\(\int \frac {x}{((1-\sqrt {3}) \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {a-b x^3}} \, dx\) [187]

Optimal result

Integrand size = 40, antiderivative size = 286 \[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}} \] Output:

-1/3*2^(1/2)*arctanh((-3+2*3^(1/2))^(1/2)*a^(1/6)*(a^(1/3)-b^(1/3)*x)/(-b* 
x^3+a)^(1/2))*3^(1/4)/a^(1/6)/b^(2/3)+2/3*(1/3*6^(1/2)+1/2*2^(1/2))*(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),I*3^(1/2)+2*I)*3^(3/4)/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 = 11.67 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.59 \[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {4 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {1}{2} \left (i \left (-3+(2+i) \sqrt {3}\right ) \sqrt [3]{a}+\left (3-(2-i) \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )+i \left (-1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}},\arcsin \left (\sqrt {-\frac {i \left (2 \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (3-(2-i) \sqrt {3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a-b x^3}} \] Input:

Integrate[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))]*(((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))]*Elliptic 
F[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])/2 + I*(-1 + Sqrt[3])*a^(1/3)*Sqrt[((- 
I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sq 
rt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(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] (warning: unable to verify)

Time = 1.19 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2566, 27, 759, 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.

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

\(\Big \downarrow \) 2566

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*(a^(1/3) - b^(1/3)*x))/Sqrt[a - 
b*x^3]])/((3 + Sqrt[3])*Sqrt[-3 + 2*Sqrt[3]]*a^(1/6)*b^(2/3)) + (2*(2 + Sq 
rt[3])^(3/2)*(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 - Sq 
rt[3])*a^(1/3) - b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 - 4*S 
qrt[3]])/(3^(1/4)*(3 + Sqrt[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 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 759

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]
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 1354, normalized size of antiderivative = 4.73 \[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(sqrt(2)*a^(1/3)*b^(4/3)*sqrt(sqrt(3)/a)*log((b^8*x^24 + 1840*a*b^7* 
x^21 + 67264*a^2*b^6*x^18 + 58624*a^3*b^5*x^15 + 504064*a^4*b^4*x^12 - 214 
0160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 - 1089536*a^7*b*x^3 + 28672*a^8 + 3 
2*(9*b^7*x^22 + 846*a*b^6*x^19 + 4617*a^2*b^5*x^16 - 5472*a^3*b^4*x^13 + 4 
3776*a^4*b^3*x^10 - 98496*a^5*b^2*x^7 + 59328*a^6*b*x^4 - 4608*a^7*x - sqr 
t(3)*(5*b^7*x^22 + 505*a*b^6*x^19 + 2130*a^2*b^5*x^16 + 4928*a^3*b^4*x^13 
- 28688*a^4*b^3*x^10 + 53760*a^5*b^2*x^7 - 35200*a^6*b*x^4 + 2560*a^7*x))* 
a^(2/3)*b^(1/3) + 8*(3*b^7*x^23 + 1077*a*b^6*x^20 + 13320*a^2*b^5*x^17 + 1 
9200*a^3*b^4*x^14 - 111360*a^4*b^3*x^11 + 345024*a^5*b^2*x^8 - 328704*a^6* 
b*x^5 + 61440*a^7*x^2 - 2*sqrt(3)*(b^7*x^23 + 299*a*b^6*x^20 + 4260*a^2*b^ 
5*x^17 - 1520*a^3*b^4*x^14 + 26720*a^4*b^3*x^11 - 105024*a^5*b^2*x^8 + 931 
84*a^6*b*x^5 - 17920*a^7*x^2))*a^(1/3)*b^(2/3) + 2*sqrt(2)*((b^7*x^22 + 11 
60*a*b^6*x^19 + 23232*a^2*b^5*x^16 + 53920*a^3*b^4*x^13 - 148288*a^4*b^3*x 
^10 + 586752*a^5*b^2*x^7 - 496640*a^6*b*x^4 + 38912*a^7*x - sqrt(3)*(b^7*x 
^22 + 632*a*b^6*x^19 + 14736*a^2*b^5*x^16 + 8416*a^3*b^4*x^13 + 105920*a^4 
*b^3*x^10 - 334848*a^5*b^2*x^7 + 286720*a^6*b*x^4 - 22528*a^7*x))*sqrt(-b* 
x^3 + a)*a^(2/3)*b^(1/3) + 12*(17*a*b^6*x^20 + 1014*a^2*b^5*x^17 + 2748*a^ 
3*b^4*x^14 + 9632*a^4*b^3*x^11 - 36096*a^5*b^2*x^8 + 53376*a^6*b*x^5 - 110 
08*a^7*x^2 - 2*sqrt(3)*(5*a*b^6*x^20 + 285*a^2*b^5*x^17 + 1038*a^3*b^4*x^1 
4 - 784*a^4*b^3*x^11 + 11424*a^5*b^2*x^8 - 15168*a^6*b*x^5 + 3200*a^7*x...
 

Sympy [F]

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

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

Output:

-Integral(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]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-2 b^{\frac {1}{3}} a^{\frac {2}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x^{2}}{4 a^{\frac {7}{3}}-8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}+8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}-b^{\frac {7}{3}} x^{7}}d x \right )+b^{\frac {2}{3}} a^{\frac {1}{3}} \sqrt {3}\, \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x^{3}}{4 a^{\frac {7}{3}}-8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}+8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}-b^{\frac {7}{3}} x^{7}}d x \right )+3 b^{\frac {2}{3}} a^{\frac {1}{3}} \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x^{3}}{4 a^{\frac {7}{3}}-8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}+8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}-b^{\frac {7}{3}} x^{7}}d x \right )-2 \sqrt {3}\, \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{4 a^{\frac {7}{3}}-8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}+8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}-b^{\frac {7}{3}} x^{7}}d x \right ) a -\left (\int \frac {\sqrt {-b \,x^{3}+a}\, x^{4}}{4 a^{\frac {7}{3}}-8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}+8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}-b^{\frac {7}{3}} x^{7}}d x \right ) b -2 \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{4 a^{\frac {7}{3}}-8 a^{\frac {4}{3}} b \,x^{3}+4 a^{\frac {1}{3}} b^{2} x^{6}+8 b^{\frac {1}{3}} a^{2} x -7 b^{\frac {4}{3}} a \,x^{4}-b^{\frac {7}{3}} x^{7}}d x \right ) a \] Input:

int(x/((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) + b**(2/3)*a**(1/3)*sqrt(3)*int((s 
qrt(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) + 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) - 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 - int((sqrt(a - b*x* 
*3)*x**4)/(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)*b - 2*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