Integrand size = 24, antiderivative size = 396 \[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {54 d^2 \left (16 c d^2-58 b d e+667 a e^2\right ) x \sqrt {d+e x^3}}{124729 e^2}+\frac {30 d \left (16 c d^2-58 b d e+667 a e^2\right ) x \left (d+e x^3\right )^{3/2}}{124729 e^2}+\frac {2 \left (16 c d^2-58 b d e+667 a e^2\right ) x \left (d+e x^3\right )^{5/2}}{11339 e^2}-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} d^3 \left (16 c d^2-58 b d e+667 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{124729 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \]
30/124729*d*(667*a*e^2-58*b*d*e+16*c*d^2)*x*(e*x^3+d)^(3/2)/e^2+2/11339*(6 67*a*e^2-58*b*d*e+16*c*d^2)*x*(e*x^3+d)^(5/2)/e^2-2/667*(-29*b*e+8*c*d)*x* (e*x^3+d)^(7/2)/e^2+2/29*c*x^4*(e*x^3+d)^(7/2)/e+54/124729*d^2*(667*a*e^2- 58*b*d*e+16*c*d^2)*x*(e*x^3+d)^(1/2)/e^2+54/124729*3^(3/4)*d^3*(667*a*e^2- 58*b*d*e+16*c*d^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-3^( 1/2)))/(e^(1/3)*x+d^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)+1/2*2^( 1/2))*((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+3^(1/ 2)))^2)^(1/2)/e^(7/3)/(e*x^3+d)^(1/2)/(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3 )*x+d^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.26 \[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {x \sqrt {d+e x^3} \left (-2 \left (d+e x^3\right )^3 \left (8 c d-29 b e-23 c e x^3\right )+\frac {\left (16 c d^4+29 d^2 e (-2 b d+23 a e)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{3},\frac {4}{3},-\frac {e x^3}{d}\right )}{\sqrt {1+\frac {e x^3}{d}}}\right )}{667 e^2} \]
(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)^3*(8*c*d - 29*b*e - 23*c*e*x^3) + ((16* c*d^4 + 29*d^2*e*(-2*b*d + 23*a*e))*Hypergeometric2F1[-5/2, 1/3, 4/3, -((e *x^3)/d)])/Sqrt[1 + (e*x^3)/d]))/(667*e^2)
Time = 0.42 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1741, 27, 913, 748, 748, 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.
| \(\displaystyle \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \left (e x^3+d\right )^{5/2} \left (29 a e-(8 c d-29 b e) x^3\right )dx}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \left (e x^3+d\right )^{5/2} \left (29 a e-(8 c d-29 b e) x^3\right )dx}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-29 e (2 b d-23 a e)\right ) \int \left (e x^3+d\right )^{5/2}dx}{23 e}-\frac {2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{23 e}}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-29 e (2 b d-23 a e)\right ) \left (\frac {15}{17} d \int \left (e x^3+d\right )^{3/2}dx+\frac {2}{17} x \left (d+e x^3\right )^{5/2}\right )}{23 e}-\frac {2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{23 e}}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-29 e (2 b d-23 a e)\right ) \left (\frac {15}{17} d \left (\frac {9}{11} d \int \sqrt {e x^3+d}dx+\frac {2}{11} x \left (d+e x^3\right )^{3/2}\right )+\frac {2}{17} x \left (d+e x^3\right )^{5/2}\right )}{23 e}-\frac {2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{23 e}}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-29 e (2 b d-23 a e)\right ) \left (\frac {15}{17} d \left (\frac {9}{11} d \left (\frac {3}{5} d \int \frac {1}{\sqrt {e x^3+d}}dx+\frac {2}{5} x \sqrt {d+e x^3}\right )+\frac {2}{11} x \left (d+e x^3\right )^{3/2}\right )+\frac {2}{17} x \left (d+e x^3\right )^{5/2}\right )}{23 e}-\frac {2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{23 e}}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-29 e (2 b d-23 a e)\right ) \left (\frac {15}{17} d \left (\frac {9}{11} d \left (\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2}{5} x \sqrt {d+e x^3}\right )+\frac {2}{11} x \left (d+e x^3\right )^{3/2}\right )+\frac {2}{17} x \left (d+e x^3\right )^{5/2}\right )}{23 e}-\frac {2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{23 e}}{29 e}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}\) |
(2*c*x^4*(d + e*x^3)^(7/2))/(29*e) + ((-2*(8*c*d - 29*b*e)*x*(d + e*x^3)^( 7/2))/(23*e) + ((16*c*d^2 - 29*e*(2*b*d - 23*a*e))*((2*x*(d + e*x^3)^(5/2) )/17 + (15*d*((2*x*(d + e*x^3)^(3/2))/11 + (9*d*((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]*Ellipt icF[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))/17))/(23* e))/(29*e)
3.1.35.3.1 Defintions of rubi rules used
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 = 2.22 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {2 x \left (4301 e^{4} c \,x^{12}+5423 b \,e^{4} x^{9}+11407 d \,e^{3} c \,x^{9}+7337 a \,e^{4} x^{6}+15631 b d \,e^{3} x^{6}+8591 c \,d^{2} e^{2} x^{6}+24679 d \,e^{3} a \,x^{3}+14123 b \,d^{2} e^{2} x^{3}+405 d^{3} e c \,x^{3}+35351 e^{2} d^{2} a +2349 d^{3} e b -648 d^{4} c \right ) \sqrt {e \,x^{3}+d}}{124729 e^{2}}-\frac {54 i d^{3} \left (667 a \,e^{2}-58 b d e +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{124729 e^{3} \sqrt {e \,x^{3}+d}}\) | \(434\) |
