Integrand size = 29, antiderivative size = 187 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\left (e+f+\sqrt {3} f\right ) \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2+\sqrt {3}} \left (e+\left (1-\sqrt {3}\right ) f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]
-1/3*(1-x)*EllipticF((1-x-3^(1/2))/(1-x+3^(1/2)),I*3^(1/2)+2*I)*(e+f*(1-3^ (1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))*((x^2+x+1)/(1-x+3^(1/2))^2)^(1/2)*3^(1/4 )/(-x^3+1)^(1/2)/((1-x)/(1-x+3^(1/2))^2)^(1/2)-arctan((1-x)*(3+2*3^(1/2))^ (1/2)/(-x^3+1)^(1/2))*(e+f+f*3^(1/2))/(9+6*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 20.52 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-3 i f \sqrt {-i+\sqrt {3}-2 i x} \left (-i \left ((2+i)+\sqrt {3}\right )+\left ((2-i)+\sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \left (\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1-x^3}} \]
(2*Sqrt[2/3]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*f*Sqrt[-I + Sqr t[3] - (2*I)*x]*((-I)*((2 + I) + Sqrt[3]) + ((2 - I) + Sqrt[3])*x)*Ellipti cF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*(Sqrt[3]*e + (3 + Sqrt[3])*f)*Sqrt[I + Sqrt[3] + (2*I)*x] *Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSi n[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[ 3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3 ])
Time = 0.49 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2566, 27, 759, 2565, 216}
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.
| \(\displaystyle \int \frac {e+f x}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}} \, dx\) |
| \(\Big \downarrow \) 2566 |
| \(\displaystyle \frac {1}{12} \left (\frac {e+f}{\sqrt {3}}+f\right ) \int -\frac {6 \left (-x-\sqrt {3}+1\right )}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx-\frac {1}{2} \left (f-\frac {e+f}{\sqrt {3}}\right ) \int \frac {1}{\sqrt {1-x^3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} \left (f-\frac {e+f}{\sqrt {3}}\right ) \int \frac {1}{\sqrt {1-x^3}}dx-\frac {1}{2} \left (\frac {e+f}{\sqrt {3}}+f\right ) \int \frac {-x-\sqrt {3}+1}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (f-\frac {e+f}{\sqrt {3}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {1}{2} \left (\frac {e+f}{\sqrt {3}}+f\right ) \int \frac {-x-\sqrt {3}+1}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle \frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (f-\frac {e+f}{\sqrt {3}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\left (\frac {e+f}{\sqrt {3}}+f\right ) \int \frac {1}{\frac {\left (3+2 \sqrt {3}\right ) (1-x)^2}{1-x^3}+1}d\frac {1-x}{\sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (f-\frac {e+f}{\sqrt {3}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right ) \left (\frac {e+f}{\sqrt {3}}+f\right )}{\sqrt {3+2 \sqrt {3}}}\) |
-(((f + (e + f)/Sqrt[3])*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt[1 - x^3 ]])/Sqrt[3 + 2*Sqrt[3]]) + (Sqrt[2 + Sqrt[3]]*(f - (e + f)/Sqrt[3])*(1 - x )*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])
3.2.28.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d) S ubst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*(x/c))/Sqrt[a + b*x ^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c ^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^ 3), 0]
Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x _Symbol] :> Simp[-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d ^3)) Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/(c*d*(b*c^3 - 28*a* d^3)) Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3) , 0]
Time = 1.47 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(264\) |
| elliptic | \(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(264\) |
2/3*I*f*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I *3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*E llipticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(- 3/2+1/2*I*3^(1/2)))^(1/2))-2/3*I*(-e-f-f*3^(1/2))*3^(1/2)*(I*(x+1/2-1/2*I* 3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2* I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)/(-3/2-3^(1/2)+1/2*I*3^(1/2))*Elli pticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-3/2 -3^(1/2)+1/2*I*3^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.93 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\left [-\frac {1}{3} \, {\left (\sqrt {3} {\left (i \, e + i \, f\right )} - 3 i \, f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{12} \, \sqrt {3 \, e^{2} - 6 \, e f - 