Integrand size = 17, antiderivative size = 193 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}-\frac {5 \arctan \left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}+\frac {5 \arctan \left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}} \]
5/4*arctan(-1+2^(1/2)*(-1+2*x^(1/2))^(1/4))*2^(1/2)+5/4*arctan(1+2^(1/2)*( -1+2*x^(1/2))^(1/4))*2^(1/2)-5/8*ln(1-2^(1/2)*(-1+2*x^(1/2))^(1/4)+(-1+2*x ^(1/2))^(1/2))*2^(1/2)+5/8*ln(1+2^(1/2)*(-1+2*x^(1/2))^(1/4)+(-1+2*x^(1/2) )^(1/2))*2^(1/2)-5/2*(-1+2*x^(1/2))^(1/4)/x^(1/2)-(-1+2*x^(1/2))^(5/4)/x
Time = 0.36 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=\frac {2 \left (2-9 \sqrt {x}\right ) \sqrt [4]{-1+2 \sqrt {x}}+5 \sqrt {2} x \arctan \left (\frac {-1+\sqrt {-1+2 \sqrt {x}}}{\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}}\right )+5 \sqrt {2} x \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}}{1+\sqrt {-1+2 \sqrt {x}}}\right )}{4 x} \]
(2*(2 - 9*Sqrt[x])*(-1 + 2*Sqrt[x])^(1/4) + 5*Sqrt[2]*x*ArcTan[(-1 + Sqrt[ -1 + 2*Sqrt[x]])/(Sqrt[2]*(-1 + 2*Sqrt[x])^(1/4))] + 5*Sqrt[2]*x*ArcTanh[( Sqrt[2]*(-1 + 2*Sqrt[x])^(1/4))/(1 + Sqrt[-1 + 2*Sqrt[x]])])/(4*x)
Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {798, 51, 51, 73, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {\left (2 \sqrt {x}-1\right )^{5/4}}{x^2} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (2 \sqrt {x}-1\right )^{5/4}}{x^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle 2 \left (\frac {5}{4} \int \frac {\sqrt [4]{2 \sqrt {x}-1}}{x}d\sqrt {x}-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{\left (2 \sqrt {x}-1\right )^{3/4} \sqrt {x}}d\sqrt {x}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\int \frac {1}{\frac {x^2}{2}+\frac {1}{2}}d\sqrt [4]{2 \sqrt {x}-1}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {2 (1-x)}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}+\frac {1}{2} \int \frac {2 (x+1)}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\int \frac {1-x}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}+\int \frac {x+1}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\int \frac {1-x}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}+\frac {1}{2} \int \frac {1}{x-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}+\frac {1}{2} \int \frac {1}{x+\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\int \frac {1-x}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}+\frac {\int \frac {1}{-x-1}d\left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-1}d\left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\int \frac {1-x}{x^2+1}d\sqrt [4]{2 \sqrt {x}-1}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt [4]{2 \sqrt {x}-1}}{x-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{x+\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}}{2 \sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{2 \sqrt {x}-1}}{x-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{x+\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}}{2 \sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{2 \sqrt {x}-1}}{x-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}{x+\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1}d\sqrt [4]{2 \sqrt {x}-1}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {5}{4} \left (-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 \sqrt {x}-1}}{\sqrt {x}}-\frac {\log \left (x-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{2 \sqrt {2}}+\frac {\log \left (x+\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{2 \sqrt {2}}\right )-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{2 x}\right )\) |
2*(-1/2*(-1 + 2*Sqrt[x])^(5/4)/x + (5*(-((-1 + 2*Sqrt[x])^(1/4)/Sqrt[x]) - ArcTan[1 - Sqrt[2]*(-1 + 2*Sqrt[x])^(1/4)]/Sqrt[2] + ArcTan[1 + Sqrt[2]*( -1 + 2*Sqrt[x])^(1/4)]/Sqrt[2] - Log[1 - Sqrt[2]*(-1 + 2*Sqrt[x])^(1/4) + x]/(2*Sqrt[2]) + Log[1 + Sqrt[2]*(-1 + 2*Sqrt[x])^(1/4) + x]/(2*Sqrt[2]))) /4)
3.3.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.44
| method | result | size |
| meijerg | \(\frac {5 \operatorname {signum}\left (-1+2 \sqrt {x}\right )^{\frac {5}{4}} \left (-\frac {2 \Gamma \left (\frac {3}{4}\right )}{5 x}+\frac {2 \Gamma \left (\frac {3}{4}\right )}{\sqrt {x}}+\frac {\left (-2 \ln \left (2\right )+\frac {\pi }{2}-\frac {3}{2}+\frac {\ln \left (x \right )}{2}+i \pi \right ) \Gamma \left (\frac {3}{4}\right )}{2}+\frac {\Gamma \left (\frac {3}{4}\right ) \sqrt {x}\, {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {7}{4};2,4;2 \sqrt {x}\right )}{4}\right )}{2 \Gamma \left (\frac {3}{4}\right ) \left (-\operatorname {signum}\left (-1+2 \sqrt {x}\right )\right )^{\frac {5}{4}}}\) | \(85\) |
