\(\int \frac {x}{((1-\sqrt {3}) \sqrt [3]{a}+\sqrt [3]{b} x) \sqrt {-a-b x^3}} \, dx\) [189]

Optimal result

Integrand size = 41, antiderivative size = 278 \[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {\sqrt {2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {-a-b x^3}} \] Output:

-1/3*2^(1/2)*arctan((-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)+1/3*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),2*I-I*3^(1/2))*3^(1/4)/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 = 11.98 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.61 \[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=-\frac {4 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {1}{2} i \left (\left (-3+(2+i) \sqrt {3}\right ) \sqrt [3]{a}+\left (3 i-(1+2 i) \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{a}-\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )+i \left (-1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}} \sqrt {1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}},\arcsin \left (\sqrt {\frac {-2 i \sqrt [3]{a}+\left (i+\sqrt {3}\right ) \sqrt [3]{b} x}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right ),\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left (3-(2-i) \sqrt {3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {-a-b x^3}} \] 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/2)*((-3 + ( 
2 + I)*Sqrt[3])*a^(1/3) + (3*I - (1 + 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))]*E 
llipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + S 
qrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2] + I*(-1 + Sqrt[3])*a^(1/3)*Sqrt[((- 
2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[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[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/ 
3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqr 
t[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 = 1.22 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, 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.

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

\(\Big \downarrow \) 2566

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(-2*ArcTan[(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*Sqrt[2 
- Sqrt[3]]*(2 + Sqrt[3])*(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[Arc 
Sin[((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^(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 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]

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

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 = 0.88 (sec) , antiderivative size = 1348, normalized size of antiderivative = 4.85 \[ \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\text {Too large to display} \] 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 + 21 
40160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 + 1089536*a^7*b*x^3 + 28672*a^8 - 
4*sqrt(2)*(13*a*b^7*x^21 - 2090*a^2*b^6*x^18 + 19776*a^3*b^5*x^15 + 5216*a 
^4*b^4*x^12 + 135872*a^5*b^3*x^9 + 349824*a^6*b^2*x^6 + 142336*a^7*b*x^3 + 
 4096*a^8 - sqrt(3)*(7*a*b^7*x^21 - 1250*a^2*b^6*x^18 + 9984*a^3*b^5*x^15 
- 19456*a^4*b^4*x^12 - 82624*a^5*b^3*x^9 - 193920*a^6*b^2*x^6 - 84992*a^7* 
b*x^3 - 2048*a^8))*sqrt(-b*x^3 - a)*sqrt(-sqrt(3)/a) + 2*(144*b^7*x^22 - 1 
3536*a*b^6*x^19 + 73872*a^2*b^5*x^16 + 87552*a^3*b^4*x^13 + 700416*a^4*b^3 
*x^10 + 1575936*a^5*b^2*x^7 + 949248*a^6*b*x^4 + 73728*a^7*x + sqrt(2)*(b^ 
7*x^22 - 1160*a*b^6*x^19 + 23232*a^2*b^5*x^16 - 53920*a^3*b^4*x^13 - 14828 
8*a^4*b^3*x^10 - 586752*a^5*b^2*x^7 - 496640*a^6*b*x^4 - 38912*a^7*x - sqr 
t(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)*sqrt(-sqrt(3)/a) - 16*sqrt(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 - 19200*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 - 3*sqr 
t(2)*(17*a*b^6*x^20 - 1014*a^2*b^5*x^17 + 2748*a^3*b^4*x^14 - 9632*a^4*...
 

Sympy [F]

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

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

Output:

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

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

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

Output:

i*( - 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 
((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) - 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)