Integrand size = 24, antiderivative size = 316 \[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\frac {2 \left (16 c d^2-34 b d e+187 a e^2\right ) x \sqrt {d+e x^3}}{935 e^2}-\frac {2 (8 c d-17 b e) x \left (d+e x^3\right )^{3/2}}{187 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (16 c d^2-34 b d e+187 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{935 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:
2/935*(187*a*e^2-34*b*d*e+16*c*d^2)*x*(e*x^3+d)^(1/2)/e^2-2/187*(-17*b*e+8 *c*d)*x*(e*x^3+d)^(3/2)/e^2+2/17*c*x^4*(e*x^3+d)^(3/2)/e+2/935*3^(3/4)*(1/ 2*6^(1/2)+1/2*2^(1/2))*d*(187*a*e^2-34*b*d*e+16*c*d^2)*(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)/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 = 6.71 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.31 \[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\frac {x \sqrt {d+e x^3} \left (-2 \left (d+e x^3\right ) \left (8 c d-17 b e-11 c e x^3\right )+\frac {\left (16 c d^2+17 e (-2 b d+11 a e)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {e x^3}{d}\right )}{\sqrt {1+\frac {e x^3}{d}}}\right )}{187 e^2} \] Input:
Integrate[Sqrt[d + e*x^3]*(a + b*x^3 + c*x^6),x]
Output:
(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)*(8*c*d - 17*b*e - 11*c*e*x^3) + ((16*c* d^2 + 17*e*(-2*b*d + 11*a*e))*Hypergeometric2F1[-1/2, 1/3, 4/3, -((e*x^3)/ d)])/Sqrt[1 + (e*x^3)/d]))/(187*e^2)
Time = 0.37 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1741, 27, 913, 748, 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.
| \(\displaystyle \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {e x^3+d} \left (17 a e-(8 c d-17 b e) x^3\right )dx}{17 e}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {e x^3+d} \left (17 a e-(8 c d-17 b e) x^3\right )dx}{17 e}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-17 e (2 b d-11 a e)\right ) \int \sqrt {e x^3+d}dx}{11 e}-\frac {2 x \left (d+e x^3\right )^{3/2} (8 c d-17 b e)}{11 e}}{17 e}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-17 e (2 b d-11 a e)\right ) \left (\frac {3}{5} d \int \frac {1}{\sqrt {e x^3+d}}dx+\frac {2}{5} x \sqrt {d+e x^3}\right )}{11 e}-\frac {2 x \left (d+e x^3\right )^{3/2} (8 c d-17 b e)}{11 e}}{17 e}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-17 e (2 b d-11 a e)\right ) \left (\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2}{5} x \sqrt {d+e x^3}\right )}{11 e}-\frac {2 x \left (d+e x^3\right )^{3/2} (8 c d-17 b e)}{11 e}}{17 e}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}\) |
Input:
Int[Sqrt[d + e*x^3]*(a + b*x^3 + c*x^6),x]
Output:
(2*c*x^4*(d + e*x^3)^(3/2))/(17*e) + ((-2*(8*c*d - 17*b*e)*x*(d + e*x^3)^( 3/2))/(11*e) + ((16*c*d^2 - 17*e*(2*b*d - 11*a*e))*((2*x*Sqrt[d + e*x^3])/ 5 + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*d*(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]*Ell ipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3]])/(5*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])))/(11*e))/(17 *e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[c*x^(n + 1)*((d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1))) , x] + Simp[1/(e*(n*(q + 2) + 1)) Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0]
Time = 0.64 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {2 x \left (55 c \,e^{2} x^{6}+85 b \,e^{2} x^{3}+15 c d e \,x^{3}+187 a \,e^{2}+51 b d e -24 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{935 e^{2}}-\frac {2 i d \left (187 a \,e^{2}-34 b d e +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{935 e^{3} \sqrt {e \,x^{3}+d}}\) | \(362\) |
