Integrand size = 51, antiderivative size = 223 \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\frac {(-1-2 x) \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{2 (1+x)}+\frac {\sqrt [4]{\frac {-1+x}{1+2 x}} (1+2 x)}{6 (1+x)}-\frac {1}{72} \sqrt {24420+55819 \sqrt {2}} \arctan \left (\frac {\sqrt [4]{\frac {-1+x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {2}{9} \arctan \left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{-1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right )+\frac {1}{72} \sqrt {-24420+55819 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {-1+x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {10}{9} \text {arctanh}\left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right ) \]
(-1-2*x)*((-1+x)/(1+2*x))^(3/4)/(2+2*x)+((-1+x)/(1+2*x))^(1/4)*(1+2*x)/(6+ 6*x)-1/72*(24420+55819*2^(1/2))^(1/2)*arctan(1/2*((-1+x)/(1+2*x))^(1/4)*2^ (3/4))+2/9*arctan(2*((-1+x)/(1+2*x))^(1/4)/(-1+2*((-1+x)/(1+2*x))^(1/2)))+ 1/72*(-24420+55819*2^(1/2))^(1/2)*arctanh(1/2*((-1+x)/(1+2*x))^(1/4)*2^(3/ 4))+10/9*arctanh(2*((-1+x)/(1+2*x))^(1/4)/(1+2*((-1+x)/(1+2*x))^(1/2)))
Time = 0.68 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {-12 \sqrt [4]{\frac {-1+x}{1+2 x}}-24 x \sqrt [4]{\frac {-1+x}{1+2 x}}+36 \left (\frac {-1+x}{1+2 x}\right )^{3/4}+72 x \left (\frac {-1+x}{1+2 x}\right )^{3/4}+37 \sqrt [4]{2} \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+165\ 2^{3/4} \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+37 \sqrt [4]{2} x \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+165\ 2^{3/4} x \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+16 \arctan \left (\frac {-\frac {1}{2}+\sqrt {\frac {-1+x}{1+2 x}}}{\sqrt [4]{\frac {-1+x}{1+2 x}}}\right )+16 x \arctan \left (\frac {-\frac {1}{2}+\sqrt {\frac {-1+x}{1+2 x}}}{\sqrt [4]{\frac {-1+x}{1+2 x}}}\right )-\sqrt [4]{2} \left (-37+165 \sqrt {2}\right ) (1+x) \text {arctanh}\left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )-80 (1+x) \text {arctanh}\left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right )}{72 (1+x)} \]
Integrate[(((-1 + x)/(1 + 2*x))^(1/4) - 3*((-1 + x)/(1 + 2*x))^(3/4))/((-1 + x)*(1 + x)^2*(-1 + 2*x)),x]
-1/72*(-12*((-1 + x)/(1 + 2*x))^(1/4) - 24*x*((-1 + x)/(1 + 2*x))^(1/4) + 36*((-1 + x)/(1 + 2*x))^(3/4) + 72*x*((-1 + x)/(1 + 2*x))^(3/4) + 37*2^(1/ 4)*ArcTan[((-1 + x)/(2 + 4*x))^(1/4)] + 165*2^(3/4)*ArcTan[((-1 + x)/(2 + 4*x))^(1/4)] + 37*2^(1/4)*x*ArcTan[((-1 + x)/(2 + 4*x))^(1/4)] + 165*2^(3/ 4)*x*ArcTan[((-1 + x)/(2 + 4*x))^(1/4)] + 16*ArcTan[(-1/2 + Sqrt[(-1 + x)/ (1 + 2*x)])/((-1 + x)/(1 + 2*x))^(1/4)] + 16*x*ArcTan[(-1/2 + Sqrt[(-1 + x )/(1 + 2*x)])/((-1 + x)/(1 + 2*x))^(1/4)] - 2^(1/4)*(-37 + 165*Sqrt[2])*(1 + x)*ArcTanh[((-1 + x)/(2 + 4*x))^(1/4)] - 80*(1 + x)*ArcTanh[(2*((-1 + x )/(1 + 2*x))^(1/4))/(1 + 2*Sqrt[(-1 + x)/(1 + 2*x)])])/(1 + x)
Time = 1.38 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.71, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {7268, 25, 7276, 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.
