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

Optimal. Leaf size=48 \[ \frac{1}{3} \left (x^2+x\right )^{3/2}-\frac{1}{8} (2 x+1) \sqrt{x^2+x}+\frac{1}{8} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

[Out]

-((1 + 2*x)*Sqrt[x + x^2])/8 + (x + x^2)^(3/2)/3 + ArcTanh[x/Sqrt[x + x^2]]/8

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Rubi [A]  time = 0.0340763, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{1}{3} \left (x^2+x\right )^{3/2}-\frac{1}{8} (2 x+1) \sqrt{x^2+x}+\frac{1}{8} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[x + x^2],x]

[Out]

-((1 + 2*x)*Sqrt[x + x^2])/8 + (x + x^2)^(3/2)/3 + ArcTanh[x/Sqrt[x + x^2]]/8

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Rubi in Sympy [A]  time = 3.10529, size = 37, normalized size = 0.77 \[ - \frac{\left (2 x + 1\right ) \sqrt{x^{2} + x}}{8} + \frac{\left (x^{2} + x\right )^{\frac{3}{2}}}{3} + \frac{\operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} + x}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(x**2+x)**(1/2),x)

[Out]

-(2*x + 1)*sqrt(x**2 + x)/8 + (x**2 + x)**(3/2)/3 + atanh(x/sqrt(x**2 + x))/8

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Mathematica [A]  time = 0.0510891, size = 43, normalized size = 0.9 \[ \frac{1}{24} \sqrt{x (x+1)} \left (8 x^2+2 x+\frac{3 \sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x+1} \sqrt{x}}-3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[x + x^2],x]

[Out]

(Sqrt[x*(1 + x)]*(-3 + 2*x + 8*x^2 + (3*ArcSinh[Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x]))
)/24

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Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \[{\frac{1}{3} \left ({x}^{2}+x \right ) ^{{\frac{3}{2}}}}-{\frac{1+2\,x}{8}\sqrt{{x}^{2}+x}}+{\frac{1}{16}\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(x^2+x)^(1/2),x)

[Out]

1/3*(x^2+x)^(3/2)-1/8*(1+2*x)*(x^2+x)^(1/2)+1/16*ln(1/2+x+(x^2+x)^(1/2))

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Maxima [A]  time = 0.714126, size = 62, normalized size = 1.29 \[ \frac{1}{3} \,{\left (x^{2} + x\right )}^{\frac{3}{2}} - \frac{1}{4} \, \sqrt{x^{2} + x} x - \frac{1}{8} \, \sqrt{x^{2} + x} + \frac{1}{16} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} + x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + x)*x,x, algorithm="maxima")

[Out]

1/3*(x^2 + x)^(3/2) - 1/4*sqrt(x^2 + x)*x - 1/8*sqrt(x^2 + x) + 1/16*log(2*x + 2
*sqrt(x^2 + x) + 1)

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Fricas [A]  time = 0.219606, size = 207, normalized size = 4.31 \[ -\frac{2048 \, x^{6} + 4608 \, x^{5} + 2688 \, x^{4} - 384 \, x^{3} - 576 \, x^{2} + 12 \,{\left (32 \, x^{3} + 48 \, x^{2} - 2 \,{\left (16 \, x^{2} + 16 \, x + 3\right )} \sqrt{x^{2} + x} + 18 \, x + 1\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) - 2 \,{\left (1024 \, x^{5} + 1792 \, x^{4} + 576 \, x^{3} - 320 \, x^{2} - 128 \, x + 3\right )} \sqrt{x^{2} + x} - 54 \, x + 5}{192 \,{\left (32 \, x^{3} + 48 \, x^{2} - 2 \,{\left (16 \, x^{2} + 16 \, x + 3\right )} \sqrt{x^{2} + x} + 18 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + x)*x,x, algorithm="fricas")

[Out]

-1/192*(2048*x^6 + 4608*x^5 + 2688*x^4 - 384*x^3 - 576*x^2 + 12*(32*x^3 + 48*x^2
- 2*(16*x^2 + 16*x + 3)*sqrt(x^2 + x) + 18*x + 1)*log(-2*x + 2*sqrt(x^2 + x) -
1) - 2*(1024*x^5 + 1792*x^4 + 576*x^3 - 320*x^2 - 128*x + 3)*sqrt(x^2 + x) - 54*
x + 5)/(32*x^3 + 48*x^2 - 2*(16*x^2 + 16*x + 3)*sqrt(x^2 + x) + 18*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{x \left (x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(x**2+x)**(1/2),x)

[Out]

Integral(x*sqrt(x*(x + 1)), x)

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GIAC/XCAS [A]  time = 0.21024, size = 51, normalized size = 1.06 \[ \frac{1}{24} \,{\left (2 \,{\left (4 \, x + 1\right )} x - 3\right )} \sqrt{x^{2} + x} - \frac{1}{16} \,{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + x)*x,x, algorithm="giac")

[Out]

1/24*(2*(4*x + 1)*x - 3)*sqrt(x^2 + x) - 1/16*ln(abs(-2*x + 2*sqrt(x^2 + x) - 1)
)