\(\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 (\frac {-1+x}{1+2 x})^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx\) [2583]
Optimal result
Integrand size = 51, antiderivative size = 223 \[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\frac {(-1-2 x) \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{2 (1+x)}+\frac {\sqrt [4]{\frac {-1+x}{1+2 x}} (1+2 x)}{6 (1+x)}-\frac {1}{72} \sqrt {24420+55819 \sqrt {2}} \arctan \left (\frac {\sqrt [4]{\frac {-1+x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {2}{9} \arctan \left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{-1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right )+\frac {1}{72} \sqrt {-24420+55819 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {-1+x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {10}{9} \text {arctanh}\left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right )
\]
[Out]
(-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)))
Rubi [A] (verified)
Time = 0.84 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.67, number of
steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6857, 648, 631, 210, 642,
1193, 1181, 213, 209} \[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {8}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )+\frac {3 \left (1+\sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac {2}{9} \arctan \left (1-2 \sqrt [4]{-\frac {1-x}{2 x+1}}\right )-\frac {2}{9} \arctan \left (2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )+\frac {3 \left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {8}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )+\frac {\sqrt [4]{-\frac {1-x}{2 x+1}} (2 x+1) \left (1-3 \sqrt {-\frac {1-x}{2 x+1}}\right )}{6 (x+1)}-\frac {5}{9} \log \left (2 \sqrt {-\frac {1-x}{2 x+1}}-2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )+\frac {5}{9} \log \left (2 \sqrt {-\frac {1-x}{2 x+1}}+2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )
\]
[In]
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]
[Out]
((-((1 - x)/(1 + 2*x)))^(1/4)*(1 + 2*x)*(1 - 3*Sqrt[-((1 - x)/(1 + 2*x))]))/(6*(1 + x)) + (3*(1 + Sqrt[2])*Arc
Tan[(-((1 - x)/(1 + 2*x)))^(1/4)/2^(1/4)])/(4*2^(3/4)) - (8*2^(1/4)*(1 + 3*Sqrt[2])*ArcTan[(-((1 - x)/(1 + 2*x
)))^(1/4)/2^(1/4)])/9 + (2*ArcTan[1 - 2*(-((1 - x)/(1 + 2*x)))^(1/4)])/9 - (2*ArcTan[1 + 2*(-((1 - x)/(1 + 2*x
)))^(1/4)])/9 - (8*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)])/(4*2^(3/4)) - (5*Log[1 - 2*(-((1 - x)/(1 + 2*x)))^(1/4) + 2*Sq
rt[-((1 - x)/(1 + 2*x))]])/9 + (5*Log[1 + 2*(-((1 - x)/(1 + 2*x)))^(1/4) + 2*Sqrt[-((1 - x)/(1 + 2*x))]])/9
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 210
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])
Rule 213
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 631
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]
Rule 648
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 1181
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q))
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]
Rule 1193
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)*((a + c*x^4)^(p + 1)/
(4*a*(p + 1))), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x],
x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Rule 6857
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rubi steps \begin{align*}
\text {integral}& = -\left (4 \text {Subst}\left (\int \frac {\left (-1+3 x^2\right ) \left (1-2 x^4\right )^2}{\left (-2+x^4\right )^2 \left (1+4 x^4\right )} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \left (\frac {-2+5 x}{9 \left (1-2 x+2 x^2\right )}+\frac {-2-5 x}{9 \left (1+2 x+2 x^2\right )}+\frac {-1+3 x^2}{\left (-2+x^4\right )^2}+\frac {8 \left (-1+3 x^2\right )}{9 \left (-2+x^4\right )}\right ) \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\right ) \\ & = -\left (\frac {4}{9} \text {Subst}\left (\int \frac {-2+5 x}{1-2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\right )-\frac {4}{9} \text {Subst}\left (\int \frac {-2-5 x}{1+2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {32}{9} \text {Subst}\left (\int \frac {-1+3 x^2}{-2+x^4} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-4 \text {Subst}\left (\int \frac {-1+3 x^2}{\left (-2+x^4\right )^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right ) \\ & = \frac {\sqrt [4]{-\frac {1-x}{1+2 x}} (1+2 x) \left (1-3 \sqrt {-\frac {1-x}{1+2 x}}\right )}{6 (1+x)}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{1-2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {2}{9} \text {Subst}\left (\int \frac {1}{1+2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {3-3 x^2}{-2+x^4} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {5}{9} \text {Subst}\left (\int \frac {-2+4 x}{1-2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {5}{9} \text {Subst}\left (\int \frac {2+4 x}{1+2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {1}{9} \left (8 \left (6-\sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {1}{9} \left (8 \left (6+\sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right ) \\ & = \frac {\sqrt [4]{-\frac {1-x}{1+2 x}} (1+2 x) \left (1-3 \sqrt {-\frac {1-x}{1+2 x}}\right )}{6 (1+x)}-\frac {8}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )-\frac {8}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )-\frac {5}{9} \log \left (1-2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )+\frac {5}{9} \log \left (1+2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )-\frac {2}{9} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-2 \sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {2}{9} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+2 \sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {1}{8} \left (3 \left (2-\sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {1}{8} \left (3 \left (2+\sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right ) \\ & = \frac {\sqrt [4]{-\frac {1-x}{1+2 x}} (1+2 x) \left (1-3 \sqrt {-\frac {1-x}{1+2 x}}\right )}{6 (1+x)}+\frac {3 \left (1+\sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {8}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {2}{9} \arctan \left (1-2 \sqrt [4]{-\frac {1-x}{1+2 x}}\right )-\frac {2}{9} \arctan \left (1+2 \sqrt [4]{-\frac {1-x}{1+2 x}}\right )-\frac {8}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {3 \left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {5}{9} \log \left (1-2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )+\frac {5}{9} \log \left (1+2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right ) \\
\end{align*}
Mathematica [A] (verified)
Time = 0.68 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.48
\[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {-12 \sqrt [4]{\frac {-1+x}{1+2 x}}-24 x \sqrt [4]{\frac {-1+x}{1+2 x}}+36 \left (\frac {-1+x}{1+2 x}\right )^{3/4}+72 x \left (\frac {-1+x}{1+2 x}\right )^{3/4}+37 \sqrt [4]{2} \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+165\ 2^{3/4} \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+37 \sqrt [4]{2} x \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+165\ 2^{3/4} x \arctan \left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )+16 \arctan \left (\frac {-\frac {1}{2}+\sqrt {\frac {-1+x}{1+2 x}}}{\sqrt [4]{\frac {-1+x}{1+2 x}}}\right )+16 x \arctan \left (\frac {-\frac {1}{2}+\sqrt {\frac {-1+x}{1+2 x}}}{\sqrt [4]{\frac {-1+x}{1+2 x}}}\right )-\sqrt [4]{2} \left (-37+165 \sqrt {2}\right ) (1+x) \text {arctanh}\left (\sqrt [4]{\frac {-1+x}{2+4 x}}\right )-80 (1+x) \text {arctanh}\left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right )}{72 (1+x)}
\]
[In]
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]
[Out]
-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)
Maple [C] (warning: unable to verify)
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
| | |
| method | result | size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(1758\) |
| | |
|
|
|
|
[In]
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)
[Out]
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(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))*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+2*x))-2/3*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*ln((
-291962880*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*(-(1-x)/(1+2*x))^(1/2)*x-145981440*RootOf(4246732
8*_Z^4+225054720*_Z^2-2817592561)^3*(-(1-x)/(1+2*x))^(1/2)+5925856678*(-(1-x)/(1+2*x))^(3/4)*x-1170163776*Root
Of(42467328*_Z^4+225054720*_Z^2-2817592561)*(-(1-x)/(1+2*x))^(1/2)*x+978388992*(-(1-x)/(1+2*x))^(1/4)*RootOf(4
2467328*_Z^4+225054720*_Z^2-2817592561)^2*x-40919040*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*x+29629
28339*(-(1-x)/(1+2*x))^(3/4)-585081888*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*(-(1-x)/(1+2*x))^(1/2)+
489194496*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2-8183808*RootOf(42467328*_Z^
4+225054720*_Z^2-2817592561)^3+2592476040*(-(1-x)/(1+2*x))^(1/4)*x-2318857200*RootOf(42467328*_Z^4+225054720*_
Z^2-2817592561)*x+1296238020*(-(1-x)/(1+2*x))^(1/4)-463771440*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561))
/(1+x))-1/72*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*ln((6082560*RootOf(_Z^2
+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*(-(1-x)/(1+2*x))^(1/2)*RootOf(42467328*_Z^4+225
054720*_Z^2-2817592561)^2*x+3041280*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*
(-(1-x)/(1+2*x))^(1/2)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+5925856678*(-(1-x)/(1+2*x))^(3/4)*x+7
855988*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*(-(1-x)/(1+2*x))^(1/2)*x-9783
88992*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2*x+852480*RootOf(42467328*_Z^4+2
25054720*_Z^2-2817592561)^2*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*x+296292
8339*(-(1-x)/(1+2*x))^(3/4)+3927994*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*
(-(1-x)/(1+2*x))^(1/2)-489194496*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+1704
96*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817
592561)^2+12210)-2592476040*(-(1-x)/(1+2*x))^(1/4)*x-43791825*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*
_Z^2-2817592561)^2+12210)*x-1296238020*(-(1-x)/(1+2*x))^(1/4)-8758365*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+22
5054720*_Z^2-2817592561)^2+12210))/(1+x))
Fricas [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.62
