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

Optimal. Leaf size=32 \[ \frac{2}{5} (x-1)^{5/2}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]

[Out]

2*Sqrt[-1 + x] + (4*(-1 + x)^(3/2))/3 + (2*(-1 + x)^(5/2))/5

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Rubi [A]  time = 0.0180979, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{5} (x-1)^{5/2}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[-1 + x],x]

[Out]

2*Sqrt[-1 + x] + (4*(-1 + x)^(3/2))/3 + (2*(-1 + x)^(5/2))/5

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Rubi in Sympy [A]  time = 1.28766, size = 27, normalized size = 0.84 \[ \frac{2 \left (x - 1\right )^{\frac{5}{2}}}{5} + \frac{4 \left (x - 1\right )^{\frac{3}{2}}}{3} + 2 \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x - 1)**(5/2)/5 + 4*(x - 1)**(3/2)/3 + 2*sqrt(x - 1)

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Mathematica [A]  time = 0.00798838, size = 21, normalized size = 0.66 \[ \frac{2}{15} \sqrt{x-1} \left (3 x^2+4 x+8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[-1 + x],x]

[Out]

(2*Sqrt[-1 + x]*(8 + 4*x + 3*x^2))/15

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Maple [A]  time = 0.006, size = 18, normalized size = 0.6 \[{\frac{6\,{x}^{2}+8\,x+16}{15}\sqrt{-1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-1+x)^(1/2),x)

[Out]

2/15*(-1+x)^(1/2)*(3*x^2+4*x+8)

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Maxima [A]  time = 1.35459, size = 30, normalized size = 0.94 \[ \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(x - 1),x, algorithm="maxima")

[Out]

2/5*(x - 1)^(5/2) + 4/3*(x - 1)^(3/2) + 2*sqrt(x - 1)

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Fricas [A]  time = 0.226445, size = 23, normalized size = 0.72 \[ \frac{2}{15} \,{\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(x - 1),x, algorithm="fricas")

[Out]

2/15*(3*x^2 + 4*x + 8)*sqrt(x - 1)

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Sympy [A]  time = 1.95417, size = 76, normalized size = 2.38 \[ \begin{cases} \frac{2 x^{2} \sqrt{x - 1}}{5} + \frac{8 x \sqrt{x - 1}}{15} + \frac{16 \sqrt{x - 1}}{15} & \text{for}\: \left |{x}\right | > 1 \\\frac{2 i x^{2} \sqrt{- x + 1}}{5} + \frac{8 i x \sqrt{- x + 1}}{15} + \frac{16 i \sqrt{- x + 1}}{15} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-1+x)**(1/2),x)

[Out]

Piecewise((2*x**2*sqrt(x - 1)/5 + 8*x*sqrt(x - 1)/15 + 16*sqrt(x - 1)/15, Abs(x)
> 1), (2*I*x**2*sqrt(-x + 1)/5 + 8*I*x*sqrt(-x + 1)/15 + 16*I*sqrt(-x + 1)/15,
True))

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GIAC/XCAS [A]  time = 0.207487, size = 30, normalized size = 0.94 \[ \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(x - 1),x, algorithm="giac")

[Out]

2/5*(x - 1)^(5/2) + 4/3*(x - 1)^(3/2) + 2*sqrt(x - 1)