Optimal. Leaf size=26 \[ \sqrt{x^2+2 x}-\tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]
[Out]
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Rubi [A] time = 0.0455806, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \sqrt{x^2+2 x}-\tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2*x + x^2]/(1 + x),x]
[Out]
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Rubi in Sympy [A] time = 7.56477, size = 20, normalized size = 0.77 \[ \sqrt{x^{2} + 2 x} - \operatorname{atan}{\left (\sqrt{x^{2} + 2 x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+2*x)**(1/2)/(1+x),x)
[Out]
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Mathematica [A] time = 0.0645454, size = 38, normalized size = 1.46 \[ \sqrt{x (x+2)} \left (1-\frac{2 \tan ^{-1}\left (\sqrt{\frac{x}{x+2}}\right )}{\sqrt{x} \sqrt{x+2}}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[2*x + x^2]/(1 + x),x]
[Out]
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Maple [A] time = 0.009, size = 21, normalized size = 0.8 \[ \sqrt{ \left ( 1+x \right ) ^{2}-1}+\arctan \left ({\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-1}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+2*x)^(1/2)/(1+x),x)
[Out]
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Maxima [A] time = 0.771874, size = 23, normalized size = 0.88 \[ \sqrt{x^{2} + 2 \, x} + \arcsin \left (\frac{1}{{\left | x + 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2*x)/(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219098, size = 101, normalized size = 3.88 \[ -\frac{2 \, x^{2} + 4 \,{\left (x - \sqrt{x^{2} + 2 \, x} + 1\right )} \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) - \sqrt{x^{2} + 2 \, x}{\left (2 \, x + 1\right )} + 3 \, x - 1}{2 \,{\left (x - \sqrt{x^{2} + 2 \, x} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2*x)/(x + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (x + 2\right )}}{x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+2*x)**(1/2)/(1+x),x)
[Out]
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GIAC/XCAS [A] time = 0.214455, size = 36, normalized size = 1.38 \[ \sqrt{x^{2} + 2 \, x} - 2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2*x)/(x + 1),x, algorithm="giac")
[Out]