3.309 \(\int \frac{\sqrt{2 x+x^2}}{1+x} \, dx\)

Optimal. Leaf size=26 \[ \sqrt{x^2+2 x}-\tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]

[Out]

Sqrt[2*x + x^2] - ArcTan[Sqrt[2*x + x^2]]

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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]

Sqrt[2*x + x^2] - ArcTan[Sqrt[2*x + x^2]]

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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]

sqrt(x**2 + 2*x) - atan(sqrt(x**2 + 2*x))

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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]

Sqrt[x*(2 + x)]*(1 - (2*ArcTan[Sqrt[x/(2 + x)]])/(Sqrt[x]*Sqrt[2 + x]))

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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]

((1+x)^2-1)^(1/2)+arctan(1/((1+x)^2-1)^(1/2))

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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]

sqrt(x^2 + 2*x) + arcsin(1/abs(x + 1))

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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]

-1/2*(2*x^2 + 4*(x - sqrt(x^2 + 2*x) + 1)*arctan(-x + sqrt(x^2 + 2*x) - 1) - sqr
t(x^2 + 2*x)*(2*x + 1) + 3*x - 1)/(x - sqrt(x^2 + 2*x) + 1)

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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]

Integral(sqrt(x*(x + 2))/(x + 1), x)

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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]

sqrt(x^2 + 2*x) - 2*arctan(-x + sqrt(x^2 + 2*x) - 1)