\(\int \frac {(-1+x)^2 (x-2 x^2+2 x^3)}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} (-2+4 x+3 x^2-4 x^3+2 x^4)} \, dx\) [1222]

Optimal result

Integrand size = 66, antiderivative size = 89 \[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\frac {\sqrt {\frac {1-2 x}{1+2 x^2}} \left (1-x+2 x^2-2 x^3\right )}{3 (-1+2 x)}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {3}}+\frac {x}{\sqrt {3}}}{\sqrt {\frac {1-2 x}{1+2 x^2}}}\right )}{\sqrt {3}} \]

[Out]

((1-2*x)/(2*x^2+1))^(1/2)*(-2*x^3+2*x^2-x+1)/(-3+6*x)-1/3*arctanh((-1/3*3^(1/2)+1/3*x*3^(1/2))/((1-2*x)/(2*x^2
+1))^(1/2))*3^(1/2)

Rubi [F]

\[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx \]

[In]

Int[((-1 + x)^2*(x - 2*x^2 + 2*x^3))/((-1 + 2*x)*Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(-2 + 4*x + 3*x^2 - 4*x^3 + 2*x^4
)),x]

[Out]

1/(3*Sqrt[(1 - 2*x)/(1 + 2*x^2)]) + (4*(1 - 2*x))/(3*(1 + Sqrt[3] - 2*x)*Sqrt[(1 - 2*x)/(1 + 2*x^2)]) - (2 - x
)/(3*Sqrt[(1 - 2*x)/(1 + 2*x^2)]) + ((I/3)*Sqrt[2]*(1 - 2*x)*EllipticE[ArcSin[Sqrt[1 - I*Sqrt[2]*x]/Sqrt[2]],
(-2*Sqrt[2])/(I - Sqrt[2])])/(Sqrt[(1 - 2*x)/(1 + I*Sqrt[2])]*Sqrt[(1 - 2*x)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2]) - (
2*Sqrt[2]*Sqrt[1 - 2*x]*(1 + Sqrt[3] - 2*x)*Sqrt[(1 + 2*x^2)/(1 + Sqrt[3] - 2*x)^2]*EllipticE[2*ArcTan[Sqrt[1
- 2*x]/3^(1/4)], (3 + Sqrt[3])/6])/(3^(3/4)*Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(1 + 2*x^2)) - (Sqrt[2]*(1 - Sqrt[3])*
Sqrt[1 - 2*x]*(1 + Sqrt[3] - 2*x)*Sqrt[(1 + 2*x^2)/(1 + Sqrt[3] - 2*x)^2]*EllipticF[2*ArcTan[Sqrt[1 - 2*x]/3^(
1/4)], (3 + Sqrt[3])/6])/(3*3^(1/4)*Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(1 + 2*x^2)) + (8*Sqrt[2]*Sqrt[1 - 2*x]*Defer[
Subst][Defer[Int][Sqrt[3 - 2*x^2 + x^4]/(3 - 20*x^2 + x^8), x], x, Sqrt[1 - 2*x]])/(3*Sqrt[(1 - 2*x)/(1 + 2*x^
2)]*Sqrt[1 + 2*x^2]) - (5*Sqrt[2]*Sqrt[1 - 2*x]*Defer[Subst][Defer[Int][(x^2*Sqrt[3 - 2*x^2 + x^4])/(3 - 20*x^
2 + x^8), x], x, Sqrt[1 - 2*x]])/(Sqrt[(1 - 2*x)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2]) - (Sqrt[2]*Sqrt[1 - 2*x]*Defer[
Subst][Defer[Int][(x^6*Sqrt[3 - 2*x^2 + x^4])/(3 - 20*x^2 + x^8), x], x, Sqrt[1 - 2*x]])/(3*Sqrt[(1 - 2*x)/(1
+ 2*x^2)]*Sqrt[1 + 2*x^2])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-1+x)^2 x \left (1-2 x+2 x^2\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx \\ & = \frac {\sqrt {1-2 x} \int \frac {(-1+x)^2 x \sqrt {1+2 x^2} \left (1-2 x+2 x^2\right )}{\sqrt {1-2 x} (-1+2 x) \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = -\frac {\sqrt {1-2 x} \int \frac {(-1+x)^2 x \sqrt {1+2 x^2} \left (1-2 x+2 x^2\right )}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = -\frac {\sqrt {1-2 x} \int \left (-\frac {\sqrt {1+2 x^2}}{(1-2 x)^{3/2}}+\frac {x \sqrt {1+2 x^2}}{(1-2 x)^{3/2}}-\frac {\sqrt {1+2 x^2} \left (2-7 x+5 x^2\right )}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )}\right ) \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {\sqrt {1-2 x} \int \frac {\sqrt {1+2 x^2}}{(1-2 x)^{3/2}} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\sqrt {1-2 x} \int \frac {x \sqrt {1+2 x^2}}{(1-2 x)^{3/2}} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\sqrt {1-2 x} \int \frac {\sqrt {1+2 x^2} \left (2-7 x+5 x^2\right )}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{\sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {\sqrt {1-2 x} \int \frac {2+8 x}{\sqrt {1-2 x} \sqrt {1+2 x^2}} \, dx}{6 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\sqrt {1-2 x} \int \left (\frac {2 \sqrt {1+2 x^2}}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )}-\frac {7 x \sqrt {1+2 x^2}}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )}+\frac {5 x^2 \sqrt {1+2 x^2}}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )}\right ) \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (2 \sqrt {1-2 x}\right ) \int \frac {x}{\sqrt {1-2 x} \sqrt {1+2 x^2}} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{\sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {\left (2 \sqrt {1-2 x}\right ) \int \frac {\sqrt {1-2 x}}{\sqrt {1+2 x^2}} \, dx}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\sqrt {1-2 x} \int \frac {\sqrt {1-2 x}}{\sqrt {1+2 x^2}} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (2 \sqrt {1-2 x}\right ) \int \frac {\sqrt {1+2 x^2}}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (5 \sqrt {1-2 x}\right ) \int \frac {x^2 \sqrt {1+2 x^2}}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (7 \sqrt {1-2 x}\right ) \int \frac {x \sqrt {1+2 x^2}}{(1-2 x)^{3/2} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{\sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\left (-1+x^2\right )^2 \sqrt {3-2 x^2+x^4}}{x^2 \left (3-20 x^2+x^8\right )} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{x^2 \left (3-20 x^2+x^8\right )} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\left (-1+x^2\right ) \sqrt {3-2 x^2+x^4}}{x^2 \left (3-20 x^2+x^8\right )} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {(2 i (1-2 x)) \text {Subst}\left (\int \frac {\sqrt {1-\frac {4 i \sqrt {2} x^2}{2+2 i \sqrt {2}}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )}{3 \sqrt {\frac {1-2 x}{2+2 i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {(i (1-2 x)) \text {Subst}\left (\int \frac {\sqrt {1-\frac {4 i \sqrt {2} x^2}{2+2 i \sqrt {2}}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )}{\sqrt {\frac {1-2 x}{2+2 i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{\sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {i \sqrt {2} (1-2 x) E\left (\arcsin \left (\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {2}}{i-\sqrt {2}}\right )}{3 \sqrt {\frac {1-2 x}{1+i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {3-2 x^2+x^4}}{3 x^2}+\frac {\sqrt {3-2 x^2+x^4} \left (14+3 x^2-x^6\right )}{3 \left (3-20 x^2+x^8\right )}\right ) \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {3-2 x^2+x^4}}{3 x^2}+\frac {\sqrt {3-2 x^2+x^4} \left (20-x^6\right )}{3 \left (3-20 x^2+x^8\right )}\right ) \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {3-2 x^2+x^4}}{3 x^2}+\frac {\sqrt {3-2 x^2+x^4} \left (-17+x^6\right )}{3 \left (3-20 x^2+x^8\right )}\right ) \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{\sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {i \sqrt {2} (1-2 x) E\left (\arcsin \left (\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {2}}{i-\sqrt {2}}\right )}{3 \sqrt {\frac {1-2 x}{1+i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{x^2} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4} \left (14+3 x^2-x^6\right )}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{x^2} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4} \left (20-x^6\right )}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{x^2} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4} \left (-17+x^6\right )}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {i \sqrt {2} (1-2 x) E\left (\arcsin \left (\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {2}}{i-\sqrt {2}}\right )}{3 \sqrt {\frac {1-2 x}{1+i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {-2+2 x^2}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \left (\frac {14 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}+\frac {3 x^2 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}-\frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}\right ) \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {-2+2 x^2}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \left (\frac {20 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}-\frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}\right ) \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {-2+2 x^2}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \left (-\frac {17 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}+\frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8}\right ) \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {i \sqrt {2} (1-2 x) E\left (\arcsin \left (\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {2}}{i-\sqrt {2}}\right )}{3 \sqrt {\frac {1-2 x}{1+i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (10 \sqrt {\frac {2}{3}} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {3}}}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (16 \sqrt {\frac {2}{3}} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {3}}}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (28 \sqrt {\frac {2}{3}} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {3}}}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (70 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (160 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (238 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (10 \sqrt {2} \left (1-\sqrt {3}\right ) \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (16 \sqrt {2} \left (1-\sqrt {3}\right ) \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (28 \sqrt {2} \left (1-\sqrt {3}\right ) \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-2 x^2+x^4}} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ & = \frac {1}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {4 (1-2 x)}{3 \left (1+\sqrt {3}-2 x\right ) \sqrt {\frac {1-2 x}{1+2 x^2}}}-\frac {2-x}{3 \sqrt {\frac {1-2 x}{1+2 x^2}}}+\frac {i \sqrt {2} (1-2 x) E\left (\arcsin \left (\frac {\sqrt {1-i \sqrt {2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {2}}{i-\sqrt {2}}\right )}{3 \sqrt {\frac {1-2 x}{1+i \sqrt {2}}} \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {2 \sqrt {2} \sqrt {1-2 x} \left (1+\sqrt {3}-2 x\right ) \sqrt {\frac {1+2 x^2}{\left (1+\sqrt {3}-2 x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt [4]{3}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{3^{3/4} \sqrt {\frac {1-2 x}{1+2 x^2}} \left (1+2 x^2\right )}-\frac {\sqrt {2} \left (1-\sqrt {3}\right ) \sqrt {1-2 x} \left (1+\sqrt {3}-2 x\right ) \sqrt {\frac {1+2 x^2}{\left (1+\sqrt {3}-2 x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt [4]{3}}\right ),\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-2 x}{1+2 x^2}} \left (1+2 x^2\right )}+\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (14 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (5 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{\sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (70 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (160 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (238 \sqrt {2} \sqrt {1-2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {3-2 x^2+x^4}}{3-20 x^2+x^8} \, dx,x,\sqrt {1-2 x}\right )}{3 \sqrt {\frac {1-2 x}{1+2 x^2}} \sqrt {1+2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.85 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.81 \[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\frac {\sqrt {\frac {1-2 x}{1+2 x^2}} \left (-1+x-2 x^2+2 x^3\right )}{3-6 x}-\frac {\text {arctanh}\left (\frac {-1+x}{\sqrt {\frac {3-6 x}{1+2 x^2}}}\right )}{\sqrt {3}} \]

