Integrand size = 35, antiderivative size = 316 \[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \] Output:
-1/36*(2+3^(1/2))*arctan(1/2*3^(1/4)*(1+3^(1/2))*a^(1/6)*(a^(1/3)-b^(1/3)* x)*2^(1/2)/(-b*x^3+a)^(1/2))*2^(1/2)*3^(3/4)/a^(5/6)/b^(2/3)-1/18*(2+3^(1/ 2))*arctan(1/2*3^(1/4)*a^(1/6)*((1-3^(1/2))*a^(1/3)+2*b^(1/3)*x)*2^(1/2)/( -b*x^3+a)^(1/2))*2^(1/2)*3^(3/4)/a^(5/6)/b^(2/3)+1/12*(2+3^(1/2))*arctanh( 1/2*3^(1/4)*(1-3^(1/2))*a^(1/6)*(a^(1/3)-b^(1/3)*x)*2^(1/2)/(-b*x^3+a)^(1/ 2))*2^(1/2)*3^(1/4)/a^(5/6)/b^(2/3)+1/18*(2+3^(1/2))*arctanh(1/6*(1+3^(1/2 ))*(-b*x^3+a)^(1/2)*2^(1/2)*3^(1/4)/a^(1/2))*2^(1/2)*3^(1/4)/a^(5/6)/b^(2/ 3)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.26 \[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\frac {x^2 \sqrt {1-\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},\frac {b x^3}{a},\frac {b x^3}{10 a-6 \sqrt {3} a}\right )}{\left (20 a-12 \sqrt {3} a\right ) \sqrt {a-b x^3}} \] Input:
Integrate[x/(Sqrt[a - b*x^3]*(2*(5 - 3*Sqrt[3])*a - b*x^3)),x]
Output:
(x^2*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)])/((20*a - 12*Sqrt[3]*a)*Sqrt[a - b*x^3])
Time = 0.49 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {989}
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.
| \(\displaystyle \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 989 |
| \(\displaystyle -\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a-b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a-b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\) |
Input:
Int[x/(Sqrt[a - b*x^3]*(2*(5 - 3*Sqrt[3])*a - b*x^3)),x]
Output:
-1/6*((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/ 3)*x))/(Sqrt[2]*Sqrt[a - b*x^3])])/(Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) - ((2 + Sqrt[3])*ArcTan[(3^(1/4)*a^(1/6)*((1 - Sqrt[3])*a^(1/3) + 2*b^(1/3)*x)) /(Sqrt[2]*Sqrt[a - b*x^3])])/(3*Sqrt[2]*3^(1/4)*a^(5/6)*b^(2/3)) + ((2 + S qrt[3])*ArcTanh[(3^(1/4)*(1 - Sqrt[3])*a^(1/6)*(a^(1/3) - b^(1/3)*x))/(Sqr t[2]*Sqrt[a - b*x^3])])/(2*Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3)) + ((2 + Sqrt[3 ])*ArcTanh[((1 + Sqrt[3])*Sqrt[a - b*x^3])/(Sqrt[2]*3^(3/4)*Sqrt[a])])/(3* Sqrt[2]*3^(3/4)*a^(5/6)*b^(2/3))
Int[(x_)/(Sqrt[(a_) + (b_.)*(x_)^3]*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wi th[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, Simp[(-q)*(2 - r )*(ArcTan[(1 - r)*(Sqrt[a + b*x^3]/(Sqrt[2]*Rt[a, 2]*r^(3/2)))]/(3*Sqrt[2]* Rt[a, 2]*d*r^(3/2))), x] + (-Simp[q*(2 - r)*(ArcTan[Rt[a, 2]*Sqrt[r]*(1 + r )*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(2*Sqrt[2]*Rt[a, 2]*d*r^(3/2))), x ] - Simp[q*(2 - r)*(ArcTanh[Rt[a, 2]*Sqrt[r]*((1 + r - 2*q*x)/(Sqrt[2]*Sqrt [a + b*x^3]))]/(3*Sqrt[2]*Rt[a, 2]*d*Sqrt[r])), x] - Simp[q*(2 - r)*(ArcTan h[Rt[a, 2]*(1 - r)*Sqrt[r]*((1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3]))]/(6*Sqrt[2 ]*Rt[a, 2]*d*Sqrt[r])), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && PosQ[a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.94 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.61
| method | result | size |
| default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+6 \sqrt {3}\, a -10 a \right )}{\sum }\frac {\left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i b \left (2 x +\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (-3 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +4 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}+3 i \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -2 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +6 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+6 i \left (a \,b^{2}\right )^{\frac {2}{3}}-2 \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-3 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 \left (a \,b^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {-2 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, b +i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}-4 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b +2 i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -2 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, a b -3 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +2 i a b +2 \sqrt {3}\, a b +3 a b}{6 b a}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {-b \,x^{3}+a}}\right )}{27 a \,b^{3}}\) | \(509\) |
| elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+6 \sqrt {3}\, a -10 a \right )}{\sum }\frac {\left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i b \left (2 x +\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (-3 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +4 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}+3 i \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -2 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +6 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+6 i \left (a \,b^{2}\right )^{\frac {2}{3}}-2 \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-3 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 \left (a \,b^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {-2 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, b +i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}-4 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b +2 i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -2 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, a b -3 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +2 i a b +2 \sqrt {3}\, a b +3 a b}{6 b a}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {-b \,x^{3}+a}}\right )}{27 a \,b^{3}}\) | \(509\) |
Input:
int(x/(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x,method=_RETURNVERBOSE)
Output:
-1/27*I/a/b^3*2^(1/2)*sum(1/_alpha*(a*b^2)^(1/3)*(-1/2*I*b*(2*x+1/b*(I*3^( 1/2)*(a*b^2)^(1/3)+(a*b^2)^(1/3)))/(a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(a*b^2)^ (1/3))/(-3*(a*b^2)^(1/3)-I*3^(1/2)*(a*b^2)^(1/3)))^(1/2)*(1/2*I*b*(2*x+1/b *(-I*3^(1/2)*(a*b^2)^(1/3)+(a*b^2)^(1/3)))/(a*b^2)^(1/3))^(1/2)/(-b*x^3+a) ^(1/2)*(-3*I*(a*b^2)^(1/3)*_alpha*3^(1/2)*b+4*b^2*_alpha^2*3^(1/2)+3*I*(a* b^2)^(2/3)*3^(1/2)-6*I*(a*b^2)^(1/3)*_alpha*b-2*(a*b^2)^(1/3)*_alpha*3^(1/ 2)*b+6*_alpha^2*b^2+6*I*(a*b^2)^(2/3)-2*(a*b^2)^(2/3)*3^(1/2)-3*(a*b^2)^(1 /3)*_alpha*b-3*(a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^ (1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2),1/6/b* (-2*I*(a*b^2)^(1/3)*_alpha^2*3^(1/2)*b+I*(a*b^2)^(2/3)*_alpha*3^(1/2)-4*I* (a*b^2)^(1/3)*_alpha^2*b+2*I*(a*b^2)^(2/3)*_alpha-2*(a*b^2)^(2/3)*_alpha*3 ^(1/2)+I*3^(1/2)*a*b-3*(a*b^2)^(2/3)*_alpha+2*I*a*b+2*3^(1/2)*a*b+3*a*b)/a ,(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2) ^(1/3)))^(1/2)),_alpha=RootOf(b*_Z^3+6*3^(1/2)*a-10*a))
Leaf count of result is larger than twice the leaf count of optimal. 5667 vs. \(2 (219) = 438\).
Time = 4.54 (sec) , antiderivative size = 5667, normalized size of antiderivative = 17.93 \[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\text {Too large to display} \] Input:
integrate(x/(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x, algorithm="frica s")
Output:
Too large to include
\[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=- \int \frac {x}{- 10 a \sqrt {a - b x^{3}} + 6 \sqrt {3} a \sqrt {a - b x^{3}} + b x^{3} \sqrt {a - b x^{3}}}\, dx \] Input:
integrate(x/(-b*x**3+a)**(1/2)/(2*(5-3*3**(1/2))*a-b*x**3),x)
Output:
-Integral(x/(-10*a*sqrt(a - b*x**3) + 6*sqrt(3)*a*sqrt(a - b*x**3) + b*x** 3*sqrt(a - b*x**3)), x)
\[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\int { -\frac {x}{{\left (b x^{3} + 2 \, a {\left (3 \, \sqrt {3} - 5\right )}\right )} \sqrt {-b x^{3} + a}} \,d x } \] Input:
integrate(x/(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x, algorithm="maxim a")
Output:
-integrate(x/((b*x^3 + 2*a*(3*sqrt(3) - 5))*sqrt(-b*x^3 + a)), x)
Exception generated. \[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x/(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x, algorithm="giac" )
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:index.cc index_m operator + Error: Bad Argument Value
Timed out. \[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=\int -\frac {x}{\sqrt {a-b\,x^3}\,\left (b\,x^3+2\,a\,\left (3\,\sqrt {3}-5\right )\right )} \,d x \] Input:
int(-x/((a - b*x^3)^(1/2)*(b*x^3 + 2*a*(3*3^(1/2) - 5))),x)
Output:
int(-x/((a - b*x^3)^(1/2)*(b*x^3 + 2*a*(3*3^(1/2) - 5))), x)
\[ \int \frac {x}{\sqrt {a-b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a-b x^3\right )} \, dx=-6 \sqrt {3}\, \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{b^{3} x^{9}-21 a \,b^{2} x^{6}+12 a^{2} b \,x^{3}+8 a^{3}}d x \right ) a +\left (\int \frac {\sqrt {-b \,x^{3}+a}\, x^{4}}{b^{3} x^{9}-21 a \,b^{2} x^{6}+12 a^{2} b \,x^{3}+8 a^{3}}d x \right ) b -10 \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{b^{3} x^{9}-21 a \,b^{2} x^{6}+12 a^{2} b \,x^{3}+8 a^{3}}d x \right ) a \] Input:
int(x/(-b*x^3+a)^(1/2)/(2*(5-3*3^(1/2))*a-b*x^3),x)
Output:
- 6*sqrt(3)*int((sqrt(a - b*x**3)*x)/(8*a**3 + 12*a**2*b*x**3 - 21*a*b**2 *x**6 + b**3*x**9),x)*a + int((sqrt(a - b*x**3)*x**4)/(8*a**3 + 12*a**2*b* x**3 - 21*a*b**2*x**6 + b**3*x**9),x)*b - 10*int((sqrt(a - b*x**3)*x)/(8*a **3 + 12*a**2*b*x**3 - 21*a*b**2*x**6 + b**3*x**9),x)*a