3.742 \(\int \left (1+\frac{2 x}{1+x^2}\right )^{3/2} \, dx\)

Optimal. Leaf size=90 \[ -(1-x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)-\frac{x \left (x^2+1\right ) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}+\frac{3 \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]

[Out]

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

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Rubi [A]  time = 0.130463, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -(1-x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)-\frac{x \left (x^2+1\right ) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}+\frac{3 \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]

Antiderivative was successfully verified.

[In]  Int[(1 + (2*x)/(1 + x^2))^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.73701, size = 76, normalized size = 0.84 \[ - \frac{x \left (x^{2} + 1\right ) \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{x + 1} - \frac{\left (- 2 x + 2\right ) \left (x + 1\right ) \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{2} + \frac{3 \sqrt{x^{2} + 1} \sqrt{\frac{2 x}{x^{2} + 1} + 1} \operatorname{asinh}{\left (x \right )}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x/(x**2+1))**(3/2),x)

[Out]

-x*(x**2 + 1)*sqrt(2*x/(x**2 + 1) + 1)/(x + 1) - (-2*x + 2)*(x + 1)*sqrt(2*x/(x*
*2 + 1) + 1)/2 + 3*sqrt(x**2 + 1)*sqrt(2*x/(x**2 + 1) + 1)*asinh(x)/(x + 1)

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Mathematica [A]  time = 0.0343863, size = 44, normalized size = 0.49 \[ \frac{\sqrt{\frac{(x+1)^2}{x^2+1}} \left (x^2+3 \sqrt{x^2+1} \sinh ^{-1}(x)-2 x-1\right )}{x+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + (2*x)/(1 + x^2))^(3/2),x]

[Out]

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

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Maple [A]  time = 0.014, size = 49, normalized size = 0.5 \[{\frac{{x}^{2}+1}{ \left ( 1+x \right ) ^{3}} \left ({\frac{{x}^{2}+2\,x+1}{{x}^{2}+1}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{\it Arcsinh} \left ( x \right ) \sqrt{{x}^{2}+1}+{x}^{2}-2\,x-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x/(x^2+1))^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.787713, size = 47, normalized size = 0.52 \[ \frac{x^{2}}{\sqrt{x^{2} + 1}} - \frac{2 \, x}{\sqrt{x^{2} + 1}} - \frac{1}{\sqrt{x^{2} + 1}} + 3 \, \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.269331, size = 123, normalized size = 1.37 \[ -\frac{3 \,{\left (x + 1\right )} \log \left (-\frac{x \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} - x - 1}{\sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}\right ) -{\left (x^{2} - 2 \, x - 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 2 \, x + 2}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{2 x}{x^{2} + 1} + 1\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x/(x**2+1))**(3/2),x)

[Out]

Integral((2*x/(x**2 + 1) + 1)**(3/2), x)

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GIAC/XCAS [A]  time = 0.263801, size = 90, normalized size = 1. \[ -{\left (\sqrt{2} - 3 \,{\rm ln}\left (\sqrt{2} + 1\right )\right )}{\rm sign}\left (x + 1\right ) - 3 \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ){\rm sign}\left (x + 1\right ) + \frac{{\left (x{\rm sign}\left (x + 1\right ) - 2 \,{\rm sign}\left (x + 1\right )\right )} x -{\rm sign}\left (x + 1\right )}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(3/2),x, algorithm="giac")

[Out]

-(sqrt(2) - 3*ln(sqrt(2) + 1))*sign(x + 1) - 3*ln(-x + sqrt(x^2 + 1))*sign(x + 1
) + ((x*sign(x + 1) - 2*sign(x + 1))*x - sign(x + 1))/sqrt(x^2 + 1)