\(\int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{(1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x) \sqrt {a+b x^3}} \, dx\) [155]

Optimal result

Integrand size = 52, antiderivative size = 73 \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1+\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \] 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 = 10.91 (sec) , antiderivative size = 663, normalized size of antiderivative = 9.08 \[ \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*Sqrt[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*Appel 
lF1[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))])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, 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.

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 219

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

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 = 0.57 (sec) , antiderivative size = 1273, normalized size of antiderivative = 17.44 \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\text {Too large to display} \] 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. \[ \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {b\,x^3+a}\,\left (x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1\right )} \,d x \] Input:

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

Output:

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