| derivativedivides | \(\frac {-\frac {9 \left (-1+2 \sqrt {x}\right )^{\frac {5}{4}}}{4}-\frac {5 \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}}{4}}{x}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}{1-\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}\right )+2 \arctan \left (1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )+2 \arctan \left (-1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )\right )}{8}\) | \(125\) |
| default | \(\frac {-\frac {9 \left (-1+2 \sqrt {x}\right )^{\frac {5}{4}}}{4}-\frac {5 \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}}{4}}{x}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}{1-\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}\right )+2 \arctan \left (1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )+2 \arctan \left (-1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )\right )}{8}\) | \(125\) |
5/2/GAMMA(3/4)*signum(-1+2*x^(1/2))^(5/4)/(-signum(-1+2*x^(1/2)))^(5/4)*(- 2/5*GAMMA(3/4)/x+2*GAMMA(3/4)/x^(1/2)+1/2*(-2*ln(2)+1/2*Pi-3/2+1/2*ln(x)+I *Pi)*GAMMA(3/4)+1/4*GAMMA(3/4)*x^(1/2)*hypergeom([1,1,7/4],[2,4],2*x^(1/2) ))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=\frac {\left (5 i + 5\right ) \, \sqrt {2} x \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right ) - \left (5 i - 5\right ) \, \sqrt {2} x \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right ) + \left (5 i - 5\right ) \, \sqrt {2} x \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right ) - \left (5 i + 5\right ) \, \sqrt {2} x \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (9 \, \sqrt {x} - 2\right )} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}}{8 \, x} \]
1/8*((5*I + 5)*sqrt(2)*x*log((I + 1)*sqrt(2) + 2*(2*sqrt(x) - 1)^(1/4)) - (5*I - 5)*sqrt(2)*x*log(-(I - 1)*sqrt(2) + 2*(2*sqrt(x) - 1)^(1/4)) + (5*I - 5)*sqrt(2)*x*log((I - 1)*sqrt(2) + 2*(2*sqrt(x) - 1)^(1/4)) - (5*I + 5) *sqrt(2)*x*log(-(I + 1)*sqrt(2) + 2*(2*sqrt(x) - 1)^(1/4)) - 4*(9*sqrt(x) - 2)*(2*sqrt(x) - 1)^(1/4))/x
Result contains complex when optimal does not.
Time = 2.75 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=- \frac {4 \cdot \sqrt [4]{2} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{2 \sqrt {x}}} \right )}}{x^{\frac {3}{8}} \Gamma \left (\frac {7}{4}\right )} \]
-4*2**(1/4)*gamma(3/4)*hyper((-5/4, 3/4), (7/4,), exp_polar(2*I*pi)/(2*sqr t(x)))/(x**(3/8)*gamma(7/4))
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {5}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {9 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {5}{4}} + 5 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, \sqrt {x} - 1\right )}^{2} + 4 \, \sqrt {x} - 1} \]
5/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(2*sqrt(x) - 1)^(1/4))) + 5/4* sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(2*sqrt(x) - 1)^(1/4))) + 5/8*sqr t(2)*log(sqrt(2)*(2*sqrt(x) - 1)^(1/4) + sqrt(2*sqrt(x) - 1) + 1) - 5/8*sq rt(2)*log(-sqrt(2)*(2*sqrt(x) - 1)^(1/4) + sqrt(2*sqrt(x) - 1) + 1) - (9*( 2*sqrt(x) - 1)^(5/4) + 5*(2*sqrt(x) - 1)^(1/4))/((2*sqrt(x) - 1)^2 + 4*sqr t(x) - 1)
Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {5}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {9 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {5}{4}} + 5 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}}{4 \, x} \]
5/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(2*sqrt(x) - 1)^(1/4))) + 5/4* sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(2*sqrt(x) - 1)^(1/4))) + 5/8*sqr t(2)*log(sqrt(2)*(2*sqrt(x) - 1)^(1/4) + sqrt(2*sqrt(x) - 1) + 1) - 5/8*sq rt(2)*log(-sqrt(2)*(2*sqrt(x) - 1)^(1/4) + sqrt(2*sqrt(x) - 1) + 1) - 1/4* (9*(2*sqrt(x) - 1)^(5/4) + 5*(2*sqrt(x) - 1)^(1/4))/x
Time = 1.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.40 \[ \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx=-\frac {5\,{\left (2\,\sqrt {x}-1\right )}^{1/4}}{4\,x}-\frac {9\,{\left (2\,\sqrt {x}-1\right )}^{5/4}}{4\,x}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (2\,\sqrt {x}-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{4}+\frac {5}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (2\,\sqrt {x}-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{4}-\frac {5}{4}{}\mathrm {i}\right ) \]