| \(\displaystyle \int \frac {\sqrt [4]{\frac {x-1}{2 x+1}}-3 \left (\frac {x-1}{2 x+1}\right )^{3/4}}{(x-1) (x+1)^2 (2 x-1)} \, dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle -4 \int -\frac {\left (1-3 \sqrt {-\frac {1-x}{2 x+1}}\right ) \left (\frac {2 (1-x)}{2 x+1}+1\right )^2}{\left (1-\frac {4 (1-x)}{2 x+1}\right ) \left (\frac {1-x}{2 x+1}+2\right )^2}d\sqrt [4]{-\frac {1-x}{2 x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \int \frac {\left (1-3 \sqrt {-\frac {1-x}{2 x+1}}\right ) \left (\frac {2 (1-x)}{2 x+1}+1\right )^2}{\left (1-\frac {4 (1-x)}{2 x+1}\right ) \left (\frac {1-x}{2 x+1}+2\right )^2}d\sqrt [4]{-\frac {1-x}{2 x+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 4 \int \left (\frac {2-5 \sqrt [4]{-\frac {1-x}{2 x+1}}}{9 \left (2 \sqrt {-\frac {1-x}{2 x+1}}-2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )}+\frac {5 \sqrt [4]{-\frac {1-x}{2 x+1}}+2}{9 \left (2 \sqrt {-\frac {1-x}{2 x+1}}+2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )}-\frac {8 \left (3 \sqrt {-\frac {1-x}{2 x+1}}-1\right )}{9 \left (-\frac {1-x}{2 x+1}-2\right )}+\frac {1-3 \sqrt {-\frac {1-x}{2 x+1}}}{\left (-\frac {1-x}{2 x+1}-2\right )^2}\right )d\sqrt [4]{-\frac {1-x}{2 x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (\frac {2}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )-\frac {3 \left (1+\sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )}{16\ 2^{3/4}}-\frac {1}{18} \arctan \left (1-2 \sqrt [4]{-\frac {1-x}{2 x+1}}\right )+\frac {1}{18} \arctan \left (2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )-\frac {3 \left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )}{16\ 2^{3/4}}+\frac {2}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )-\frac {\sqrt [4]{-\frac {1-x}{2 x+1}} \left (1-3 \sqrt {-\frac {1-x}{2 x+1}}\right )}{8 \left (\frac {1-x}{2 x+1}+2\right )}+\frac {5}{36} \log \left (2 \sqrt {-\frac {1-x}{2 x+1}}-2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )-\frac {5}{36} \log \left (2 \sqrt {-\frac {1-x}{2 x+1}}+2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )\right )\) |
Int[(((-1 + x)/(1 + 2*x))^(1/4) - 3*((-1 + x)/(1 + 2*x))^(3/4))/((-1 + x)* (1 + x)^2*(-1 + 2*x)),x]
