Optimal. Leaf size=50 \[ \frac{1}{3} \left (x^2+2 x+4\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{x^2+2 x+4}-\frac{3}{2} \sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]
[Out]
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Rubi [A] time = 0.0497145, 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 \[ \frac{1}{3} \left (x^2+2 x+4\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{x^2+2 x+4}-\frac{3}{2} \sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[4 + 2*x + x^2],x]
[Out]
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Rubi in Sympy [A] time = 4.18478, size = 54, normalized size = 1.08 \[ - \frac{\left (2 x + 2\right ) \sqrt{x^{2} + 2 x + 4}}{4} + \frac{\left (x^{2} + 2 x + 4\right )^{\frac{3}{2}}}{3} - \frac{3 \operatorname{atanh}{\left (\frac{2 x + 2}{2 \sqrt{x^{2} + 2 x + 4}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(x**2+2*x+4)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0374262, 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.
[In] Integrate[x*Sqrt[4 + 2*x + x^2],x]
[Out]
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Maple [A] time = 0.006, size = 42, normalized size = 0.8 \[{\frac{1}{3} \left ({x}^{2}+2\,x+4 \right ) ^{{\frac{3}{2}}}}-{\frac{2+2\,x}{4}\sqrt{{x}^{2}+2\,x+4}}-{\frac{3}{2}{\it Arcsinh} \left ({\frac{ \left ( 1+x \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(x^2+2*x+4)^(1/2),x)
[Out]
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Maxima [A] time = 0.753721, size = 66, normalized size = 1.32 \[ \frac{1}{3} \,{\left (x^{2} + 2 \, x + 4\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x^{2} + 2 \, x + 4} x - \frac{1}{2} \, \sqrt{x^{2} + 2 \, x + 4} - \frac{3}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2*x + 4)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234482, size = 220, normalized size = 4.4 \[ -\frac{32 \, x^{6} + 144 \, x^{5} + 456 \, x^{4} + 816 \, x^{3} + 1104 \, x^{2} - 36 \,{\left (4 \, x^{3} + 12 \, x^{2} -{\left (4 \, x^{2} + 8 \, x + 7\right )} \sqrt{x^{2} + 2 \, x + 4} + 21 \, x + 13\right )} \log \left (-x + \sqrt{x^{2} + 2 \, x + 4} - 1\right ) -{\left (32 \, x^{5} + 112 \, x^{4} + 296 \, x^{3} + 400 \, x^{2} + 416 \, x + 211\right )} \sqrt{x^{2} + 2 \, x + 4} + 885 \, x + 469}{24 \,{\left (4 \, x^{3} + 12 \, x^{2} -{\left (4 \, x^{2} + 8 \, x + 7\right )} \sqrt{x^{2} + 2 \, x + 4} + 21 \, x + 13\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2*x + 4)*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{x^{2} + 2 x + 4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(x**2+2*x+4)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.207038, size = 54, normalized size = 1.08 \[ \frac{1}{6} \,{\left ({\left (2 \, x + 1\right )} x + 5\right )} \sqrt{x^{2} + 2 \, x + 4} + \frac{3}{2} \,{\rm ln}\left (-x + \sqrt{x^{2} + 2 \, x + 4} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2*x + 4)*x,x, algorithm="giac")
[Out]