| elliptic | \(\frac {2 c \,e^{2} x^{13} \sqrt {e \,x^{3}+d}}{29}+\frac {2 \left (b \,e^{3}+\frac {61}{29} c d \,e^{2}\right ) x^{10} \sqrt {e \,x^{3}+d}}{23 e}+\frac {2 \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (b \,e^{3}+\frac {61}{29} c d \,e^{2}\right )}{23 e}\right ) x^{7} \sqrt {e \,x^{3}+d}}{17 e}+\frac {2 \left (3 d \,e^{2} a +3 b \,d^{2} e +d^{3} c -\frac {14 d \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (b \,e^{3}+\frac {61}{29} c d \,e^{2}\right )}{23 e}\right )}{17 e}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (3 a \,d^{2} e +b \,d^{3}-\frac {8 d \left (3 d \,e^{2} a +3 b \,d^{2} e +d^{3} c -\frac {14 d \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (b \,e^{3}+\frac {61}{29} c d \,e^{2}\right )}{23 e}\right )}{17 e}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (a \,d^{3}-\frac {2 d \left (3 a \,d^{2} e +b \,d^{3}-\frac {8 d \left (3 d \,e^{2} a +3 b \,d^{2} e +d^{3} c -\frac {14 d \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (b \,e^{3}+\frac {61}{29} c d \,e^{2}\right )}{23 e}\right )}{17 e}\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}}}}\, F\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}}\) | \(665\) |
| default | \(\text {Expression too large to display}\) | \(1070\) |
2/124729/e^2*x*(4301*c*e^4*x^12+5423*b*e^4*x^9+11407*c*d*e^3*x^9+7337*a*e^ 4*x^6+15631*b*d*e^3*x^6+8591*c*d^2*e^2*x^6+24679*a*d*e^3*x^3+14123*b*d^2*e ^2*x^3+405*c*d^3*e*x^3+35351*a*d^2*e^2+2349*b*d^3*e-648*c*d^4)*(e*x^3+d)^( 1/2)-54/124729*I*d^3*(667*a*e^2-58*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)*Ellipti cF(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))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.43 \[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {2 \, {\left (81 \, {\left (16 \, c d^{5} - 58 \, b d^{4} e + 667 \, a d^{3} e^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) + {\left (4301 \, c e^{5} x^{13} + 187 \, {\left (61 \, c d e^{4} + 29 \, b e^{5}\right )} x^{10} + 11 \, {\left (781 \, c d^{2} e^{3} + 1421 \, b d e^{4} + 667 \, a e^{5}\right )} x^{7} + {\left (405 \, c d^{3} e^{2} + 14123 \, b d^{2} e^{3} + 24679 \, a d e^{4}\right )} x^{4} - {\left (648 \, c d^{4} e - 2349 \, b d^{3} e^{2} - 35351 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{124729 \, e^{3}} \]
2/124729*(81*(16*c*d^5 - 58*b*d^4*e + 667*a*d^3*e^2)*sqrt(e)*weierstrassPI nverse(0, -4*d/e, x) + (4301*c*e^5*x^13 + 187*(61*c*d*e^4 + 29*b*e^5)*x^10 + 11*(781*c*d^2*e^3 + 1421*b*d*e^4 + 667*a*e^5)*x^7 + (405*c*d^3*e^2 + 14 123*b*d^2*e^3 + 24679*a*d*e^4)*x^4 - (648*c*d^4*e - 2349*b*d^3*e^2 - 35351 *a*d^2*e^3)*x)*sqrt(e*x^3 + d))/e^3
Time = 3.68 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {a d^{\frac {5}{2}} 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 {2 a d^{\frac {3}{2}} e 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 {a \sqrt {d} e^{2} 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 )} + \frac {b d^{\frac {5}{2}} 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 {2 b d^{\frac {3}{2}} e 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 )} + \frac {b \sqrt {d} e^{2} x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} + \frac {c d^{\frac {5}{2}} 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 )} + \frac {2 c d^{\frac {3}{2}} e x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} + \frac {c \sqrt {d} e^{2} x^{13} \Gamma \left (\frac {13}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{3} \\ \frac {16}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {16}{3}\right )} \]
a*d**(5/2)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/ d)/(3*gamma(4/3)) + 2*a*d**(3/2)*e*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3 ,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + a*sqrt(d)*e**2*x**7*gamma(7 /3)*hyper((-1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + b*d**(5/2)*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), e*x**3*exp_polar(I *pi)/d)/(3*gamma(7/3)) + 2*b*d**(3/2)*e*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + b*sqrt(d)*e**2*x**10 *gamma(10/3)*hyper((-1/2, 10/3), (13/3,), e*x**3*exp_polar(I*pi)/d)/(3*gam ma(13/3)) + c*d**(5/2)*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), e*x**3* exp_polar(I*pi)/d)/(3*gamma(10/3)) + 2*c*d**(3/2)*e*x**10*gamma(10/3)*hype r((-1/2, 10/3), (13/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(13/3)) + c*sqr t(d)*e**2*x**13*gamma(13/3)*hyper((-1/2, 13/3), (16/3,), e*x**3*exp_polar( I*pi)/d)/(3*gamma(16/3))
\[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {5}{2}} \,d x } \]
\[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx=\int {\left (e\,x^3+d\right )}^{5/2}\,\left (c\,x^6+b\,x^3+a\right ) \,d x \]