\(\int \frac {e+f x}{(2^{2/3}-x) \sqrt {-1+x^3}} \, dx\) [104]

Optimal result

Integrand size = 26, antiderivative size = 178 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \left (e+2^{2/3} f\right ) \text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2} e-f\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \] Output:

-2/9*(e+2^(2/3)*f)*arctanh(3^(1/2)*(1-2^(1/3)*x)/(x^3-1)^(1/2))*3^(1/2)-2/ 
9*(1/2*6^(1/2)-1/2*2^(1/2))*(2^(1/3)*e-f)*(1-x)*((x^2+x+1)/(1-3^(1/2)-x)^2 
)^(1/2)*EllipticF((1+3^(1/2)-x)/(1-3^(1/2)-x),2*I-I*3^(1/2))*3^(3/4)/(-(1- 
x)/(1-3^(1/2)-x)^2)^(1/2)/(x^3-1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.74 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.90 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\frac {2 \sqrt [6]{2} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \left (-i f \sqrt {-i+\sqrt {3}-2 i x} \left (-6 i-3 i \sqrt [3]{2}+2 \sqrt {3}-\sqrt [3]{2} \sqrt {3}+\left (-3 i \sqrt [3]{2}+4 \sqrt {3}+\sqrt [3]{2} \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt [3]{2} e+2 f\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\sqrt {3} \left (i+2 i 2^{2/3}+\sqrt {3}\right ) \sqrt {i+\sqrt {3}+2 i x} \sqrt {-1+x^3}} \] Input:

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

Output:

(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-I)*f*Sqrt[-I + Sqrt[3] 
 - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)* 
2^(1/3) + 4*Sqrt[3] + 2^(1/3)*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]* 
(2^(1/3)*e + 2*f)*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi 
[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I 
)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I 
)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[-1 + x^3])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2564, 27, 760, 2562, 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.

\(\displaystyle \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {x^3-1}} \, dx\)

\(\Big \downarrow \) 2564

\(\displaystyle \frac {1}{3} \left (\sqrt [3]{2} e-f\right ) \int \frac {1}{\sqrt {x^3-1}}dx+\frac {1}{6} \left (\sqrt [3]{2} e+2 f\right ) \int \frac {2^{2/3} \left (\sqrt [3]{2} x+1\right )}{\left (2^{2/3}-x\right ) \sqrt {x^3-1}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\sqrt [3]{2} e-f\right ) \int \frac {1}{\sqrt {x^3-1}}dx+\frac {\left (\sqrt [3]{2} e+2 f\right ) \int \frac {\sqrt [3]{2} x+1}{\left (2^{2/3}-x\right ) \sqrt {x^3-1}}dx}{3 \sqrt [3]{2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {\left (\sqrt [3]{2} e+2 f\right ) \int \frac {\sqrt [3]{2} x+1}{\left (2^{2/3}-x\right ) \sqrt {x^3-1}}dx}{3 \sqrt [3]{2}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\)

\(\Big \downarrow \) 2562

\(\displaystyle -\frac {1}{3} 2^{2/3} \left (\sqrt [3]{2} e+2 f\right ) \int \frac {1}{1-\frac {3 \left (1-\sqrt [3]{2} x\right )^2}{x^3-1}}d\frac {1-\sqrt [3]{2} x}{\sqrt {x^3-1}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {x^3-1}}\right ) \left (\sqrt [3]{2} e+2 f\right )}{3 \sqrt {3}}\)

Input:

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

Output:

-1/3*(2^(2/3)*(2^(1/3)*e + 2*f)*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 
+ x^3]])/Sqrt[3] - (2*Sqrt[2 - Sqrt[3]]*(2^(1/3)*e - 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^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2) 
]*Sqrt[-1 + x^3])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 760

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]
 

rule 2562

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[2*(e/d)   Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*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*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

rule 2564

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> Simp[(2*d*e + c*f)/(3*c*d)   Int[1/Sqrt[a + b*x^3], x], x] + Si 
mp[(d*e - c*f)/(3*c*d)   Int[(c - 2*d*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*c^3 - 4*a* 
d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
 

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.50

method	result	size
default	\(-\frac {2 \left (e +2^{\frac {2}{3}} f \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}-\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(267\)
elliptic	\(-\frac {2 f \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-e -2^{\frac {2}{3}} f \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(270\)

Input:

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

Output:

-2*(e+2^(2/3)*f)*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*( 
(x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3 
/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)/(-2^(2/3)+1)*EllipticPi(((x-1)/(-3/ 
2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(-2^(2/3)+1),((3/2+1/2*I*3^(1/ 
2))/(3/2-1/2*I*3^(1/2)))^(1/2))-2*f*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2* 
I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/ 
2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((x-1) 
/(-3/2-1/2*I*3^(1/2)))^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/ 
2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (122) = 244\).

