\(\int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{(1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x) \sqrt {-a+b x^3}} \, dx\) [169]

Optimal result

Integrand size = 56, antiderivative size = 76 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {a} \left (1-\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {-a+b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 10.96 (sec) , antiderivative size = 650, normalized size of antiderivative = 8.55 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {-a+b x^3}} \, dx=\frac {x \left (-12 \left (3+\sqrt {3}\right ) \sqrt [3]{\frac {b}{a}} x \sqrt {1-\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )-8 \left (\frac {b}{a}\right )^{2/3} x^2 \sqrt {3-\frac {3 b x^3}{a}} \operatorname {AppellF1}\left (1,\frac {1}{2},1,2,\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )-\frac {3 \left (18176 a^3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )+10496 \sqrt {3} a^3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )+b x^3 \left (2 \left (5+3 \sqrt {3}\right ) a-b x^3\right ) \sqrt {1-\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right ) \left (8 \left (5+3 \sqrt {3}\right ) a \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )+3 b x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )+\left (5+3 \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )\right )\right )\right )}{a \left (2 \left (5+3 \sqrt {3}\right ) a-b x^3\right ) \left (8 \left (5+3 \sqrt {3}\right ) a \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )+3 b x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )+\left (5+3 \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},\frac {b x^3}{a},\frac {b x^3}{10 a+6 \sqrt {3} a}\right )\right )\right )}\right )}{24 \left (5+3 \sqrt {3}\right ) \sqrt {-a+b x^3}} \] Input:

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

Output:

(x*(-12*(3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 
 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt 
[3 - (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt 
[3]*a)] - (3*(18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10* 
a + 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a 
, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + b*x^3*(2*(5 + 3*Sqrt[3])*a - b*x^3)*Sqrt 
[1 - (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sq 
rt[3]*a)]*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^ 
3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, 
(b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, 
 (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)]))))/(a*(2*(5 + 3*Sqrt[3])*a - b* 
x^3)*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(1 
0*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^ 
3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x 
^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])))))/(24*(5 + 3*Sqrt[3])*Sqrt[-a + b* 
x^3])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {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.

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[a]*(1 - (b/a)^(1/3)*x))/Sqrt[-a + b*x 
^3]])/(Sqrt[3 + 2*Sqrt[3]]*Sqrt[a]*(b/a)^(1/3))
 

Defintions of rubi rules used

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 1273, normalized size of antiderivative = 16.75 \[ \int \frac {1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {-a+b x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*sqrt(1/3)*sqrt((2*sqrt(3) - 3)*(b/a)^(1/3)/b)*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 - 
 2140160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 - 1089536*a^7*b*x^3 + 28672*a^8 
 - 4*sqrt(1/3)*(486*a*b^7*x^20 + 28512*a^2*b^6*x^17 + 86832*a^3*b^5*x^14 + 
 145152*a^4*b^4*x^11 - 238464*a^5*b^3*x^8 + 414720*a^6*b^2*x^5 - 82944*a^7 
*b*x^2 + (3*a*b^7*x^22 + 2688*a^2*b^6*x^19 + 56952*a^3*b^5*x^16 + 93504*a^ 
4*b^4*x^13 - 63552*a^5*b^3*x^10 + 377856*a^6*b^2*x^7 - 314880*a^7*b*x^4 + 
24576*a^8*x + 2*sqrt(3)*(a*b^7*x^22 + 764*a^2*b^6*x^19 + 16860*a^3*b^5*x^1 
6 + 19792*a^4*b^4*x^13 + 42368*a^5*b^3*x^10 - 104448*a^6*b^2*x^7 + 90880*a 
^7*b*x^4 - 7168*a^8*x))*(b/a)^(2/3) + 6*sqrt(3)*(47*a*b^7*x^20 + 2724*a^2* 
b^6*x^17 + 8976*a^3*b^5*x^14 + 4928*a^4*b^4*x^11 + 32448*a^5*b^3*x^8 - 376 
32*a^6*b^2*x^5 + 8192*a^7*b*x^2) + 2*(30*a*b^7*x^21 + 5010*a^2*b^6*x^18 + 
44640*a^3*b^5*x^15 + 21360*a^4*b^4*x^12 + 79872*a^5*b^3*x^9 - 233856*a^6*b 
^2*x^6 + 86016*a^7*b*x^3 - 3072*a^8 + sqrt(3)*(17*a*b^7*x^21 + 2920*a^2*b^ 
6*x^18 + 24864*a^3*b^5*x^15 + 26576*a^4*b^4*x^12 - 56000*a^5*b^3*x^9 + 115 
968*a^6*b^2*x^6 - 56320*a^7*b*x^3 + 1024*a^8))*(b/a)^(1/3))*sqrt(b*x^3 - a 
)*sqrt((2*sqrt(3) - 3)*(b/a)^(1/3)/b) + 8*(3*a*b^7*x^23 + 1077*a^2*b^6*x^2 
0 + 13320*a^3*b^5*x^17 + 19200*a^4*b^4*x^14 - 111360*a^5*b^3*x^11 + 345024 
*a^6*b^2*x^8 - 328704*a^7*b*x^5 + 61440*a^8*x^2 + 2*sqrt(3)*(a*b^7*x^23 + 
299*a^2*b^6*x^20 + 4260*a^3*b^5*x^17 - 1520*a^4*b^4*x^14 + 26720*a^5*b^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

6*b**(2/3)*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) - 6*b**(2/3)*a**(2/3)*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) 
- 4*a**(1/3)*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 - 2*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)*b + 8*a**(1/3 
)*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 + 4*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)*b - b**(1/3)*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 + 4*b**(1/3)* 
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