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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2617, 2615, 6865, 6874, 209, 1602, 2045, 2033, 2046, 1160, 455, 52, 65} \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\frac {32 \sqrt {2} \sqrt {x^2-x} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{3 \sqrt {x-1} \sqrt {x}}+\frac {44}{3} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )-\frac {32}{3} \sqrt {2} \arctan \left (2 \sqrt {2} \sqrt {x}\right )+\frac {4 \sqrt {x^2-x}}{3 x^{3/2}}-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}-\frac {16}{3 \sqrt {x}} \]

[In]

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

[Out]

-16/(3*Sqrt[x]) + (4*Sqrt[-x + x^2])/(3*x^(3/2)) + (32*Sqrt[2]*Sqrt[-x + x^2]*ArcTan[(2*Sqrt[2]*Sqrt[-1 + x])/
3])/(3*Sqrt[-1 + x]*Sqrt[x]) - (32*Sqrt[2]*ArcTan[2*Sqrt[2]*Sqrt[x]])/3 + (44*ArcTan[Sqrt[x]/Sqrt[-x + x^2]])/
3 - (2*Log[-1 + 4*x + 4*Sqrt[-x + x^2]])/(3*x^(3/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 1160

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(b*x^2 + c*x^4)^FracPart
[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracPart[p]), Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x], x] /; FreeQ[{b,
c, d, e, p, q}, x] &&  !IntegerQ[p]

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 2033

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rule 2045

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b*
x^n)^p/(c*(m + j*p + 1))), x] - Dist[b*p*((n - j)/(c^n*(m + j*p + 1))), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2615

Int[Log[(d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]]*((g_.)*(x_))^(m_.), x_Symbol] :> S
imp[(g*x)^(m + 1)*(Log[d + e*x + f*Sqrt[a + b*x + c*x^2]]/(g*(m + 1))), x] + Dist[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 2617

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] && QuadraticQ[u, x] &&  !QuadraticMatchQ[u, x] && (EqQ[v, 1] || MatchQ[v, ((g_.)*
x)^(m_.) /; FreeQ[{g, m}, x]])

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x^{5/2}} \, dx \\ & = -\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {16}{3} \int \frac {1}{x^{3/2} \left (-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )\right )} \, dx \\ & = -\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {32}{3} \text {Subst}\left (\int \frac {1}{x^2 \left (-4 \left (1+2 x^2\right ) \sqrt {-x^2+x^4}+8 \left (-x^2+x^4\right )\right )} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {32}{3} \text {Subst}\left (\int \left (-\frac {1}{2 x^2}+\frac {4}{1+8 x^2}-\frac {x^2}{12 \sqrt {-x^2+x^4}}+\frac {\sqrt {-x^2+x^4}}{4 x^4}-\frac {5 \sqrt {-x^2+x^4}}{4 x^2}+\frac {32 \sqrt {-x^2+x^4}}{3 \left (1+8 x^2\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {16}{3 \sqrt {x}}-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}+\frac {8}{9} \text {Subst}\left (\int \frac {x^2}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {8}{3} \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{x^4} \, dx,x,\sqrt {x}\right )+\frac {40}{3} \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{x^2} \, dx,x,\sqrt {x}\right )-\frac {128}{3} \text {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )-\frac {1024}{9} \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{1+8 x^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {128 \sqrt {-x+x^2}}{9 \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {4}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {40}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {\left (1024 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {x \sqrt {-1+x^2}}{1+8 x^2} \, dx,x,\sqrt {x}\right )}{9 \sqrt {-1+x} \sqrt {x}} \\ & = -\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {128 \sqrt {-x+x^2}}{9 \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}+\frac {4}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )+\frac {40}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {\left (512 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x}}{1+8 x} \, dx,x,x\right )}{9 \sqrt {-1+x} \sqrt {x}} \\ & = -\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \tan ^{-1}\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}}+\frac {\left (64 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} (1+8 x)} \, dx,x,x\right )}{\sqrt {-1+x} \sqrt {x}} \\ & = -\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \tan ^{-1}\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}}+\frac {\left (128 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{9+8 x^2} \, dx,x,\sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}} \\ & = -\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {32 \sqrt {2} \sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{3 \sqrt {-1+x} \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \tan ^{-1}\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.85 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=-\frac {2 \left (8 \sqrt {-(-1+x)^2} x-2 \sqrt {-(-1+x)^2} \sqrt {(-1+x) x}+8 \sqrt {2-2 x} x \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}-i \sqrt {x}}{3 \sqrt {-1+x}}\right )+8 \sqrt {2-2 x} x \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}+i \sqrt {x}}{3 \sqrt {-1+x}}\right )+24 \sqrt {1-x} x \sqrt {(-1+x) x} \arctan \left (\sqrt {-1+x}\right )+16 \sqrt {2} \sqrt {-(-1+x)^2} x^{3/2} \arctan \left (2 \sqrt {2} \sqrt {x}\right )-2 \sqrt {-1+x} x \sqrt {(-1+x) x} \text {arctanh}\left (\sqrt {1-x}\right )+\sqrt {-(-1+x)^2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )\right )}{3 \sqrt {-(-1+x)^2} x^{3/2}} \]

[In]

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

[Out]

(-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))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=-\frac {2 \, {\left (16 \, \sqrt {2} x^{2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + 16 \, \sqrt {2} x^{2} \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 22 \, x^{2} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^{2} - x}}\right ) + 8 \, x^{\frac {3}{2}} + \sqrt {x} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - 2 \, \sqrt {x^{2} - x} \sqrt {x}\right )}}{3 \, x^{2}} \]

[In]

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

[Out]

-2/3*(16*sqrt(2)*x^2*arctan(2*sqrt(2)*sqrt(x)) + 16*sqrt(2)*x^2*arctan(3/4*sqrt(2)*sqrt(x)/sqrt(x^2 - x)) - 22
*x^2*arctan(sqrt(x)/sqrt(x^2 - 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. \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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)

none

Time = 0.40 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.20 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx=\frac {22}{3} \, \pi - \frac {16}{3} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} - \frac {32}{3} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + \frac {8 \, {\left (\sqrt {x - 1} - \sqrt {x} - \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}}{3 \, {\left ({\left (\sqrt {x - 1} - \sqrt {x} - \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}^{2} + 4\right )}} - \frac {16}{3 \, \sqrt {x}} - \frac {2 \, \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right )}{3 \, x^{\frac {3}{2}}} - \frac {44}{3} \, \arctan \left (\frac {{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right ) \]

[In]

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

[Out]

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) - 4
4/3*arctan(1/2*((sqrt(x - 1) - sqrt(x))^2 - 1)/(sqrt(x - 1) - sqrt(x)))

Mupad [F(-1)]

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

[In]

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

[Out]

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