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

Optimal. Leaf size=19 \[ 2 \log \left (\sqrt{x-4}+\sqrt{x-1}+1\right ) \]

[Out]

2*Log[1 + Sqrt[-4 + x] + Sqrt[-1 + x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.817248, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 66, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ 2 \log \left (\sqrt{x-4}+\sqrt{x-1}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[1 + Sqrt[-4 + x] + Sqrt[-1 + x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.864, size = 17, normalized size = 0.89 \[ 2 \log{\left (\sqrt{x - 4} + \sqrt{x - 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-(-4+x)**(1/2)+x*(-4+x)**(1/2)-4*(-1+x)**(1/2)+x*(-1+x)**(1/2))/(x**2-5*x+4)/(1+(-4+x)**(1/2)+(-1+x)**(1/2)),x)

[Out]

2*log(sqrt(x - 4) + sqrt(x - 1) + 1)

_______________________________________________________________________________________

Mathematica [B]  time = 0.0747189, size = 75, normalized size = 3.95 \[ \frac{1}{2} \log \left (-5 x-4 \sqrt{x-4} \sqrt{x-1}+17\right )+\frac{1}{2} \log \left (-2 x-2 \sqrt{x-4} \sqrt{x-1}+5\right )-\tanh ^{-1}\left (\sqrt{x-4}\right )+\tanh ^{-1}\left (\frac{\sqrt{x-1}}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[Sqrt[-4 + x]] + ArcTanh[Sqrt[-1 + x]/2] + Log[17 - 4*Sqrt[-4 + x]*Sqrt[
-1 + x] - 5*x]/2 + Log[5 - 2*Sqrt[-4 + x]*Sqrt[-1 + x] - 2*x]/2

_______________________________________________________________________________________

Maple [B]  time = 0.074, size = 147, normalized size = 7.7 \[{\frac{\ln \left ( -5+x \right ) }{2}}+{\frac{1}{2}\ln \left ( -1+\sqrt{x-4} \right ) }-{\frac{1}{2}\ln \left ( 1+\sqrt{x-4} \right ) }+{\frac{1}{2}\ln \left ( \sqrt{-1+x}+2 \right ) }-{\frac{1}{2}\ln \left ( \sqrt{-1+x}-2 \right ) }+{\frac{7}{4}\sqrt{x-4}\sqrt{-1+x}{\it Artanh} \left ({\frac{5\,x-17}{4}{\frac{1}{\sqrt{{x}^{2}-5\,x+4}}}} \right ){\frac{1}{\sqrt{{x}^{2}-5\,x+4}}}}+{\frac{1}{4}\sqrt{x-4}\sqrt{-1+x} \left ( 2\,\ln \left ( -5/2+x+\sqrt{{x}^{2}-5\,x+4} \right ) -5\,{\it Artanh} \left ( 1/4\,{\frac{5\,x-17}{\sqrt{{x}^{2}-5\,x+4}}} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-5\,x+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(-5+x)+1/2*ln(-1+(x-4)^(1/2))-1/2*ln(1+(x-4)^(1/2))+1/2*ln((-1+x)^(1/2)+2)
-1/2*ln((-1+x)^(1/2)-2)+7/4*(x-4)^(1/2)*(-1+x)^(1/2)/(x^2-5*x+4)^(1/2)*arctanh(1
/4*(5*x-17)/(x^2-5*x+4)^(1/2))+1/4*(x-4)^(1/2)*(-1+x)^(1/2)*(2*ln(-5/2+x+(x^2-5*
x+4)^(1/2))-5*arctanh(1/4*(5*x-17)/(x^2-5*x+4)^(1/2)))/(x^2-5*x+4)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.771392, size = 127, normalized size = 6.68 \[ \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (\frac{2 \, x^{2} + 2 \,{\left ({\left (x - 1\right )} \sqrt{x - 4} + 2 \, x - 6\right )} \sqrt{x - 1} + 2 \,{\left (2 \, x - 3\right )} \sqrt{x - 4} - 7 \, x + 3}{2 \,{\left ({\left (x - 1\right )} \sqrt{x - 4} + 2 \, x - 6\right )}}\right ) + \frac{1}{2} \, \log \left (\frac{{\left (x - 1\right )} \sqrt{x - 4} + 2 \, x - 6}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x - 1) + 1/2*log(1/2*(2*x^2 + 2*((x - 1)*sqrt(x - 4) + 2*x - 6)*sqrt(x -
1) + 2*(2*x - 3)*sqrt(x - 4) - 7*x + 3)/((x - 1)*sqrt(x - 4) + 2*x - 6)) + 1/2*
log(((x - 1)*sqrt(x - 4) + 2*x - 6)/(x - 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.308698, size = 130, normalized size = 6.84 \[ -\frac{1}{2} \, \log \left (-{\left (4 \, x - 11\right )} \sqrt{x - 1} \sqrt{x - 4} + 4 \, x^{2} - 21 \, x + 23\right ) + \frac{1}{2} \, \log \left (\sqrt{x - 1} \sqrt{x - 4} - x + 7\right ) + \frac{1}{2} \, \log \left (x - 5\right ) + \frac{1}{2} \, \log \left (\sqrt{x - 1} + 2\right ) - \frac{1}{2} \, \log \left (\sqrt{x - 1} - 2\right ) - \frac{1}{2} \, \log \left (\sqrt{x - 4} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{x - 4} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(-(4*x - 11)*sqrt(x - 1)*sqrt(x - 4) + 4*x^2 - 21*x + 23) + 1/2*log(sqrt
(x - 1)*sqrt(x - 4) - x + 7) + 1/2*log(x - 5) + 1/2*log(sqrt(x - 1) + 2) - 1/2*l
og(sqrt(x - 1) - 2) - 1/2*log(sqrt(x - 4) + 1) + 1/2*log(sqrt(x - 4) - 1)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.37642, size = 81, normalized size = 4.26 \[{\rm ln}\left (\sqrt{x - 1} + 2\right ) -{\rm ln}\left ({\left | -\sqrt{x - 1} + \sqrt{x - 4} \right |}\right ) -{\rm ln}\left ({\left | -\sqrt{x - 1} + \sqrt{x - 4} - 1 \right |}\right ) +{\rm ln}\left ({\left | -\sqrt{x - 1} + \sqrt{x - 4} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(sqrt(x - 1) + 2) - ln(abs(-sqrt(x - 1) + sqrt(x - 4))) - ln(abs(-sqrt(x - 1)
+ sqrt(x - 4) - 1)) + ln(abs(-sqrt(x - 1) + sqrt(x - 4) - 3))