\(\int \frac {e+f x}{((1-\sqrt {3}) \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {-a+b x^3}} \, dx\) [179]

Optimal result

Integrand size = 45, antiderivative size = 345 \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {\left (\sqrt [3]{b} e+\left (1-\sqrt {3}\right ) \sqrt [3]{a} f\right ) \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {-a+b x^3}}\right )}{\sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {a} b^{2/3}}+\frac {\sqrt {2-\sqrt {3}} \left (\sqrt [3]{b} e+\left (1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.38 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.35 \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=-\frac {4 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {1}{2} f \left (i \left (-3+(2+i) \sqrt {3}\right ) \sqrt [3]{a}+\left (3-(2-i) \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )-i \left (\sqrt [3]{b} e-\left (-1+\sqrt {3}\right ) \sqrt [3]{a} f\right ) \sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}},\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (3-(2-i) \sqrt {3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a+b x^3}} \] Input:

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

Output:

(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((f*(I*(-3 + (2 
 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - I)*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt 
[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Ellipt 
icF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sq 
rt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 - I*(b^(1/3)*e - (-1 + Sqrt[3])*a 
^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqr 
t[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*Elli 
pticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3 
) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[ 
3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3) 
*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2566, 27, 760, 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.

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

\(\Big \downarrow \) 2566

\(\displaystyle \frac {\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {6 a b \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {b x^3-a}}dx}{12 \sqrt {3} a^{4/3} b^{4/3}}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \int \frac {1}{\sqrt {b x^3-a}}dx}{2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {b x^3-a}}\right ) \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right )}{\sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {a} b^{2/3}}\)

Input:

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

Output:

((b^(1/3)*e + (1 - Sqrt[3])*a^(1/3)*f)*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6 
)*(a^(1/3) - b^(1/3)*x))/Sqrt[-a + b*x^3]])/(Sqrt[3*(-3 + 2*Sqrt[3])]*Sqrt 
[a]*b^(2/3)) + (Sqrt[2 - Sqrt[3]]*(b^(1/3)*e + (1 + Sqrt[3])*a^(1/3)*f)*(a 
^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - 
 Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) 
- b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(3^(3/ 
4)*a^(1/3)*b^(2/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - Sqrt[3])*a 
^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*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 216

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])
 

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 2565

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]
 

rule 2566

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (245) = 490\).

Time = 27.38 (sec) , antiderivative size = 7009, normalized size of antiderivative = 20.32 \[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x, algor 
ithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

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

Output:

-Integral(e/(-a**(1/3)*sqrt(-a + b*x**3) + sqrt(3)*a**(1/3)*sqrt(-a + b*x* 
*3) + b**(1/3)*x*sqrt(-a + b*x**3)), x) - Integral(f*x/(-a**(1/3)*sqrt(-a 
+ b*x**3) + sqrt(3)*a**(1/3)*sqrt(-a + b*x**3) + b**(1/3)*x*sqrt(-a + b*x* 
*3)), x)
 

Maxima [F]

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

integrate((f*x+e)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x, algor 
ithm="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate((f*x+e)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)/(b*x^3-a)^(1/2),x, algor 
ithm="giac")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {e+f x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a+b x^3}} \, dx =\text {Too large to display} \] Input:

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

Output:

2*b**(1/3)*a**(2/3)*sqrt(3)*int((sqrt( - a + b*x**3)*x**2)/(4*a**(1/3)*a** 
2 - 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 + 8*b**(1/3)*a**2*x - 7*b** 
(1/3)*a*b*x**4 - b**(1/3)*b**2*x**7),x)*f + 2*b**(1/3)*a**(2/3)*sqrt(3)*in 
t((sqrt( - a + b*x**3)*x)/(4*a**(1/3)*a**2 - 8*a**(1/3)*a*b*x**3 + 4*a**(1 
/3)*b**2*x**6 + 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 - b**(1/3)*b**2*x* 
*7),x)*e - b**(2/3)*a**(1/3)*sqrt(3)*int((sqrt( - a + b*x**3)*x**3)/(4*a** 
(1/3)*a**2 - 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 + 8*b**(1/3)*a**2* 
x - 7*b**(1/3)*a*b*x**4 - b**(1/3)*b**2*x**7),x)*f - b**(2/3)*a**(1/3)*sqr 
t(3)*int((sqrt( - a + b*x**3)*x**2)/(4*a**(1/3)*a**2 - 8*a**(1/3)*a*b*x**3 
 + 4*a**(1/3)*b**2*x**6 + 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 - b**(1/ 
3)*b**2*x**7),x)*e - 3*b**(2/3)*a**(1/3)*int((sqrt( - a + b*x**3)*x**3)/(4 
*a**(1/3)*a**2 - 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 + 8*b**(1/3)*a 
**2*x - 7*b**(1/3)*a*b*x**4 - b**(1/3)*b**2*x**7),x)*f - 3*b**(2/3)*a**(1/ 
3)*int((sqrt( - a + b*x**3)*x**2)/(4*a**(1/3)*a**2 - 8*a**(1/3)*a*b*x**3 + 
 4*a**(1/3)*b**2*x**6 + 8*b**(1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 - b**(1/3) 
*b**2*x**7),x)*e + 2*sqrt(3)*int(sqrt( - a + b*x**3)/(4*a**(1/3)*a**2 - 8* 
a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 + 8*b**(1/3)*a**2*x - 7*b**(1/3)* 
a*b*x**4 - b**(1/3)*b**2*x**7),x)*a*e + 2*sqrt(3)*int((sqrt( - a + b*x**3) 
*x)/(4*a**(1/3)*a**2 - 8*a**(1/3)*a*b*x**3 + 4*a**(1/3)*b**2*x**6 + 8*b**( 
1/3)*a**2*x - 7*b**(1/3)*a*b*x**4 - b**(1/3)*b**2*x**7),x)*a*f + 2*int(...