3.340 \(\int x \sqrt{4+2 x+x^2} \, dx\)

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]

-((1 + x)*Sqrt[4 + 2*x + x^2])/2 + (4 + 2*x + x^2)^(3/2)/3 - (3*ArcSinh[(1 + x)/
Sqrt[3]])/2

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Rubi [A]  time = 0.0335192, 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]

-((1 + x)*Sqrt[4 + 2*x + x^2])/2 + (4 + 2*x + x^2)^(3/2)/3 - (3*ArcSinh[(1 + x)/
Sqrt[3]])/2

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Rubi in Sympy [A]  time = 1.92478, 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]

-(2*x + 2)*sqrt(x**2 + 2*x + 4)/4 + (x**2 + 2*x + 4)**(3/2)/3 - 3*atanh((2*x + 2
)/(2*sqrt(x**2 + 2*x + 4)))/2

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Mathematica [A]  time = 0.032022, 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]

(Sqrt[4 + 2*x + x^2]*(5 + x + 2*x^2) - 9*ArcSinh[(1 + x)/Sqrt[3]])/6

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Maple [A]  time = 0.007, size = 42, normalized size = 0.8 \[{\frac{1}{3} \left ({x}^{2}+2\,x+4 \right ) ^{{\frac{3}{2}}}}-{\frac{2\,x+2}{4}\sqrt{{x}^{2}+2\,x+4}}-{\frac{3}{2}{\it Arcsinh} \left ({\frac{ \left ( 1+x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(x^2+2*x+4)^(1/2),x)

[Out]

1/3*(x^2+2*x+4)^(3/2)-1/4*(2*x+2)*(x^2+2*x+4)^(1/2)-3/2*arcsinh(1/3*(1+x)*3^(1/2
))

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Maxima [A]  time = 1.52057, 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]

1/3*(x^2 + 2*x + 4)^(3/2) - 1/2*sqrt(x^2 + 2*x + 4)*x - 1/2*sqrt(x^2 + 2*x + 4)
- 3/2*arcsinh(1/3*sqrt(3)*(x + 1))

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Fricas [A]  time = 0.207473, 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]

-1/24*(32*x^6 + 144*x^5 + 456*x^4 + 816*x^3 + 1104*x^2 - 36*(4*x^3 + 12*x^2 - (4
*x^2 + 8*x + 7)*sqrt(x^2 + 2*x + 4) + 21*x + 13)*log(-x + sqrt(x^2 + 2*x + 4) -
1) - (32*x^5 + 112*x^4 + 296*x^3 + 400*x^2 + 416*x + 211)*sqrt(x^2 + 2*x + 4) +
885*x + 469)/(4*x^3 + 12*x^2 - (4*x^2 + 8*x + 7)*sqrt(x^2 + 2*x + 4) + 21*x + 13
)

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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]

Integral(x*sqrt(x**2 + 2*x + 4), x)

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GIAC/XCAS [A]  time = 0.213015, 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]

1/6*((2*x + 1)*x + 5)*sqrt(x^2 + 2*x + 4) + 3/2*ln(-x + sqrt(x^2 + 2*x + 4) - 1)