Integrand size = 24, antiderivative size = 309 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac {2 \left (11 c d^2-2 b d e-7 a e^2\right ) x}{27 d^2 e^2 \sqrt {d+e x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2+e (2 b d+7 a e)\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} d^2 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:
2/9*(a*e^2-b*d*e+c*d^2)*x/d/e^2/(e*x^3+d)^(3/2)-2/27*(-7*a*e^2-2*b*d*e+11* c*d^2)*x/d^2/e^2/(e*x^3+d)^(1/2)+2/81*(1/2*6^(1/2)+1/2*2^(1/2))*(16*c*d^2+ e*(7*a*e+2*b*d))*(d^(1/3)+e^(1/3)*x)*((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x ^2)/((1+3^(1/2))*d^(1/3)+e^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*d^(1/3 )+e^(1/3)*x)/((1+3^(1/2))*d^(1/3)+e^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/d^2/e^ (7/3)/(d^(1/3)*(d^(1/3)+e^(1/3)*x)/((1+3^(1/2))*d^(1/3)+e^(1/3)*x)^2)^(1/2 )/(e*x^3+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.42 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\frac {-2 x \left (c d^2 \left (8 d+11 e x^3\right )+e \left (b d \left (d-2 e x^3\right )-a e \left (10 d+7 e x^3\right )\right )\right )+\left (16 c d^2+e (2 b d+7 a e)\right ) x \left (d+e x^3\right ) \sqrt {1+\frac {e x^3}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {e x^3}{d}\right )}{27 d^2 e^2 \left (d+e x^3\right )^{3/2}} \] Input:
Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2),x]
Output:
(-2*x*(c*d^2*(8*d + 11*e*x^3) + e*(b*d*(d - 2*e*x^3) - a*e*(10*d + 7*e*x^3 ))) + (16*c*d^2 + e*(2*b*d + 7*a*e))*x*(d + e*x^3)*Sqrt[1 + (e*x^3)/d]*Hyp ergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)])/(27*d^2*e^2*(d + e*x^3)^(3/2) )
Time = 0.38 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1739, 27, 910, 759}
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.
| \(\displaystyle \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1739 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac {2 \int \frac {-9 c d e x^3+2 c d^2-e (2 b d+7 a e)}{2 \left (e x^3+d\right )^{3/2}}dx}{9 d e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac {\int \frac {-9 c d e x^3+2 c d^2-7 a e^2-2 b d e}{\left (e x^3+d\right )^{3/2}}dx}{9 d e^2}\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac {\frac {2 x \left (-7 a e^2-2 b d e+11 c d^2\right )}{3 d \sqrt {d+e x^3}}-\frac {\left (e (7 a e+2 b d)+16 c d^2\right ) \int \frac {1}{\sqrt {e x^3+d}}dx}{3 d}}{9 d e^2}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac {\frac {2 x \left (-7 a e^2-2 b d e+11 c d^2\right )}{3 d \sqrt {d+e x^3}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (e (7 a e+2 b d)+16 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}}{9 d e^2}\) |
Input:
Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2),x]
Output:
(2*(c*d^2 - b*d*e + a*e^2)*x)/(9*d*e^2*(d + e*x^3)^(3/2)) - ((2*(11*c*d^2 - 2*b*d*e - 7*a*e^2)*x)/(3*d*Sqrt[d + e*x^3]) - (2*Sqrt[2 + Sqrt[3]]*(16*c *d^2 + e*(2*b*d + 7*a*e))*(d^(1/3) + e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/3)*e^ (1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*EllipticF[Ar cSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)* x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*d*e^(1/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^(1/3 )*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt[d + e*x^3]))/(9*d*e^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] & & PosQ[a]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[(-(c*d^2 - b*d*e + a*e^2))*x*((d + e*x^n)^(q + 1)/(d* e^2*n*(q + 1))), x] + Simp[1/(n*(q + 1)*d*e^2) Int[(d + e*x^n)^(q + 1)*Si mp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && N eQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]
Time = 0.72 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.30
| method | result | size |
| elliptic | \(\frac {2 x \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{9 d \,e^{4} \left (x^{3}+\frac {d}{e}\right )^{2}}+\frac {2 x \left (7 a \,e^{2}+2 b d e -11 c \,d^{2}\right )}{27 e^{2} d^{2} \sqrt {\left (x^{3}+\frac {d}{e}\right ) e}}-\frac {2 i \left (\frac {c}{e^{2}}+\frac {7 a \,e^{2}+2 b d e -11 c \,d^{2}}{27 d^{2} e^{2}}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\) | \(401\) |
| default | \(\text {Expression too large to display}\) | \(1005\) |
Input:
int((c*x^6+b*x^3+a)/(e*x^3+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9*x/d/e^4*(a*e^2-b*d*e+c*d^2)*(e*x^3+d)^(1/2)/(x^3+d/e)^2+2/27/e^2*x/d^2 *(7*a*e^2+2*b*d*e-11*c*d^2)/((x^3+d/e)*e)^(1/2)-2/3*I*(c/e^2+1/27/d^2/e^2* (7*a*e^2+2*b*d*e-11*c*d^2))*3^(1/2)/e*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^ (1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x- 1/e*(-d*e^2)^(1/3))/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)) )^(1/2)*(-I*(x+1/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2 )*e/(-d*e^2)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/ e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3)) ^(1/2),(I*3^(1/2)/e*(-d*e^2)^(1/3)/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e* (-d*e^2)^(1/3)))^(1/2))
