3.537 \(\int \sqrt{-x} \left (\sqrt{-x}+x\right ) \, dx\)

Optimal. Leaf size=19 \[ \frac{2}{5} (-x)^{5/2}-\frac{x^2}{2} \]

[Out]

(2*(-x)^(5/2))/5 - x^2/2

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Rubi [A]  time = 0.0111821, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2}{5} (-x)^{5/2}-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-x]*(Sqrt[-x] + x),x]

[Out]

(2*(-x)^(5/2))/5 - x^2/2

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Rubi in Sympy [A]  time = 3.49258, size = 14, normalized size = 0.74 \[ - \frac{x^{2}}{2} + \frac{2 \left (- x\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2/2 + 2*(-x)**(5/2)/5

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Mathematica [A]  time = 0.00753848, size = 19, normalized size = 1. \[ \frac{2}{5} (-x)^{5/2}-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-x]*(Sqrt[-x] + x),x]

[Out]

(2*(-x)^(5/2))/5 - x^2/2

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \[{\frac{2}{5} \left ( -x \right ) ^{{\frac{5}{2}}}}-{\frac{{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x)^(1/2)*(x+(-x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.717931, size = 18, normalized size = 0.95 \[ \frac{2}{5} \, \left (-x\right )^{\frac{5}{2}} - \frac{1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(-x)^(5/2) - 1/2*x^2

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Fricas [A]  time = 0.260425, size = 22, normalized size = 1.16 \[ \frac{2}{5} \, \sqrt{-x} x^{2} - \frac{1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*sqrt(-x)*x^2 - 1/2*x^2

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Sympy [A]  time = 0.478577, size = 14, normalized size = 0.74 \[ \frac{2 i x^{\frac{5}{2}}}{5} - \frac{x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I*x**(5/2)/5 - x**2/2

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GIAC/XCAS [A]  time = 0.284895, size = 22, normalized size = 1.16 \[ \frac{2}{5} \, \sqrt{-x} x^{2} - \frac{1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*sqrt(-x)*x^2 - 1/2*x^2