Integrand size = 55, antiderivative size = 75 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a-b x^3}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1-\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \] 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)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.93 (sec) , antiderivative size = 648, normalized size of antiderivative = 8.64 \[ \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*Sqr t[3 - (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqr t[3]*a)] - (3*(-18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(1 0*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)*S qrt[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*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)])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[ a - b*x^3])
Time = 0.58 (sec) , antiderivative size = 75, 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.
\(\displaystyle \int \frac {x \left (-\sqrt [3]{\frac {b}{a}}\right )+\sqrt {3}+1}{\left (x \left (-\sqrt [3]{\frac {b}{a}}\right )-\sqrt {3}+1\right ) \sqrt {a-b x^3}} \, dx\) |
\(\Big \downarrow \) 2565 |
\(\displaystyle \frac {2 \int \frac {1}{\frac {\left (3-2 \sqrt {3}\right ) a \left (1-\sqrt [3]{\frac {b}{a}} x\right )^2}{a-b x^3}+1}d\frac {1-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}}}{\sqrt [3]{\frac {b}{a}}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (1-x \sqrt [3]{\frac {b}{a}}\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}}\) |
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))
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])
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]
\[\int \frac {1+\sqrt {3}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x}{\left (1-\sqrt {3}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}+a}}d x\]
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 )
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (57) = 114\).
Time = 0.53 (sec) , antiderivative size = 1330, normalized size of antiderivative = 17.73 \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a-b x^3}} \, dx=\text {Too large to display} \] 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)*((3*a*b^7*x^22 + 2688*a^2*b^6*x^19 + 56952*a^3*b^5*x^16 + 9 3504*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^16 + 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))*sqrt(-b*x^3 + a)*(b/a)^(2/3) + 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^1 2 - 56000*a^5*b^3*x^9 + 115968*a^6*b^2*x^6 - 56320*a^7*b*x^3 + 1024*a^8))* sqrt(-b*x^3 + a)*(b/a)^(1/3) + 6*(81*a*b^7*x^20 + 4752*a^2*b^6*x^17 + 1447 2*a^3*b^5*x^14 + 24192*a^4*b^4*x^11 - 39744*a^5*b^3*x^8 + 69120*a^6*b^2*x^ 5 - 13824*a^7*b*x^2 - 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 - 37632*a^6*b^2*x^5 + 8 192*a^7*b*x^2))*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^20 + 13320*a^3*b^5*x^17 + 19200*a^4*b^4*x^1 4 - 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 - ...
\[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a-b x^3}} \, dx=\int \frac {x \sqrt [3]{\frac {b}{a}} - \sqrt {3} - 1}{\sqrt {a - b x^{3}} \left (x \sqrt [3]{\frac {b}{a}} - 1 + \sqrt {3}\right )}\, dx \] 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) - sqrt(3) - 1)/(sqrt(a - b*x**3)*(x*(b/a)**(1/3) - 1 + sqrt(3))), x)
\[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a-b x^3}} \, dx=\int { \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} - 1}{\sqrt {-b x^{3} + a} {\left (x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} - 1\right )}} \,d x } \] 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)
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
Timed out. \[ \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a-b x^3}} \, dx=\int -\frac {\sqrt {3}-x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {a-b\,x^3}\,\left (\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}-1\right )} \,d x \] Input:
int(-(3^(1/2) - x*(b/a)^(1/3) + 1)/((a - b*x^3)^(1/2)*(3^(1/2) + x*(b/a)^( 1/3) - 1)),x)
Output:
int(-(3^(1/2) - x*(b/a)^(1/3) + 1)/((a - b*x^3)^(1/2)*(3^(1/2) + x*(b/a)^( 1/3) - 1)), x)
\[ \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