 4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05913, size = 54, normalized size = 1.08 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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