3.3.8 \(\int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 \left ((x-2)^2\right )^{3/4} (2 x-7)}{3 (x-2)} \]

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.57, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {640, 609} \begin {gather*} \frac {2 (2-x)}{\sqrt [4]{x^2-4 x+4}}+\frac {4}{3} \left (x^2-4 x+4\right )^{3/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + 2*x)/(4 - 4*x + x^2)^(1/4),x]

[Out]

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rule 640

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 (x-2) (2 x-7)}{3 \sqrt [4]{(x-2)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 2*x)/(4 - 4*x + x^2)^(1/4),x]

[Out]

(2*(-2 + x)*(-7 + 2*x))/(3*((-2 + x)^2)^(1/4))

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IntegrateAlgebraic [A]  time = 4.46, size = 39, normalized size = 1.70 \begin {gather*} \frac {2 \left (-3 \sqrt {-2+x}+2 (-2+x)^{3/2}\right ) \left ((-2+x)^2\right )^{3/4}}{3 (-2+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-5 + 2*x)/(4 - 4*x + x^2)^(1/4),x]

[Out]

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

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fricas [A]  time = 0.45, size = 17, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, {\left (x^{2} - 4 \, x + 4\right )}^{\frac {1}{4}} {\left (2 \, x - 7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(x^2-4*x+4)^(1/4),x, algorithm="fricas")

[Out]

2/3*(x^2 - 4*x + 4)^(1/4)*(2*x - 7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x - 5}{{\left (x^{2} - 4 \, x + 4\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(x^2-4*x+4)^(1/4),x, algorithm="giac")

[Out]

integrate((2*x - 5)/(x^2 - 4*x + 4)^(1/4), x)

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maple [A]  time = 0.08, size = 18, normalized size = 0.78

method	result	size
risch	\(\frac {2 \left (-2+x \right ) \left (-7+2 x \right )}{3 \left (\left (-2+x \right )^{2}\right )^{\frac {1}{4}}}\)	\(18\)
gosper	\(\frac {2 \left (-2+x \right ) \left (-7+2 x \right )}{3 \left (x^{2}-4 x +4\right )^{\frac {1}{4}}}\)	\(21\)
meijerg	\(\frac {5 \sqrt {2}\, \sqrt {-\mathrm {signum}\left (-2+x \right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-\frac {x}{2}}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-2+x \right )}}+\frac {4 \sqrt {2}\, \sqrt {-\mathrm {signum}\left (-2+x \right )}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (8+2 x \right ) \sqrt {1-\frac {x}{2}}}{6}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-2+x \right )}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5+2*x)/(x^2-4*x+4)^(1/4),x,method=_RETURNVERBOSE)

[Out]

2/3/((-2+x)^2)^(1/4)*(-2+x)*(-7+2*x)

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maxima [A]  time = 0.32, size = 23, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (x^{2} + 2 \, x - 8\right )}}{3 \, \sqrt {x - 2}} - 10 \, \sqrt {x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(x^2-4*x+4)^(1/4),x, algorithm="maxima")

[Out]

4/3*(x^2 + 2*x - 8)/sqrt(x - 2) - 10*sqrt(x - 2)

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mupad [B]  time = 0.11, size = 24, normalized size = 1.04 \begin {gather*} \frac {2\,\left (2\,x-7\right )\,{\left (x^2-4\,x+4\right )}^{3/4}}{3\,\left (x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 5)/(x^2 - 4*x + 4)^(1/4),x)

[Out]

(2*(2*x - 7)*(x^2 - 4*x + 4)^(3/4))/(3*(x - 2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x - 5}{\sqrt [4]{\left (x - 2\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(x**2-4*x+4)**(1/4),x)

[Out]

Integral((2*x - 5)/((x - 2)**2)**(1/4), x)

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