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

Optimal. Leaf size=23 \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2} \]

[Out]

(-2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5

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Rubi [A]  time = 0.0114164, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[1 + x],x]

[Out]

(-2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5

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Rubi in Sympy [A]  time = 0.980611, size = 19, normalized size = 0.83 \[ \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{5} - \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + 1)**(5/2)/5 - 2*(x + 1)**(3/2)/3

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Mathematica [A]  time = 0.00503717, size = 16, normalized size = 0.7 \[ \frac{2}{15} (x+1)^{3/2} (3 x-2) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[1 + x],x]

[Out]

(2*(1 + x)^(3/2)*(-2 + 3*x))/15

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Maple [A]  time = 0., size = 13, normalized size = 0.6 \[{\frac{-4+6\,x}{15} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(1+x)^(1/2),x)

[Out]

2/15*(1+x)^(3/2)*(-2+3*x)

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Maxima [A]  time = 1.34049, size = 20, normalized size = 0.87 \[ \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)

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Fricas [A]  time = 0.208882, size = 20, normalized size = 0.87 \[ \frac{2}{15} \,{\left (3 \, x^{2} + x - 2\right )} \sqrt{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*x^2 + x - 2)*sqrt(x + 1)

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Sympy [A]  time = 1.44474, size = 34, normalized size = 1.48 \[ \frac{2 x^{2} \sqrt{x + 1}}{5} + \frac{2 x \sqrt{x + 1}}{15} - \frac{4 \sqrt{x + 1}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(1+x)**(1/2),x)

[Out]

2*x**2*sqrt(x + 1)/5 + 2*x*sqrt(x + 1)/15 - 4*sqrt(x + 1)/15

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GIAC/XCAS [A]  time = 0.198518, size = 20, normalized size = 0.87 \[ \frac{2}{5} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)