### 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 = 0.0054953, 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, Rules used = {43} $\frac{2}{5} (x-1)^{5/2}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1}$

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{-1+x}} \, dx &=\int \left (\frac{1}{\sqrt{-1+x}}+2 \sqrt{-1+x}+(-1+x)^{3/2}\right ) \, dx\\ &=2 \sqrt{-1+x}+\frac{4}{3} (-1+x)^{3/2}+\frac{2}{5} (-1+x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0055076, 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 = 0.003, size = 18, normalized size = 0.6 \begin{align*}{\frac{6\,{x}^{2}+8\,x+16}{15}\sqrt{-1+x}} \end{align*}

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 = 0.925678, size = 30, normalized size = 0.94 \begin{align*} \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-1+x)^(1/2),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 = 1.79603, size = 49, normalized size = 1.53 \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^2 + 4*x + 8)*sqrt(x - 1)

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Sympy [A]  time = 1.07552, size = 76, normalized size = 2.38 \begin{align*} \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{1 - x}}{5} + \frac{8 i x \sqrt{1 - x}}{15} + \frac{16 i \sqrt{1 - x}}{15} & \text{otherwise} \end{cases} \end{align*}

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

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Giac [A]  time = 1.05024, size = 30, normalized size = 0.94 \begin{align*} \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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