\[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\frac {{\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} - 24420} \log \left (\sqrt {55819 \, \sqrt {2} - 24420} {\left (165 \, \sqrt {2} + 37\right )} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - {\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} - 24420} \log \left (-\sqrt {55819 \, \sqrt {2} - 24420} {\left (165 \, \sqrt {2} + 37\right )} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - {\left (x + 1\right )} \sqrt {-55819 \, \sqrt {2} - 24420} \log \left ({\left (165 \, \sqrt {2} - 37\right )} \sqrt {-55819 \, \sqrt {2} - 24420} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + {\left (x + 1\right )} \sqrt {-55819 \, \sqrt {2} - 24420} \log \left (-{\left (165 \, \sqrt {2} - 37\right )} \sqrt {-55819 \, \sqrt {2} - 24420} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - 32 \, {\left (x + 1\right )} \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - 32 \, {\left (x + 1\right )} \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + 80 \, {\left (x + 1\right )} \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - 80 \, {\left (x + 1\right )} \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - 72 \, {\left (2 \, x + 1\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} + 24 \, {\left (2 \, x + 1\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{144 \, {\left (x + 1\right )}}
\]
[In]
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")
[Out]
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(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(-55819*sqrt(2) - 24420)*log((165*sqrt(2) - 37)*sqrt(-55819*sq
rt(2) - 24420) + 53081*((x - 1)/(2*x + 1))^(1/4)) + (x + 1)*sqrt(-55819*sqrt(2) - 24420)*log(-(165*sqrt(2) - 3
7)*sqrt(-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)
Sympy [F(-1)]
Timed out. \[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\text {Timed out}
\]
[In]
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)
[Out]
Timed out
Maxima [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.09
\[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {1}{72} \cdot 2^{\frac {1}{4}} {\left (165 \, \sqrt {2} + 37\right )} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - \frac {1}{144} \cdot 2^{\frac {1}{4}} {\left (165 \, \sqrt {2} - 37\right )} \log \left (-\frac {2^{\frac {1}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}\right ) + \frac {3 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2 \, {\left (\frac {x - 1}{2 \, x + 1} - 2\right )}} - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right )
\]
[In]
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")
[Out]
-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)
Giac [A] (verification not implemented)
none
Time = 0.38 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.12
\[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=-\frac {1}{72} \, \sqrt {55819 \, \sqrt {2} + 24420} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + \frac {1}{144} \, \sqrt {55819 \, \sqrt {2} - 24420} \log \left (2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - \frac {1}{144} \, \sqrt {55819 \, \sqrt {2} - 24420} \log \left ({\left | -2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} \right |}\right ) + \frac {3 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2 \, {\left (\frac {x - 1}{2 \, x + 1} - 2\right )}} - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right )
\]
[In]
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")
[Out]
-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) - 2
4420)*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)
Mupad [B] (verification not implemented)
Time = 7.05 (sec) , antiderivative size = 1310, normalized size of antiderivative = 5.87
\[
\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx=\text {Too large to display}
\]
[In]
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)
[Out]
(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)*(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)/(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)*(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)/((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)/(2*x + 1))^(1/4)))/288))/288))/288))/288)*(37*2^(1/2) - 330))
/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)*(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))/288))*(37*2^(1/2) - 330)*1i)/144
- (2^(3/4)*atan(((2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 - (2^(3/4)*(37*2^(1/2) + 330)*((2
5646402817*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)*(3
7*2^(1/2) + 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) + 330)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i - 6
34967019*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330))/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)*((70
0891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) + 330)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i + 634967019*2^(3/4)*((x
- 1)/(2*x + 1))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330))/288)/((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)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i - 634967019*2^(3/4)
*((x - 1)/(2*x + 1))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330)*1i)/288 - (2^(3/4)*((139370
28229*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)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i + 634967019*2^(3/4)*((x - 1)/(2*x + 1
))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330)*1i)/288))*(37*2^(1/2) + 330))/144