\(\int \frac {\log (-1+4 x+4 \sqrt {(-1+x) x})}{x^{5/2}} \, dx\) [112]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=-\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {32 \sqrt {2} \sqrt {-x+x^2} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{3 \sqrt {-1+x} \sqrt {x}}-\frac {32}{3} \sqrt {2} \arctan \left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \arctan \left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}} \] Output:

-16/3/x^(1/2)+4/3*(x^2-x)^(1/2)/x^(3/2)+32/3*(x^2-x)^(1/2)*arctan(2/3*2^(1 
/2)*(-1+x)^(1/2))*2^(1/2)/(-1+x)^(1/2)/x^(1/2)-32/3*arctan(2*2^(1/2)*x^(1/ 
2))*2^(1/2)+44/3*arctan(x^(1/2)/(x^2-x)^(1/2))-2/3*ln(-1+4*x+4*(x^2-x)^(1/ 
2))/x^(3/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.09 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.85 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=-\frac {2 \left (8 \sqrt {-(-1+x)^2} x-2 \sqrt {-(-1+x)^2} \sqrt {(-1+x) x}+8 \sqrt {2-2 x} x \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}-i \sqrt {x}}{3 \sqrt {-1+x}}\right )+8 \sqrt {2-2 x} x \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}+i \sqrt {x}}{3 \sqrt {-1+x}}\right )+24 \sqrt {1-x} x \sqrt {(-1+x) x} \arctan \left (\sqrt {-1+x}\right )+16 \sqrt {2} \sqrt {-(-1+x)^2} x^{3/2} \arctan \left (2 \sqrt {2} \sqrt {x}\right )-2 \sqrt {-1+x} x \sqrt {(-1+x) x} \text {arctanh}\left (\sqrt {1-x}\right )+\sqrt {-(-1+x)^2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )\right )}{3 \sqrt {-(-1+x)^2} x^{3/2}} \] Input:

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

Output:

(-2*(8*Sqrt[-(-1 + x)^2]*x - 2*Sqrt[-(-1 + x)^2]*Sqrt[(-1 + x)*x] + 8*Sqrt 
[2 - 2*x]*x*Sqrt[(-1 + x)*x]*ArcTan[(2*Sqrt[2] - I*Sqrt[x])/(3*Sqrt[-1 + x 
])] + 8*Sqrt[2 - 2*x]*x*Sqrt[(-1 + x)*x]*ArcTan[(2*Sqrt[2] + I*Sqrt[x])/(3 
*Sqrt[-1 + x])] + 24*Sqrt[1 - x]*x*Sqrt[(-1 + x)*x]*ArcTan[Sqrt[-1 + x]] + 
 16*Sqrt[2]*Sqrt[-(-1 + x)^2]*x^(3/2)*ArcTan[2*Sqrt[2]*Sqrt[x]] - 2*Sqrt[- 
1 + x]*x*Sqrt[(-1 + x)*x]*ArcTanh[Sqrt[1 - x]] + Sqrt[-(-1 + x)^2]*Log[-1 
+ 4*x + 4*Sqrt[(-1 + x)*x]]))/(3*Sqrt[-(-1 + x)^2]*x^(3/2))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3017, 3015, 27, 2035, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3017

\(\displaystyle \int \frac {\log \left (4 \sqrt {x^2-x}+4 x-1\right )}{x^{5/2}}dx\)

\(\Big \downarrow \) 3015

\(\displaystyle -\frac {16}{3} \int -\frac {1}{4 x^{3/2} \left (\sqrt {x^2-x} (2 x+1)+2 \left (x-x^2\right )\right )}dx-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{3} \int \frac {1}{x^{3/2} \left (\sqrt {x^2-x} (2 x+1)+2 \left (x-x^2\right )\right )}dx-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {8}{3} \int \frac {1}{x \left (\sqrt {x^2-x} (2 x+1)+2 \left (x-x^2\right )\right )}d\sqrt {x}-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {8}{3} \int \left (\frac {x}{3 \sqrt {x^2-x}}+\frac {128 \sqrt {x^2-x}}{3 (-8 x-1)}-\frac {16}{8 x+1}+\frac {5 \sqrt {x^2-x}}{x}+\frac {2}{x}-\frac {\sqrt {x^2-x}}{x^2}\right )d\sqrt {x}-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8}{3} \left (\frac {4 \sqrt {2} \sqrt {x^2-x} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{\sqrt {x-1} \sqrt {x}}+\frac {11}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )-4 \sqrt {2} \arctan \left (2 \sqrt {2} \sqrt {x}\right )+\frac {\sqrt {x^2-x}}{2 x^{3/2}}-\frac {2}{\sqrt {x}}\right )-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}\)

Input:

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

Output:

(8*(-2/Sqrt[x] + Sqrt[-x + x^2]/(2*x^(3/2)) + (4*Sqrt[2]*Sqrt[-x + x^2]*Ar 
cTan[(2*Sqrt[2]*Sqrt[-1 + x])/3])/(Sqrt[-1 + x]*Sqrt[x]) - 4*Sqrt[2]*ArcTa 
n[2*Sqrt[2]*Sqrt[x]] + (11*ArcTan[Sqrt[x]/Sqrt[-x + x^2]])/2))/3 - (2*Log[ 
-1 + 4*x + 4*Sqrt[-x + x^2]])/(3*x^(3/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 3015

