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