3.21 \(\int \frac{\sqrt [5]{1-2 x+x^2}}{1-x} \, dx\)

Optimal. Leaf size=16 \[ -\frac{5}{2} \sqrt [5]{x^2-2 x+1} \]

[Out]

(-5*(1 - 2*x + x^2)^(1/5))/2

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Rubi [A]  time = 0.0079128, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {643, 629} \[ -\frac{5}{2} \sqrt [5]{x^2-2 x+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x + x^2)^(1/5)/(1 - x),x]

[Out]

(-5*(1 - 2*x + x^2)^(1/5))/2

Rule 643

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0058255, size = 13, normalized size = 0.81 \[ -\frac{5}{2} \sqrt [5]{(x-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x + x^2)^(1/5)/(1 - x),x]

[Out]

(-5*((-1 + x)^2)^(1/5))/2

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{5}{2}\sqrt [5]{{x}^{2}-2\,x+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-2*x+1)^(1/5)/(1-x),x)

[Out]

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

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Maxima [A]  time = 0.967288, size = 9, normalized size = 0.56 \begin{align*} -\frac{5}{2} \,{\left (x - 1\right )}^{\frac{2}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-2*x+1)^(1/5)/(1-x),x, algorithm="maxima")

[Out]

-5/2*(x - 1)^(2/5)

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Fricas [A]  time = 0.418288, size = 38, normalized size = 2.38 \begin{align*} -\frac{5}{2} \,{\left (x^{2} - 2 \, x + 1\right )}^{\frac{1}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-2*x+1)^(1/5)/(1-x),x, algorithm="fricas")

[Out]

-5/2*(x^2 - 2*x + 1)^(1/5)

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Sympy [A]  time = 1.21885, size = 15, normalized size = 0.94 \begin{align*} - \frac{5 \sqrt [5]{x^{2} - 2 x + 1}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-2*x+1)**(1/5)/(1-x),x)

[Out]

-5*(x**2 - 2*x + 1)**(1/5)/2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (x^{2} - 2 \, x + 1\right )}^{\frac{1}{5}}}{x - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-2*x+1)^(1/5)/(1-x),x, algorithm="giac")

[Out]

integrate(-(x^2 - 2*x + 1)^(1/5)/(x - 1), x)