3.693 \(\int \sqrt {-x} (\sqrt {-x}+x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-x]*(Sqrt[-x] + x),x]

[Out]

(2*(-x)^(5/2))/5 - x^2/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx &=\int \left (-(-x)^{3/2}-x\right ) \, dx\\ &=\frac {2}{5} (-x)^{5/2}-\frac {x^2}{2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \]

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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fricas [A]  time = 0.44, size = 16, normalized size = 0.84 \[ \frac {2}{5} \, \sqrt {-x} x^{2} - \frac {1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^(1/2)*(x+(-x)^(1/2)),x, algorithm="fricas")

[Out]

2/5*sqrt(-x)*x^2 - 1/2*x^2

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giac [A]  time = 0.31, size = 16, normalized size = 0.84 \[ \frac {2}{5} \, \sqrt {-x} x^{2} - \frac {1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^(1/2)*(x+(-x)^(1/2)),x, algorithm="giac")

[Out]

2/5*sqrt(-x)*x^2 - 1/2*x^2

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maple [A]  time = 0.00, 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]

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 = 0.67, size = 13, normalized size = 0.68 \[ \frac {2}{5} \, \left (-x\right )^{\frac {5}{2}} - \frac {1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^(1/2)*(x+(-x)^(1/2)),x, algorithm="maxima")

[Out]

2/5*(-x)^(5/2) - 1/2*x^2

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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \[ \frac {2\,{\left (-x\right )}^{5/2}}{5}-\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x)^(1/2)*(x + (-x)^(1/2)),x)

[Out]

(2*(-x)^(5/2))/5 - x^2/2

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sympy [C]  time = 0.19, 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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