-4*(-1/8*((-((1 - x)/(1 + 2*x)))^(1/4)*(1 - 3*Sqrt[-((1 - x)/(1 + 2*x))])) /(2 + (1 - x)/(1 + 2*x)) - (3*(1 + Sqrt[2])*ArcTan[(-((1 - x)/(1 + 2*x)))^ (1/4)/2^(1/4)])/(16*2^(3/4)) + (2*2^(1/4)*(1 + 3*Sqrt[2])*ArcTan[(-((1 - x )/(1 + 2*x)))^(1/4)/2^(1/4)])/9 - ArcTan[1 - 2*(-((1 - x)/(1 + 2*x)))^(1/4 )]/18 + ArcTan[1 + 2*(-((1 - x)/(1 + 2*x)))^(1/4)]/18 + (2*2^(1/4)*(1 - 3* Sqrt[2])*ArcTanh[(-((1 - x)/(1 + 2*x)))^(1/4)/2^(1/4)])/9 - (3*(1 - Sqrt[2 ])*ArcTanh[(-((1 - x)/(1 + 2*x)))^(1/4)/2^(1/4)])/(16*2^(3/4)) + (5*Log[1 - 2*(-((1 - x)/(1 + 2*x)))^(1/4) + 2*Sqrt[-((1 - x)/(1 + 2*x))]])/36 - (5* Log[1 + 2*(-((1 - x)/(1 + 2*x)))^(1/4) + 2*Sqrt[-((1 - x)/(1 + 2*x))]])/36 )
3.26.83.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.07 (sec) , antiderivative size = 1758, normalized size of antiderivative = 7.88
int((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2/(-1+2 *x),x,method=_RETURNVERBOSE)
1/6*(1+2*x)/(1+x)*(-(1-x)/(1+2*x))^(1/4)-1/2*(1+2*x)/(1+x)*(-(1-x)/(1+2*x) )^(3/4)-10/9*ln((8*(-(1-x)/(1+2*x))^(3/4)*x+24*RootOf(18*_Z^2+30*_Z+13)*(- (1-x)/(1+2*x))^(1/2)*x+4*(-(1-x)/(1+2*x))^(3/4)+12*RootOf(18*_Z^2+30*_Z+13 )*(-(1-x)/(1+2*x))^(1/2)+16*(-(1-x)/(1+2*x))^(1/2)*x-24*RootOf(18*_Z^2+30* _Z+13)*(-(1-x)/(1+2*x))^(1/4)*x+8*(-(1-x)/(1+2*x))^(1/2)-12*RootOf(18*_Z^2 +30*_Z+13)*(-(1-x)/(1+2*x))^(1/4)-20*(-(1-x)/(1+2*x))^(1/4)*x-6*RootOf(18* _Z^2+30*_Z+13)*x-10*(-(1-x)/(1+2*x))^(1/4)+15*RootOf(18*_Z^2+30*_Z+13)-6*x +15)/(-1+2*x))-2/3*ln((8*(-(1-x)/(1+2*x))^(3/4)*x+24*RootOf(18*_Z^2+30*_Z+ 13)*(-(1-x)/(1+2*x))^(1/2)*x+4*(-(1-x)/(1+2*x))^(3/4)+12*RootOf(18*_Z^2+30 *_Z+13)*(-(1-x)/(1+2*x))^(1/2)+16*(-(1-x)/(1+2*x))^(1/2)*x-24*RootOf(18*_Z ^2+30*_Z+13)*(-(1-x)/(1+2*x))^(1/4)*x+8*(-(1-x)/(1+2*x))^(1/2)-12*RootOf(1 8*_Z^2+30*_Z+13)*(-(1-x)/(1+2*x))^(1/4)-20*(-(1-x)/(1+2*x))^(1/4)*x-6*Root Of(18*_Z^2+30*_Z+13)*x-10*(-(1-x)/(1+2*x))^(1/4)+15*RootOf(18*_Z^2+30*_Z+1 3)-6*x+15)/(-1+2*x))*RootOf(18*_Z^2+30*_Z+13)+2/3*RootOf(18*_Z^2+30*_Z+13) *ln((8*(-(1-x)/(1+2*x))^(3/4)*x-24*RootOf(18*_Z^2+30*_Z+13)*(-(1-x)/(1+2*x ))^(1/2)*x+4*(-(1-x)/(1+2*x))^(3/4)-12*RootOf(18*_Z^2+30*_Z+13)*(-(1-x)/(1 +2*x))^(1/2)-24*(-(1-x)/(1+2*x))^(1/2)*x+24*RootOf(18*_Z^2+30*_Z+13)*(-(1- x)/(1+2*x))^(1/4)*x-12*(-(1-x)/(1+2*x))^(1/2)+12*RootOf(18*_Z^2+30*_Z+13)* (-(1-x)/(1+2*x))^(1/4)+20*(-(1-x)/(1+2*x))^(1/4)*x+6*RootOf(18*_Z^2+30*_Z+ 13)*x+10*(-(1-x)/(1+2*x))^(1/4)-15*RootOf(18*_Z^2+30*_Z+13)+4*x-10)/(-1...
