3.741 \(\int \left (1+\frac{2 x}{1+x^2}\right )^{5/2} \, dx\)

Optimal. Leaf size=133 \[ -\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)^3}{3 \left (x^2+1\right )}-\frac{4}{3} (1-2 x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)-\frac{(3 x+4) \left (x^2+1\right ) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}+\frac{5 \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]

[Out]

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

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Rubi [A]  time = 0.194981, antiderivative size = 133, 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 \[ -\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)^3}{3 \left (x^2+1\right )}-\frac{4}{3} (1-2 x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)-\frac{(3 x+4) \left (x^2+1\right ) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}+\frac{5 \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1} \]

Antiderivative was successfully verified.

[In]  Int[(1 + (2*x)/(1 + x^2))^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.30216, size = 112, normalized size = 0.84 \[ - \frac{\left (- 128 x + 64\right ) \left (x + 1\right ) \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{48} - \frac{\left (- 2 x + 2\right ) \left (x + 1\right )^{3} \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{6 \left (x^{2} + 1\right )} - \frac{\left (1152 x + 1536\right ) \left (x^{2} + 1\right ) \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{384 \left (x + 1\right )} + \frac{5 \sqrt{x^{2} + 1} \sqrt{\frac{2 x}{x^{2} + 1} + 1} \operatorname{asinh}{\left (x \right )}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x/(x**2+1))**(5/2),x)

[Out]

-(-128*x + 64)*(x + 1)*sqrt(2*x/(x**2 + 1) + 1)/48 - (-2*x + 2)*(x + 1)**3*sqrt(
2*x/(x**2 + 1) + 1)/(6*(x**2 + 1)) - (1152*x + 1536)*(x**2 + 1)*sqrt(2*x/(x**2 +
1) + 1)/(384*(x + 1)) + 5*sqrt(x**2 + 1)*sqrt(2*x/(x**2 + 1) + 1)*asinh(x)/(x +
1)

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Mathematica [A]  time = 0.0841018, size = 64, normalized size = 0.48 \[ \frac{(x+1) \left (3 x^4-8 x^3-18 x^2+15 \left (x^2+1\right )^{3/2} \sinh ^{-1}(x)-12 x-17\right )}{3 \sqrt{\frac{(x+1)^2}{x^2+1}} \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + (2*x)/(1 + x^2))^(5/2),x]

[Out]

((1 + x)*(-17 - 12*x - 18*x^2 - 8*x^3 + 3*x^4 + 15*(1 + x^2)^(3/2)*ArcSinh[x]))/
(3*Sqrt[(1 + x)^2/(1 + x^2)]*(1 + x^2)^2)

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Maple [A]  time = 0.024, size = 62, normalized size = 0.5 \[{\frac{{x}^{2}+1}{3\, \left ( 1+x \right ) ^{5}} \left ({\frac{{x}^{2}+2\,x+1}{{x}^{2}+1}} \right ) ^{{\frac{5}{2}}} \left ( 15\,{\it Arcsinh} \left ( x \right ) \left ({x}^{2}+1 \right ) ^{3/2}+3\,{x}^{4}-8\,{x}^{3}-18\,{x}^{2}-12\,x-17 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x/(x^2+1))^(5/2),x)

[Out]

1/3*((x^2+2*x+1)/(x^2+1))^(5/2)/(1+x)^5*(x^2+1)*(15*arcsinh(x)*(x^2+1)^(3/2)+3*x
^4-8*x^3-18*x^2-12*x-17)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (\frac{2 \, x}{x^{2} + 1} + 1\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*x/(x^2 + 1) + 1)^(5/2), x)

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Fricas [A]  time = 0.302351, size = 169, normalized size = 1.27 \[ -\frac{8 \, x^{3} + 8 \, x^{2} + 15 \,{\left (x^{3} + x^{2} + x + 1\right )} \log \left (-\frac{x \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} - x - 1}{\sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}\right ) -{\left (3 \, x^{4} - 8 \, x^{3} - 18 \, x^{2} - 12 \, x - 17\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 8 \, x + 8}{3 \,{\left (x^{3} + x^{2} + x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(8*x^3 + 8*x^2 + 15*(x^3 + x^2 + x + 1)*log(-(x*sqrt((x^2 + 2*x + 1)/(x^2 +
1)) - x - 1)/sqrt((x^2 + 2*x + 1)/(x^2 + 1))) - (3*x^4 - 8*x^3 - 18*x^2 - 12*x
- 17)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + 8*x + 8)/(x^3 + x^2 + x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{2 x}{x^{2} + 1} + 1\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x/(x**2+1))**(5/2),x)

[Out]

Integral((2*x/(x**2 + 1) + 1)**(5/2), x)

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GIAC/XCAS [A]  time = 0.270192, size = 116, normalized size = 0.87 \[{\left (\sqrt{2} + 5 \,{\rm ln}\left (\sqrt{2} + 1\right )\right )}{\rm sign}\left (x + 1\right ) - 5 \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ){\rm sign}\left (x + 1\right ) + \frac{{\left ({\left ({\left (3 \, x{\rm sign}\left (x + 1\right ) - 8 \,{\rm sign}\left (x + 1\right )\right )} x - 18 \,{\rm sign}\left (x + 1\right )\right )} x - 12 \,{\rm sign}\left (x + 1\right )\right )} x - 17 \,{\rm sign}\left (x + 1\right )}{3 \,{\left (x^{2} + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(5/2),x, algorithm="giac")

[Out]

(sqrt(2) + 5*ln(sqrt(2) + 1))*sign(x + 1) - 5*ln(-x + sqrt(x^2 + 1))*sign(x + 1)
+ 1/3*((((3*x*sign(x + 1) - 8*sign(x + 1))*x - 18*sign(x + 1))*x - 12*sign(x +
1))*x - 17*sign(x + 1))/(x^2 + 1)^(3/2)