Integrand size = 27, antiderivative size = 183 \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {-1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}+\frac {\sqrt {2-\sqrt {3}} \left (e-\left (1-\sqrt {3}\right ) f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
1/3*(1+x)*EllipticF((1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))*(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)+arctanh((1+x)*(3+2*3^(1/2)) ^(1/2)/(-x^3-1)^(1/2))*(e-f*(1+3^(1/2)))/(9+6*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 20.46 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.60 \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \left (3 f \sqrt {-i+\sqrt {3}+2 i x} \left ((-2-i)-\sqrt {3}+\left ((1+2 i)+i \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \left (-\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}-2 i x} \sqrt {-1-x^3}} \]
(2*Sqrt[2/3]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*f*Sqrt[-I + Sqrt[3] + (2 *I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[S qrt[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]), ArcSin[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 = 0.48 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2566, 27, 760, 2565, 219}
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.
| \(\displaystyle \int \frac {e+f x}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2566 |
| \(\displaystyle \frac {\left (e-\left (1-\sqrt {3}\right ) f\right ) \int \frac {1}{\sqrt {-x^3-1}}dx}{2 \sqrt {3}}+\frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \int -\frac {6 \left (x-\sqrt {3}+1\right )}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}}dx}{12 \sqrt {3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (e-\left (1-\sqrt {3}\right ) f\right ) \int \frac {1}{\sqrt {-x^3-1}}dx}{2 \sqrt {3}}-\frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}}dx}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {\sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (e-\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {-x^3-1}}dx}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 2565 |
| \(\displaystyle \frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \int \frac {1}{1-\frac {\left (3+2 \sqrt {3}\right ) (x+1)^2}{-x^3-1}}d\frac {x+1}{\sqrt {-x^3-1}}}{\sqrt {3}}+\frac {\sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (e-\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (e-\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {\text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {-x^3-1}}\right ) \left (e-\left (1+\sqrt {3}\right ) f\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}\) |
((e - (1 + Sqrt[3])*f)*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[-1 - x^3 ]])/Sqrt[3*(3 + 2*Sqrt[3])] + (Sqrt[2 - Sqrt[3]]*(e - (1 - Sqrt[3])*f)*(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^(3/4)*Sqrt[-((1 + x)/(1 - Sq rt[3] + x)^2)]*Sqrt[-1 - x^3])
3.2.30.3.1 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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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 ] && NegQ[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 = 1.52 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(258\) |
| elliptic | \(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i \left (e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\) | \(258\) |
-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))*Elliptic Pi(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 = 0.17 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.96 \[ \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*s qrt(-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 - 1 6*(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) *sqrt(-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))]
\[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\int \frac {e + f x}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt {3}\right )}\, dx \]
\[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { \frac {f x + e}{\sqrt {-x^{3} - 1} {\left (x + \sqrt {3} + 1\right )}} \,d x } \]
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. \[ \int \frac {e+f x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\text {Hanged} \]