Time = 0.21 (sec) , antiderivative size = 991, normalized size of antiderivative = 5.57 \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

[2/3*(2^(1/3)*e - f)*weierstrassPInverse(0, 4, x) + 1/6*sqrt(2/3*2^(2/3)*e 
*f + 2/3*2^(1/3)*f^2 + 1/3*e^2)*log(((e^3 + 4*f^3)*x^18 + 1440*(e^3 + 4*f^ 
3)*x^15 + 17400*(e^3 + 4*f^3)*x^12 - 21056*(e^3 + 4*f^3)*x^9 - 10368*(e^3 
+ 4*f^3)*x^6 + 15360*(e^3 + 4*f^3)*x^3 - 2048*e^3 - 8192*f^3 - 12*(2*e*f*x 
^16 - 17*e^2*x^15 - 252*f^2*x^14 + 620*e*f*x^13 - 1058*e^2*x^12 - 5328*f^2 
*x^11 + 4664*e*f*x^10 - 2528*e^2*x^9 - 5312*e*f*x^7 + 5408*e^2*x^6 + 9216* 
f^2*x^5 - 512*e*f*x^4 - 2560*e^2*x^3 - 4608*f^2*x^2 + 1024*e*f*x + 512*e^2 
 - 2^(2/3)*(2*f^2*x^16 - 17*e*f*x^15 + 63*e^2*x^14 + 620*f^2*x^13 - 1058*e 
*f*x^12 + 1332*e^2*x^11 + 4664*f^2*x^10 - 2528*e*f*x^9 - 5312*f^2*x^7 + 54 
08*e*f*x^6 - 2304*e^2*x^5 - 512*f^2*x^4 - 2560*e*f*x^3 + 1152*e^2*x^2 + 10 
24*f^2*x + 512*e*f) - 2^(1/3)*(e^2*x^16 + 34*f^2*x^15 - 126*e*f*x^14 + 310 
*e^2*x^13 + 2116*f^2*x^12 - 2664*e*f*x^11 + 2332*e^2*x^10 + 5056*f^2*x^9 - 
 2656*e^2*x^7 - 10816*f^2*x^6 + 4608*e*f*x^5 - 256*e^2*x^4 + 5120*f^2*x^3 
- 2304*e*f*x^2 + 512*e^2*x - 1024*f^2))*sqrt(x^3 - 1)*sqrt(2/3*2^(2/3)*e*f 
 + 2/3*2^(1/3)*f^2 + 1/3*e^2) + 24*2^(2/3)*((e^3 + 4*f^3)*x^17 + 121*(e^3 
+ 4*f^3)*x^14 + 478*(e^3 + 4*f^3)*x^11 - 1144*(e^3 + 4*f^3)*x^8 + 608*(e^3 
 + 4*f^3)*x^5 - 64*(e^3 + 4*f^3)*x^2) + 48*2^(1/3)*(5*(e^3 + 4*f^3)*x^16 + 
 176*(e^3 + 4*f^3)*x^13 + 83*(e^3 + 4*f^3)*x^10 - 680*(e^3 + 4*f^3)*x^7 + 
544*(e^3 + 4*f^3)*x^4 - 128*(e^3 + 4*f^3)*x))/(x^18 - 24*x^15 + 240*x^12 - 
 1280*x^9 + 3840*x^6 - 6144*x^3 + 4096)), 2/3*(2^(1/3)*e - f)*weierstra...
 

Sympy [F]

\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=- \int \frac {e}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx - \int \frac {f x}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx \] Input:

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

Output:

-Integral(e/(x*sqrt(x**3 - 1) - 2**(2/3)*sqrt(x**3 - 1)), x) - Integral(f* 
x/(x*sqrt(x**3 - 1) - 2**(2/3)*sqrt(x**3 - 1)), x)
 

Maxima [F]

\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\int { -\frac {f x + e}{\sqrt {x^{3} - 1} {\left (x - 2^{\frac {2}{3}}\right )}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Bad 
Argument
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e+f x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\left (\int \frac {x}{\sqrt {x^{3}-1}\, 2^{\frac {2}{3}}-\sqrt {x^{3}-1}\, x}d x \right ) f +\left (\int \frac {1}{\sqrt {x^{3}-1}\, 2^{\frac {2}{3}}-\sqrt {x^{3}-1}\, x}d x \right ) e \] Input:

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

Output:

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