\(\int \frac {e+f x}{(1+\sqrt {3}-x) \sqrt {1-x^3}} \, dx\) [128]

Optimal result

Integrand size = 29, antiderivative size = 187 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\left (e+f+\sqrt {3} f\right ) \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2+\sqrt {3}} \left (e+\left (1-\sqrt {3}\right ) f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2166, 224, 2165, 209} \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\left (1-\sqrt {3}\right ) f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right ) \left (e+\sqrt {3} f+f\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}} \]

[In]

Int[(e + f*x)/((1 + Sqrt[3] - x)*Sqrt[1 - x^3]),x]

[Out]

-(((e + f + Sqrt[3]*f)*ArcTan[(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 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

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 224

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]*Sq
rt[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]

Rule 2165

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[(1 + k)*(e/d), Subst[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]

Rule 2166

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-(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] + Dist[(d*e - c*f)/(c*d*(b*c^3 - 2
8*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 - 2
2*a*d^3), 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-e-\left (1+\sqrt {3}\right ) f\right ) \int \frac {\left (1+\sqrt {3}\right ) \left (22-\left (1+\sqrt {3}\right )^3\right )+6 x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx}{\left (1+\sqrt {3}\right ) \left (28-\left (1+\sqrt {3}\right )^3\right )}-\frac {\left (-6 e+\left (1+\sqrt {3}\right ) \left (22-\left (1+\sqrt {3}\right )^3\right ) f\right ) \int \frac {1}{\sqrt {1-x^3}} \, dx}{\left (1+\sqrt {3}\right ) \left (28-\left (1+\sqrt {3}\right )^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}} F\left (\sin ^{-1}\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}}+\frac {\left (12 \left (-e-\left (1+\sqrt {3}\right ) f\right )\right ) \text {Subst}\left (\int \frac {1}{1+\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1-x}{\sqrt {1-x^3}}\right )}{\left (1+\sqrt {3}\right ) \left (28-\left (1+\sqrt {3}\right )^3\right )} \\ & = -\frac {\left (e+f+\sqrt {3} f\right ) \tan ^{-1}\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}} F\left (\sin ^{-1}\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}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.52 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.56 \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-3 i f \sqrt {-i+\sqrt {3}-2 i x} \left (-i \left ((2+i)+\sqrt {3}\right )+\left ((2-i)+\sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \left (\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1-x^3}} \]

[In]

Integrate[(e + f*x)/((1 + Sqrt[3] - x)*Sqrt[1 - x^3]),x]

[Out]

(2*Sqrt[2/3]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*((-I)*((2 + I) + Sqr
t[3]) + ((2 - I) + Sqrt[3])*x)*EllipticF[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*S
qrt[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 + S
qrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3])

Maple [A] (verified)

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	\(\frac {2 i f \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -f -f \sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\)	\(264\)

[In]

int((f*x+e)/(1-x+3^(1/2))/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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)*EllipticF(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))*
EllipticPi(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))

Fricas [C] (verification not implemented)

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 ] \]

[In]

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

[Out]

[-1/3*(sqrt(3)*(I*e + I*f) - 3*I*f)*weierstrassPInverse(0, 4, x) + 1/12*sqrt(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))]

Sympy [F]

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

[In]

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

[Out]

-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)

Maxima [F]

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

[In]

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

[Out]

-integrate((f*x + e)/(sqrt(-x^3 + 1)*(x - sqrt(3) - 1)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e+f x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[2]%%%} / %%%{%%{[2,4]:[1,0,-3]%%},[2]%%%} Error: Bad
Argument Va

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}