Time = 0.07 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.61 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c d^{2} e^{2} + 2 \, b d e^{3} + 7 \, a e^{4}\right )} x^{6} + 16 \, c d^{4} + 2 \, b d^{3} e + 7 \, a d^{2} e^{2} + 2 \, {\left (16 \, c d^{3} e + 2 \, b d^{2} e^{2} + 7 \, a d e^{3}\right )} x^{3}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) - {\left ({\left (11 \, c d^{2} e^{2} - 2 \, b d e^{3} - 7 \, a e^{4}\right )} x^{4} + {\left (8 \, c d^{3} e + b d^{2} e^{2} - 10 \, a d e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{27 \, {\left (d^{2} e^{5} x^{6} + 2 \, d^{3} e^{4} x^{3} + d^{4} e^{3}\right )}} \] Input:
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(5/2),x, algorithm="fricas")
Output:
2/27*(((16*c*d^2*e^2 + 2*b*d*e^3 + 7*a*e^4)*x^6 + 16*c*d^4 + 2*b*d^3*e + 7 *a*d^2*e^2 + 2*(16*c*d^3*e + 2*b*d^2*e^2 + 7*a*d*e^3)*x^3)*sqrt(e)*weierst rassPInverse(0, -4*d/e, x) - ((11*c*d^2*e^2 - 2*b*d*e^3 - 7*a*e^4)*x^4 + ( 8*c*d^3*e + b*d^2*e^2 - 10*a*d*e^3)*x)*sqrt(e*x^3 + d))/(d^2*e^5*x^6 + 2*d ^3*e^4*x^3 + d^4*e^3)
Time = 43.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.39 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\frac {a x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {5}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {5}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {5}{2}} \Gamma \left (\frac {7}{3}\right )} + \frac {c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{3}, \frac {5}{2} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {5}{2}} \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(5/2),x)
Output:
a*x*gamma(1/3)*hyper((1/3, 5/2), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*d**( 5/2)*gamma(4/3)) + b*x**4*gamma(4/3)*hyper((4/3, 5/2), (7/3,), e*x**3*exp_ polar(I*pi)/d)/(3*d**(5/2)*gamma(7/3)) + c*x**7*gamma(7/3)*hyper((7/3, 5/2 ), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*d**(5/2)*gamma(10/3))
\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(5/2),x, algorithm="maxima")
Output:
integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2), x)
\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(5/2),x, algorithm="giac")
Output:
integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2), x)
Timed out. \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\int \frac {c\,x^6+b\,x^3+a}{{\left (e\,x^3+d\right )}^{5/2}} \,d x \] Input:
int((a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2),x)
Output:
int((a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2), x)
\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx=\frac {-2 \sqrt {e \,x^{3}+d}\, b e x -16 \sqrt {e \,x^{3}+d}\, c d x -14 \sqrt {e \,x^{3}+d}\, c e \,x^{4}+7 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) a \,d^{2} e^{2}+14 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) a d \,e^{3} x^{3}+7 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) a \,e^{4} x^{6}+2 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) b \,d^{3} e +4 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) b \,d^{2} e^{2} x^{3}+2 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) b d \,e^{3} x^{6}+16 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) c \,d^{4}+32 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) c \,d^{3} e \,x^{3}+16 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e^{3} x^{9}+3 d \,e^{2} x^{6}+3 d^{2} e \,x^{3}+d^{3}}d x \right ) c \,d^{2} e^{2} x^{6}}{7 e^{2} \left (e^{2} x^{6}+2 d e \,x^{3}+d^{2}\right )} \] Input:
int((c*x^6+b*x^3+a)/(e*x^3+d)^(5/2),x)
Output:
( - 2*sqrt(d + e*x**3)*b*e*x - 16*sqrt(d + e*x**3)*c*d*x - 14*sqrt(d + e*x **3)*c*e*x**4 + 7*int(sqrt(d + e*x**3)/(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x* *6 + e**3*x**9),x)*a*d**2*e**2 + 14*int(sqrt(d + e*x**3)/(d**3 + 3*d**2*e* x**3 + 3*d*e**2*x**6 + e**3*x**9),x)*a*d*e**3*x**3 + 7*int(sqrt(d + e*x**3 )/(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x**6 + e**3*x**9),x)*a*e**4*x**6 + 2*in t(sqrt(d + e*x**3)/(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x**6 + e**3*x**9),x)*b *d**3*e + 4*int(sqrt(d + e*x**3)/(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x**6 + e **3*x**9),x)*b*d**2*e**2*x**3 + 2*int(sqrt(d + e*x**3)/(d**3 + 3*d**2*e*x* *3 + 3*d*e**2*x**6 + e**3*x**9),x)*b*d*e**3*x**6 + 16*int(sqrt(d + e*x**3) /(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x**6 + e**3*x**9),x)*c*d**4 + 32*int(sqr t(d + e*x**3)/(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x**6 + e**3*x**9),x)*c*d**3 *e*x**3 + 16*int(sqrt(d + e*x**3)/(d**3 + 3*d**2*e*x**3 + 3*d*e**2*x**6 + e**3*x**9),x)*c*d**2*e**2*x**6)/(7*e**2*(d**2 + 2*d*e*x**3 + e**2*x**6))