[In]

Integrate[((-1 + x)^2*(x - 2*x^2 + 2*x^3))/((-1 + 2*x)*Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(-2 + 4*x + 3*x^2 - 4*x^3 +
2*x^4)),x]

[Out]

(Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(-1 + x - 2*x^2 + 2*x^3))/(3 - 6*x) - ArcTanh[(-1 + x)/Sqrt[(3 - 6*x)/(1 + 2*x^2)
]]/Sqrt[3]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.49 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.35

method	result	size
trager	\(-\frac {\left (2 x^{2}+1\right ) \left (x -1\right ) \sqrt {-\frac {-1+2 x}{2 x^{2}+1}}}{3 \left (-1+2 x \right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{3}+12 \sqrt {-\frac {-1+2 x}{2 x^{2}+1}}\, x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}-12 \sqrt {-\frac {-1+2 x}{2 x^{2}+1}}\, x^{2}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {-\frac {-1+2 x}{2 x^{2}+1}}\, x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )-6 \sqrt {-\frac {-1+2 x}{2 x^{2}+1}}}{2 x^{4}-4 x^{3}+3 x^{2}+4 x -2}\right )}{6}\)	\(209\)
default	\(\frac {i \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-4 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+4 \textit {\_Z} -2\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {-i \left (i \sqrt {2}+2 x \right )}\, \sqrt {-\frac {-1+2 x}{1+i \sqrt {2}}}\, \sqrt {-i \left (i \sqrt {2}-2 x \right )}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+6 \underline {\hspace {1.25 ex}}\alpha ^{2}-6 \underline {\hspace {1.25 ex}}\alpha -4+i \sqrt {2}\, \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+7\right )\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {-i \left (i \sqrt {2}+2 x \right ) \sqrt {2}}}{2}, \frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3}}{9}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {2 i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3}}{9}-\frac {2 i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {14}{9}+\frac {2 i \underline {\hspace {1.25 ex}}\alpha \sqrt {2}}{3}+\frac {4 i \sqrt {2}}{9}, \sqrt {2}\, \sqrt {\frac {i \sqrt {2}}{1+i \sqrt {2}}}\right )}{\sqrt {-4 x^{3}+2 x^{2}-2 x +1}}\right ) \sqrt {-\left (-1+2 x \right ) \left (2 x^{2}+1\right )}-9 \sqrt {-4 x^{3}+2 x^{2}-2 x +1}\, \sqrt {-\left (-1+2 x \right ) \left (2 x^{2}+1\right )}-18 x^{2}-9}{54 \sqrt {-\frac {-1+2 x}{2 x^{2}+1}}\, \left (2 x^{2}+1\right )}\)	\(314\)

[In]

int((x-1)^2*(2*x^3-2*x^2+x)/(-1+2*x)/((1-2*x)/(2*x^2+1))^(1/2)/(2*x^4-4*x^3+3*x^2+4*x-2),x,method=_RETURNVERBO
SE)

[Out]

-1/3*(2*x^2+1)*(x-1)/(-1+2*x)*(-(-1+2*x)/(2*x^2+1))^(1/2)-1/6*RootOf(_Z^2-3)*ln((2*RootOf(_Z^2-3)*x^4-4*RootOf
(_Z^2-3)*x^3+12*(-(-1+2*x)/(2*x^2+1))^(1/2)*x^3+3*RootOf(_Z^2-3)*x^2-12*(-(-1+2*x)/(2*x^2+1))^(1/2)*x^2-8*Root
Of(_Z^2-3)*x+6*(-(-1+2*x)/(2*x^2+1))^(1/2)*x+4*RootOf(_Z^2-3)-6*(-(-1+2*x)/(2*x^2+1))^(1/2))/(2*x^4-4*x^3+3*x^
2+4*x-2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (78) = 156\).

Time = 0.33 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.19 \[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\frac {\sqrt {3} {\left (2 \, x - 1\right )} \log \left (-\frac {4 \, x^{8} - 16 \, x^{7} + 28 \, x^{6} - 104 \, x^{5} + 209 \, x^{4} - 200 \, x^{3} + 172 \, x^{2} - 4 \, \sqrt {3} {\left (4 \, x^{7} - 12 \, x^{6} + 16 \, x^{5} - 28 \, x^{4} + 31 \, x^{3} - 19 \, x^{2} + 12 \, x - 4\right )} \sqrt {-\frac {2 \, x - 1}{2 \, x^{2} + 1}} - 112 \, x + 28}{4 \, x^{8} - 16 \, x^{7} + 28 \, x^{6} - 8 \, x^{5} - 31 \, x^{4} + 40 \, x^{3} + 4 \, x^{2} - 16 \, x + 4}\right ) - 4 \, {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \sqrt {-\frac {2 \, x - 1}{2 \, x^{2} + 1}}}{12 \, {\left (2 \, x - 1\right )}} \]