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}} \log \left (-\frac {{\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{8} + 16 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{7} + 112 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{6} + 16 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{5} + 112 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{4} - 224 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{3} + 64 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{2} + 4 \, {\left ({\left (2 \, e - f\right )} x^{6} + 18 \, {\left (e - f\right )} x^{5} + 6 \, {\left (7 \, e - 2 \, f\right )} x^{4} + 8 \, {\left (e - 5 \, f\right )} x^{3} + 36 \, f x^{2} - 24 \, {\left (e + f\right )} x + \sqrt {3} {\left ({\left (e - f\right )} x^{6} + 6 \, {\left (2 \, e - f\right )} x^{5} + 6 \, {\left (3 \, e - 4 \, f\right )} x^{4} + 8 \, {\left (2 \, e + f\right )} x^{3} - 12 \, {\left (e + f\right )} x^{2} + 24 \, f x - 8 \, e - 16 \, f\right )} + 8 \, e + 32 \, f\right )} \sqrt {-x^{3} + 1} \sqrt {3 \, e^{2} - 6 \, e f - 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}} + 112 \, e^{2} + 224 \, e f - 224 \, f^{2} - 128 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x + 16 \, \sqrt {3} {\left ({\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{7} + 2 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{6} + 6 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{5} - 5 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{4} + 2 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{3} - 6 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{2} - 4 \, e^{2} - 8 \, e f + 8 \, f^{2} + 4 \, {\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x\right )}}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + 16 \, x^{5} - 56 \, x^{4} - 32 \, x^{3} + 64 \, x^{2} + 64 \, x + 16}\right ), -\frac {1}{3} \, {\left (\sqrt {3} {\left (i \, e + i \, f\right )} - 3 i \, f\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{6} \, \sqrt {-3 \, e^{2} + 6 \, e f + 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}} \arctan \left (\frac {{\left (3 \, f x^{2} - 6 \, {\left (e - f\right )} x - \sqrt {3} {\left ({\left (e + f\right )} x^{2} + 2 \, {\left (2 \, e - f\right )} x - 2 \, e + 4 \, f\right )} + 6 \, e\right )} \sqrt {-x^{3} + 1} \sqrt {-3 \, e^{2} + 6 \, e f + 2 \, \sqrt {3} {\left (e^{2} - e f + f^{2}\right )}}}{6 \, {\left ({\left (e^{2} + 2 \, e f - 2 \, f^{2}\right )} x^{3} - e^{2} - 2 \, e f + 2 \, f^{2}\right )}}\right )\right ] \]
[-1/3*(sqrt(3)*(I*e + I*f) - 3*I*f)*weierstrassPInverse(0, 4, x) + 1/12*sq rt(3*e^2 - 6*e*f - 2*sqrt(3)*(e^2 - e*f + f^2))*log(-((e^2 + 2*e*f - 2*f^2 )*x^8 + 16*(e^2 + 2*e*f - 2*f^2)*x^7 + 112*(e^2 + 2*e*f - 2*f^2)*x^6 + 16* (e^2 + 2*e*f - 2*f^2)*x^5 + 112*(e^2 + 2*e*f - 2*f^2)*x^4 - 224*(e^2 + 2*e *f - 2*f^2)*x^3 + 64*(e^2 + 2*e*f - 2*f^2)*x^2 + 4*((2*e - f)*x^6 + 18*(e - f)*x^5 + 6*(7*e - 2*f)*x^4 + 8*(e - 5*f)*x^3 + 36*f*x^2 - 24*(e + f)*x + sqrt(3)*((e - f)*x^6 + 6*(2*e - f)*x^5 + 6*(3*e - 4*f)*x^4 + 8*(2*e + f)* x^3 - 12*(e + f)*x^2 + 24*f*x - 8*e - 16*f) + 8*e + 32*f)*sqrt(-x^3 + 1)*s qrt(3*e^2 - 6*e*f - 2*sqrt(3)*(e^2 - e*f + f^2)) + 112*e^2 + 224*e*f - 224 *f^2 - 128*(e^2 + 2*e*f - 2*f^2)*x + 16*sqrt(3)*((e^2 + 2*e*f - 2*f^2)*x^7 + 2*(e^2 + 2*e*f - 2*f^2)*x^6 + 6*(e^2 + 2*e*f - 2*f^2)*x^5 - 5*(e^2 + 2* e*f - 2*f^2)*x^4 + 2*(e^2 + 2*e*f - 2*f^2)*x^3 - 6*(e^2 + 2*e*f - 2*f^2)*x ^2 - 4*e^2 - 8*e*f + 8*f^2 + 4*(e^2 + 2*e*f - 2*f^2)*x))/(x^8 - 8*x^7 + 16 *x^6 + 16*x^5 - 56*x^4 - 32*x^3 + 64*x^2 + 64*x + 16)), -1/3*(sqrt(3)*(I*e + I*f) - 3*I*f)*weierstrassPInverse(0, 4, x) + 1/6*sqrt(-3*e^2 + 6*e*f + 2*sqrt(3)*(e^2 - e*f + f^2))*arctan(1/6*(3*f*x^2 - 6*(e - f)*x - sqrt(3)*( (e + f)*x^2 + 2*(2*e - f)*x - 2*e + 4*f) + 6*e)*sqrt(-x^3 + 1)*sqrt(-3*e^2 + 6*e*f + 2*sqrt(3)*(e^2 - e*f + f^2))/((e^2 + 2*e*f - 2*f^2)*x^3 - e^2 - 2*e*f + 2*f^2))]
\[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=- \int \frac {e}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx - \int \frac {f x}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx \]
-Integral(e/(x*sqrt(1 - x**3) - sqrt(3)*sqrt(1 - x**3) - sqrt(1 - x**3)), x) - Integral(f*x/(x*sqrt(1 - x**3) - sqrt(3)*sqrt(1 - x**3) - sqrt(1 - x* *3)), x)
\[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}} \,d x } \]
Exception generated. \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[2]%%%} / %%%{%%{[2,4]:[1,0,-3]%%},[2]%%%} Error: Bad Ar gument Va
Timed out. \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Hanged} \]