Int[Log[(d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]] 
*((g_.)*(x_))^(m_.), x_Symbol] :> Simp[(g*x)^(m + 1)*(Log[d + e*x + f*Sqrt[ 
a + b*x + c*x^2]]/(g*(m + 1))), x] + Simp[f^2*((b^2 - 4*a*c)/(2*g*(m + 1))) 
   Int[(g*x)^(m + 1)/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + 
(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, 
 g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]
 

rule 3017

Int[Log[(d_.) + (f_.)*Sqrt[u_] + (e_.)*(x_)]*(v_.), x_Symbol] :> Int[v*Log[ 
d + e*x + f*Sqrt[ExpandToSum[u, x]]], x] /; FreeQ[{d, e, f}, x] && Quadrati 
cQ[u, x] &&  !QuadraticMatchQ[u, x] && (EqQ[v, 1] || MatchQ[v, ((g_.)*x)^(m 
_.) /; FreeQ[{g, m}, x]])
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=-\frac {2 \, {\left (16 \, \sqrt {2} x^{2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) - 16 \, \sqrt {2} x^{2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{2} - x}}{3 \, \sqrt {x}}\right ) + 22 \, x^{2} \arctan \left (\frac {\sqrt {x^{2} - x}}{\sqrt {x}}\right ) + 8 \, x^{\frac {3}{2}} + \sqrt {x} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - 2 \, \sqrt {x^{2} - x} \sqrt {x}\right )}}{3 \, x^{2}} \] Input:

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

Output:

-2/3*(16*sqrt(2)*x^2*arctan(2*sqrt(2)*sqrt(x)) - 16*sqrt(2)*x^2*arctan(2/3 
*sqrt(2)*sqrt(x^2 - x)/sqrt(x)) + 22*x^2*arctan(sqrt(x^2 - x)/sqrt(x)) + 8 
*x^(3/2) + sqrt(x)*log(4*x + 4*sqrt(x^2 - x) - 1) - 2*sqrt(x^2 - x)*sqrt(x 
))/x^2
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

2/3/sqrt(x) - 2/3*log(4*sqrt(x - 1)*sqrt(x) + 4*x - 1)/x^(3/2) - 2/9/x^(3/ 
2) - integrate(1/3*(2*x^2 + x)/(4*x^(11/2) - 5*x^(9/2) + x^(7/2) + 4*(x^5 
- x^4)*sqrt(x - 1)), x) - 1/3*log(sqrt(x) + 1) + 1/3*log(sqrt(x) - 1)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.20 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\frac {22}{3} \, \pi - \frac {16}{3} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} - \frac {32}{3} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + \frac {8 \, {\left (\sqrt {x - 1} - \sqrt {x} - \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}}{3 \, {\left ({\left (\sqrt {x - 1} - \sqrt {x} - \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}^{2} + 4\right )}} - \frac {16}{3 \, \sqrt {x}} - \frac {2 \, \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right )}{3 \, x^{\frac {3}{2}}} - \frac {44}{3} \, \arctan \left (\frac {{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right ) \] Input:

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

Output:

22/3*pi - 16/3*sqrt(2)*(pi - 2*arctan(1/3*sqrt(2)*((sqrt(x - 1) - sqrt(x)) 
^2 - 1)/(sqrt(x - 1) - sqrt(x)))) - 32/3*sqrt(2)*arctan(2*sqrt(2)*sqrt(x)) 
 + 8/3*(sqrt(x - 1) - sqrt(x) - 1/(sqrt(x - 1) - sqrt(x)))/((sqrt(x - 1) - 
 sqrt(x) - 1/(sqrt(x - 1) - sqrt(x)))^2 + 4) - 16/3/sqrt(x) - 2/3*log(4*x 
+ 4*sqrt(x^2 - x) - 1)/x^(3/2) - 44/3*arctan(1/2*((sqrt(x - 1) - sqrt(x))^ 
2 - 1)/(sqrt(x - 1) - sqrt(x)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\int \frac {\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right )}{x^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.48 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\frac {-\frac {88 \mathit {atan} \left (\sqrt {x -1}+\sqrt {x}\right ) x^{2}}{3}+\frac {64 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x -1}+2 \sqrt {x}}{\sqrt {2}}\right ) x^{2}}{3}+\frac {4 \sqrt {x -1}\, x}{3}-\frac {2 \sqrt {x}\, \mathrm {log}\left (4 \sqrt {x}\, \sqrt {x -1}+4 x -1\right )}{3}-\frac {16 \sqrt {x}\, x}{3}}{x^{2}} \] Input:

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

Output:

(2*( - 44*atan(sqrt(x - 1) + sqrt(x))*x**2 + 32*sqrt(2)*atan((2*sqrt(x - 1 
) + 2*sqrt(x))/sqrt(2))*x**2 + 2*sqrt(x - 1)*x - sqrt(x)*log(4*sqrt(x)*sqr 
t(x - 1) + 4*x - 1) - 8*sqrt(x)*x))/(3*x**2)