[In]

integrate((-1+x)^2*(2*x^3-2*x^2+x)/(-1+2*x)/((1-2*x)/(2*x^2+1))^(1/2)/(2*x^4-4*x^3+3*x^2+4*x-2),x, algorithm="
fricas")

[Out]

1/12*(sqrt(3)*(2*x - 1)*log(-(4*x^8 - 16*x^7 + 28*x^6 - 104*x^5 + 209*x^4 - 200*x^3 + 172*x^2 - 4*sqrt(3)*(4*x
^7 - 12*x^6 + 16*x^5 - 28*x^4 + 31*x^3 - 19*x^2 + 12*x - 4)*sqrt(-(2*x - 1)/(2*x^2 + 1)) - 112*x + 28)/(4*x^8
- 16*x^7 + 28*x^6 - 8*x^5 - 31*x^4 + 40*x^3 + 4*x^2 - 16*x + 4)) - 4*(2*x^3 - 2*x^2 + x - 1)*sqrt(-(2*x - 1)/(
2*x^2 + 1)))/(2*x - 1)

Sympy [F(-1)]

Timed out. \[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((-1+x)**2*(2*x**3-2*x**2+x)/(-1+2*x)/((1-2*x)/(2*x**2+1))**(1/2)/(2*x**4-4*x**3+3*x**2+4*x-2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 2 \, x^{2} + x\right )} {\left (x - 1\right )}^{2}}{{\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} + 4 \, x - 2\right )} {\left (2 \, x - 1\right )} \sqrt {-\frac {2 \, x - 1}{2 \, x^{2} + 1}}} \,d x } \]

[In]

integrate((-1+x)^2*(2*x^3-2*x^2+x)/(-1+2*x)/((1-2*x)/(2*x^2+1))^(1/2)/(2*x^4-4*x^3+3*x^2+4*x-2),x, algorithm="
maxima")

[Out]

integrate((2*x^3 - 2*x^2 + x)*(x - 1)^2/((2*x^4 - 4*x^3 + 3*x^2 + 4*x - 2)*(2*x - 1)*sqrt(-(2*x - 1)/(2*x^2 +
1))), x)

Giac [F]

\[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 2 \, x^{2} + x\right )} {\left (x - 1\right )}^{2}}{{\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} + 4 \, x - 2\right )} {\left (2 \, x - 1\right )} \sqrt {-\frac {2 \, x - 1}{2 \, x^{2} + 1}}} \,d x } \]

[In]

integrate((-1+x)^2*(2*x^3-2*x^2+x)/(-1+2*x)/((1-2*x)/(2*x^2+1))^(1/2)/(2*x^4-4*x^3+3*x^2+4*x-2),x, algorithm="
giac")

[Out]

integrate((2*x^3 - 2*x^2 + x)*(x - 1)^2/((2*x^4 - 4*x^3 + 3*x^2 + 4*x - 2)*(2*x - 1)*sqrt(-(2*x - 1)/(2*x^2 +
1))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(-1+x)^2 \left (x-2 x^2+2 x^3\right )}{(-1+2 x) \sqrt {\frac {1-2 x}{1+2 x^2}} \left (-2+4 x+3 x^2-4 x^3+2 x^4\right )} \, dx=\int \frac {{\left (x-1\right )}^2\,\left (2\,x^3-2\,x^2+x\right )}{\left (2\,x-1\right )\,\sqrt {-\frac {2\,x-1}{2\,x^2+1}}\,\left (2\,x^4-4\,x^3+3\,x^2+4\,x-2\right )} \,d x \]

[In]

int(((x - 1)^2*(x - 2*x^2 + 2*x^3))/((2*x - 1)*(-(2*x - 1)/(2*x^2 + 1))^(1/2)*(4*x + 3*x^2 - 4*x^3 + 2*x^4 - 2
)),x)

[Out]

int(((x - 1)^2*(x - 2*x^2 + 2*x^3))/((2*x - 1)*(-(2*x - 1)/(2*x^2 + 1))^(1/2)*(4*x + 3*x^2 - 4*x^3 + 2*x^4 - 2
)), x)