\(\int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx\) [208]

Optimal result

Integrand size = 18, antiderivative size = 23 \[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\frac {2 \left ((-2+x)^2\right )^{3/4} (-7+2 x)}{3 (-2+x)} \]

[Out]

2*((-2+x)^2)^(3/4)*(-7+2*x)/(-6+3*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {654, 623} \[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\frac {2 (2-x)}{\sqrt [4]{x^2-4 x+4}}+\frac {4}{3} \left (x^2-4 x+4\right )^{3/4} \]

[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 623

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 654

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\frac {2 (-2+x) (-7+2 x)}{3 \sqrt [4]{(-2+x)^2}} \]

[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))

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\frac {2}{3} \, {\left (x^{2} - 4 \, x + 4\right )}^{\frac {1}{4}} {\left (2 \, x - 7\right )} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\frac {4 \, {\left (x^{2} + 2 \, x - 8\right )}}{3 \, \sqrt {x - 2}} - 10 \, \sqrt {x - 2} \]

[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)

Giac [F]

\[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\int { \frac {2 \, x - 5}{{\left (x^{2} - 4 \, x + 4\right )}^{\frac {1}{4}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-5+2 x}{\sqrt [4]{4-4 x+x^2}} \, dx=\frac {2\,\left (2\,x-7\right )\,{\left (x^2-4\,x+4\right )}^{3/4}}{3\,\left (x-2\right )} \]

[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))