Time = 0.29 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\frac {{\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} - 24420} \log \left (\sqrt {55819 \, \sqrt {2} - 24420} {\left (165 \, \sqrt {2} + 37\right )} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - {\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} - 24420} \log \left (-\sqrt {55819 \, \sqrt {2} - 24420} {\left (165 \, \sqrt {2} + 37\right )} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - {\left (x + 1\right )} \sqrt {-55819 \, \sqrt {2} - 24420} \log \left ({\left (165 \, \sqrt {2} - 37\right )} \sqrt {-55819 \, \sqrt {2} - 24420} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + {\left (x + 1\right )} \sqrt {-55819 \, \sqrt {2} - 24420} \log \left (-{\left (165 \, \sqrt {2} - 37\right )} \sqrt {-55819 \, \sqrt {2} - 24420} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - 32 \, {\left (x + 1\right )} \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - 32 \, {\left (x + 1\right )} \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + 80 \, {\left (x + 1\right )} \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - 80 \, {\left (x + 1\right )} \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - 72 \, {\left (2 \, x + 1\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} + 24 \, {\left (2 \, x + 1\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{144 \, {\left (x + 1\right )}} \]
integrate((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2 /(-1+2*x),x, algorithm="fricas")
1/144*((x + 1)*sqrt(55819*sqrt(2) - 24420)*log(sqrt(55819*sqrt(2) - 24420) *(165*sqrt(2) + 37) + 53081*((x - 1)/(2*x + 1))^(1/4)) - (x + 1)*sqrt(5581 9*sqrt(2) - 24420)*log(-sqrt(55819*sqrt(2) - 24420)*(165*sqrt(2) + 37) + 5 3081*((x - 1)/(2*x + 1))^(1/4)) - (x + 1)*sqrt(-55819*sqrt(2) - 24420)*log ((165*sqrt(2) - 37)*sqrt(-55819*sqrt(2) - 24420) + 53081*((x - 1)/(2*x + 1 ))^(1/4)) + (x + 1)*sqrt(-55819*sqrt(2) - 24420)*log(-(165*sqrt(2) - 37)*s qrt(-55819*sqrt(2) - 24420) + 53081*((x - 1)/(2*x + 1))^(1/4)) - 32*(x + 1 )*arctan(2*((x - 1)/(2*x + 1))^(1/4) + 1) - 32*(x + 1)*arctan(2*((x - 1)/( 2*x + 1))^(1/4) - 1) + 80*(x + 1)*log(2*sqrt((x - 1)/(2*x + 1)) + 2*((x - 1)/(2*x + 1))^(1/4) + 1) - 80*(x + 1)*log(2*sqrt((x - 1)/(2*x + 1)) - 2*(( x - 1)/(2*x + 1))^(1/4) + 1) - 72*(2*x + 1)*((x - 1)/(2*x + 1))^(3/4) + 24 *(2*x + 1)*((x - 1)/(2*x + 1))^(1/4))/(x + 1)
Timed out. \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {1}{72} \cdot 2^{\frac {1}{4}} {\left (165 \, \sqrt {2} + 37\right )} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - \frac {1}{144} \cdot 2^{\frac {1}{4}} {\left (165 \, \sqrt {2} - 37\right )} \log \left (-\frac {2^{\frac {1}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}\right ) + \frac {3 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2 \, {\left (\frac {x - 1}{2 \, x + 1} - 2\right )}} - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) \]
integrate((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2 /(-1+2*x),x, algorithm="maxima")
-1/72*2^(1/4)*(165*sqrt(2) + 37)*arctan(1/2*2^(3/4)*((x - 1)/(2*x + 1))^(1 /4)) - 1/144*2^(1/4)*(165*sqrt(2) - 37)*log(-(2^(1/4) - ((x - 1)/(2*x + 1) )^(1/4))/(2^(1/4) + ((x - 1)/(2*x + 1))^(1/4))) + 1/2*(3*((x - 1)/(2*x + 1 ))^(3/4) - ((x - 1)/(2*x + 1))^(1/4))/((x - 1)/(2*x + 1) - 2) - 2/9*arctan (2*((x - 1)/(2*x + 1))^(1/4) + 1) - 2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) - 1) + 5/9*log(2*sqrt((x - 1)/(2*x + 1)) + 2*((x - 1)/(2*x + 1))^(1/4) + 1) - 5/9*log(2*sqrt((x - 1)/(2*x + 1)) - 2*((x - 1)/(2*x + 1))^(1/4) + 1)
