3.211 \(\int \frac{x}{1+\sqrt{1+x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{2}{3} (x+1)^{3/2}-x \]

[Out]

-x + (2*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.0171658, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{3} (x+1)^{3/2}-x \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + Sqrt[1 + x]),x]

[Out]

-x + (2*(1 + x)^(3/2))/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00966893, size = 15, normalized size = 1. \[ \frac{2}{3} (x+1)^{3/2}-x \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + Sqrt[1 + x]),x]

[Out]

-x + (2*(1 + x)^(3/2))/3

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-1-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+(1+x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.36507, size = 16, normalized size = 1.07 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - x - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(x + 1) + 1),x, algorithm="maxima")

[Out]

2/3*(x + 1)^(3/2) - x - 1

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Fricas [A]  time = 0.200508, size = 15, normalized size = 1. \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(x + 1) + 1),x, algorithm="fricas")

[Out]

2/3*(x + 1)^(3/2) - x

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Sympy [A]  time = 1.3924, size = 22, normalized size = 1.47 \[ \frac{2 x \sqrt{x + 1}}{3} - x + \frac{2 \sqrt{x + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+(1+x)**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207377, size = 16, normalized size = 1.07 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - x - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(x + 1) + 1),x, algorithm="giac")

[Out]

2/3*(x + 1)^(3/2) - x - 1