| elliptic | \(\frac {2 c \,x^{7} \sqrt {e \,x^{3}+d}}{17}+\frac {2 \left (e b +\frac {3 c d}{17}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (a e +b d -\frac {8 d \left (e b +\frac {3 c d}{17}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (a d -\frac {2 d \left (a e +b d -\frac {8 d \left (e b +\frac {3 c d}{17}\right )}{11 e}\right )}{5 e}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\) | \(391\) |
| default | \(\text {Expression too large to display}\) | \(956\) |
Input:
int((e*x^3+d)^(1/2)*(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
2/935*x*(55*c*e^2*x^6+85*b*e^2*x^3+15*c*d*e*x^3+187*a*e^2+51*b*d*e-24*c*d^ 2)/e^2*(e*x^3+d)^(1/2)-2/935*I*d*(187*a*e^2-34*b*d*e+16*c*d^2)/e^3*3^(1/2) *(-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 = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.33 \[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\frac {2 \, {\left (3 \, {\left (16 \, c d^{3} - 34 \, b d^{2} e + 187 \, a d e^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) + {\left (55 \, c e^{3} x^{7} + 5 \, {\left (3 \, c d e^{2} + 17 \, b e^{3}\right )} x^{4} - {\left (24 \, c d^{2} e - 51 \, b d e^{2} - 187 \, a e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{935 \, e^{3}} \] Input:
integrate((e*x^3+d)^(1/2)*(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
2/935*(3*(16*c*d^3 - 34*b*d^2*e + 187*a*d*e^2)*sqrt(e)*weierstrassPInverse (0, -4*d/e, x) + (55*c*e^3*x^7 + 5*(3*c*d*e^2 + 17*b*e^3)*x^4 - (24*c*d^2* e - 51*b*d*e^2 - 187*a*e^3)*x)*sqrt(e*x^3 + d))/e^3
Time = 1.52 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.39 \[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\frac {a \sqrt {d} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {b \sqrt {d} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {c \sqrt {d} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate((e*x**3+d)**(1/2)*(c*x**6+b*x**3+a),x)
Output:
a*sqrt(d)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/d )/(3*gamma(4/3)) + b*sqrt(d)*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), e* x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + c*sqrt(d)*x**7*gamma(7/3)*hyper(( -1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3))
\[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d} \,d x } \] Input:
integrate((e*x^3+d)^(1/2)*(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
integrate((c*x^6 + b*x^3 + a)*sqrt(e*x^3 + d), x)
\[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )} \sqrt {e x^{3} + d} \,d x } \] Input:
integrate((e*x^3+d)^(1/2)*(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
integrate((c*x^6 + b*x^3 + a)*sqrt(e*x^3 + d), x)
Timed out. \[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\int \sqrt {e\,x^3+d}\,\left (c\,x^6+b\,x^3+a\right ) \,d x \] Input:
int((d + e*x^3)^(1/2)*(a + b*x^3 + c*x^6),x)
Output:
int((d + e*x^3)^(1/2)*(a + b*x^3 + c*x^6), x)
\[ \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx=\frac {374 \sqrt {e \,x^{3}+d}\, a \,e^{2} x +102 \sqrt {e \,x^{3}+d}\, b d e x +170 \sqrt {e \,x^{3}+d}\, b \,e^{2} x^{4}-48 \sqrt {e \,x^{3}+d}\, c \,d^{2} x +30 \sqrt {e \,x^{3}+d}\, c d e \,x^{4}+110 \sqrt {e \,x^{3}+d}\, c \,e^{2} x^{7}+561 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) a d \,e^{2}-102 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) b \,d^{2} e +48 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) c \,d^{3}}{935 e^{2}} \] Input:
int((e*x^3+d)^(1/2)*(c*x^6+b*x^3+a),x)
Output:
(374*sqrt(d + e*x**3)*a*e**2*x + 102*sqrt(d + e*x**3)*b*d*e*x + 170*sqrt(d + e*x**3)*b*e**2*x**4 - 48*sqrt(d + e*x**3)*c*d**2*x + 30*sqrt(d + e*x**3 )*c*d*e*x**4 + 110*sqrt(d + e*x**3)*c*e**2*x**7 + 561*int(sqrt(d + e*x**3) /(d + e*x**3),x)*a*d*e**2 - 102*int(sqrt(d + e*x**3)/(d + e*x**3),x)*b*d** 2*e + 48*int(sqrt(d + e*x**3)/(d + e*x**3),x)*c*d**3)/(935*e**2)