Time = 0.38 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {1}{72} \, \sqrt {55819 \, \sqrt {2} + 24420} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + \frac {1}{144} \, \sqrt {55819 \, \sqrt {2} - 24420} \log \left (2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - \frac {1}{144} \, \sqrt {55819 \, \sqrt {2} - 24420} \log \left ({\left | -2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} \right |}\right ) + \frac {3 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2 \, {\left (\frac {x - 1}{2 \, x + 1} - 2\right )}} - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) \]
integrate((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2 /(-1+2*x),x, algorithm="giac")
-1/72*sqrt(55819*sqrt(2) + 24420)*arctan(1/2*2^(3/4)*((x - 1)/(2*x + 1))^( 1/4)) + 1/144*sqrt(55819*sqrt(2) - 24420)*log(2^(1/4) + ((x - 1)/(2*x + 1) )^(1/4)) - 1/144*sqrt(55819*sqrt(2) - 24420)*log(abs(-2^(1/4) + ((x - 1)/( 2*x + 1))^(1/4))) + 1/2*(3*((x - 1)/(2*x + 1))^(3/4) - ((x - 1)/(2*x + 1)) ^(1/4))/((x - 1)/(2*x + 1) - 2) - 2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) + 1) - 2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) - 1) + 5/9*log(2*sqrt((x - 1) /(2*x + 1)) + 2*((x - 1)/(2*x + 1))^(1/4) + 1) - 5/9*log(2*sqrt((x - 1)/(2 *x + 1)) - 2*((x - 1)/(2*x + 1))^(1/4) + 1)
Time = 7.05 (sec) , antiderivative size = 1310, normalized size of antiderivative = 5.87 \[ \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\text {Too large to display} \]
int((((x - 1)/(2*x + 1))^(1/4) - 3*((x - 1)/(2*x + 1))^(3/4))/((2*x - 1)*( x - 1)*(x + 1)^2),x)
(8/27 + 10i/81)^(1/2)*atan((8/27 + 10i/81)^(1/2)*((x - 1)/(2*x + 1))^(1/4) *(18/13 - 27i/13))*2i - (8/27 - 10i/81)^(1/2)*atan((8/27 - 10i/81)^(1/2)*( (x - 1)/(2*x + 1))^(1/4)*(18/13 + 27i/13))*2i - (((x - 1)/(2*x + 1))^(1/4) - 3*((x - 1)/(2*x + 1))^(3/4))/((2*x - 2)/(2*x + 1) - 4) - (2^(3/4)*atan( ((2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 - (2^(3/4)*(3 7*2^(1/2) - 330)*((25646402817*2^(3/4))/4 + (2^(3/4)*(37*2^(1/2) - 330)*(4 15942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) - (2^(3/4)*(37*2^(1/2) - 330)*( (700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) - 330)*(3601989*2^(1/2)*(37*2 ^(1/2) - 330) - 634967019*2^(3/4)*((x - 1)/(2*x + 1))^(1/4)))/288))/288))/ 288))/288)*(37*2^(1/2) - 330)*1i)/288 + (2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 + (2^(3/4)*(37*2^(1/2) - 330)*((25646402817*2^(3 /4))/4 - (2^(3/4)*(37*2^(1/2) - 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1) )^(1/4) + (2^(3/4)*(37*2^(1/2) - 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(3 7*2^(1/2) - 330)*(3601989*2^(1/2)*(37*2^(1/2) - 330) + 634967019*2^(3/4)*( (x - 1)/(2*x + 1))^(1/4)))/288))/288))/288))/288)*(37*2^(1/2) - 330)*1i)/2 88)/((11880642501*2^(3/4))/2 + (2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2* x + 1))^(1/4))/4 - (2^(3/4)*(37*2^(1/2) - 330)*((25646402817*2^(3/4))/4 + (2^(3/4)*(37*2^(1/2) - 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) - (2^(3/4)*(37*2^(1/2) - 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) - 330)*(3601989*2^(1/2)*(37*2^(1/2) - 330) - 634967019*2^(3/4)*((x - 1...