Integrand size = 24, antiderivative size = 356 \[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {18 d \left (16 c d^2-46 b d e+391 a e^2\right ) x \sqrt {d+e x^3}}{21505 e^2}+\frac {2 \left (16 c d^2-46 b d e+391 a e^2\right ) x \left (d+e x^3\right )^{3/2}}{4301 e^2}-\frac {2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} d^2 \left (16 c d^2-46 b d e+391 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{21505 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:
18/21505*d*(391*a*e^2-46*b*d*e+16*c*d^2)*x*(e*x^3+d)^(1/2)/e^2+2/4301*(391 *a*e^2-46*b*d*e+16*c*d^2)*x*(e*x^3+d)^(3/2)/e^2-2/391*(-23*b*e+8*c*d)*x*(e *x^3+d)^(5/2)/e^2+2/23*c*x^4*(e*x^3+d)^(5/2)/e+18/21505*3^(3/4)*(1/2*6^(1/ 2)+1/2*2^(1/2))*d^2*(391*a*e^2-46*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 = 8.77 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.28 \[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {x \sqrt {d+e x^3} \left (-2 \left (d+e x^3\right )^2 \left (8 c d-23 b e-17 c e x^3\right )+\frac {\left (16 c d^3+23 d e (-2 b d+17 a e)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {e x^3}{d}\right )}{\sqrt {1+\frac {e x^3}{d}}}\right )}{391 e^2} \] Input:
Integrate[(d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6),x]
Output:
(x*Sqrt[d + e*x^3]*(-2*(d + e*x^3)^2*(8*c*d - 23*b*e - 17*c*e*x^3) + ((16* c*d^3 + 23*d*e*(-2*b*d + 17*a*e))*Hypergeometric2F1[-3/2, 1/3, 4/3, -((e*x ^3)/d)])/Sqrt[1 + (e*x^3)/d]))/(391*e^2)
Time = 0.39 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1741, 27, 913, 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.
| \(\displaystyle \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \left (e x^3+d\right )^{3/2} \left (23 a e-(8 c d-23 b e) x^3\right )dx}{23 e}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \left (e x^3+d\right )^{3/2} \left (23 a e-(8 c d-23 b e) x^3\right )dx}{23 e}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-23 e (2 b d-17 a e)\right ) \int \left (e x^3+d\right )^{3/2}dx}{17 e}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{17 e}}{23 e}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-23 e (2 b d-17 a e)\right ) \left (\frac {9}{11} d \int \sqrt {e x^3+d}dx+\frac {2}{11} x \left (d+e x^3\right )^{3/2}\right )}{17 e}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{17 e}}{23 e}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-23 e (2 b d-17 a e)\right ) \left (\frac {9}{11} d \left (\frac {3}{5} d \int \frac {1}{\sqrt {e x^3+d}}dx+\frac {2}{5} x \sqrt {d+e x^3}\right )+\frac {2}{11} x \left (d+e x^3\right )^{3/2}\right )}{17 e}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{17 e}}{23 e}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {\left (16 c d^2-23 e (2 b d-17 a e)\right ) \left (\frac {9}{11} d \left (\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2}{5} x \sqrt {d+e x^3}\right )+\frac {2}{11} x \left (d+e x^3\right )^{3/2}\right )}{17 e}-\frac {2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{17 e}}{23 e}+\frac {2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}\) |
Input:
Int[(d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6),x]
Output:
(2*c*x^4*(d + e*x^3)^(5/2))/(23*e) + ((-2*(8*c*d - 23*b*e)*x*(d + e*x^3)^( 5/2))/(17*e) + ((16*c*d^2 - 23*e*(2*b*d - 17*a*e))*((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 + S qrt[3])*d^(1/3) + e^(1/3)*x)^2]*EllipticF[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*e))/(23*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 = 0.63 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {2 x \left (935 c \,e^{3} x^{9}+1265 b \,e^{3} x^{6}+1430 c d \,e^{2} x^{6}+1955 e^{3} a \,x^{3}+2300 b d \,e^{2} x^{3}+135 d^{2} e c \,x^{3}+5474 a d \,e^{2}+621 b \,d^{2} e -216 c \,d^{3}\right ) \sqrt {e \,x^{3}+d}}{21505 e^{2}}-\frac {18 i d^{2} \left (391 a \,e^{2}-46 b d e +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{21505 e^{3} \sqrt {e \,x^{3}+d}}\) | \(398\) |
| elliptic | \(\frac {2 c e \,x^{10} \sqrt {e \,x^{3}+d}}{23}+\frac {2 \left (b \,e^{2}+\frac {26}{23} c d e \right ) x^{7} \sqrt {e \,x^{3}+d}}{17 e}+\frac {2 \left (a \,e^{2}+2 b d e +c \,d^{2}-\frac {14 d \left (b \,e^{2}+\frac {26}{23} c d e \right )}{17 e}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (2 a d e +b \,d^{2}-\frac {8 d \left (a \,e^{2}+2 b d e +c \,d^{2}-\frac {14 d \left (b \,e^{2}+\frac {26}{23} c d e \right )}{17 e}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (a \,d^{2}-\frac {2 d \left (2 a d e +b \,d^{2}-\frac {8 d \left (a \,e^{2}+2 b d e +c \,d^{2}-\frac {14 d \left (b \,e^{2}+\frac {26}{23} c d e \right )}{17 e}\right )}{11 e}\right )}{5 e}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\) | \(505\) |
| default | \(\text {Expression too large to display}\) | \(1010\) |
Input:
int((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
2/21505/e^2*x*(935*c*e^3*x^9+1265*b*e^3*x^6+1430*c*d*e^2*x^6+1955*a*e^3*x^ 3+2300*b*d*e^2*x^3+135*c*d^2*e*x^3+5474*a*d*e^2+621*b*d^2*e-216*c*d^3)*(e* x^3+d)^(1/2)-18/21505*I*d^2*(391*a*e^2-46*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 = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.38 \[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {2 \, {\left (27 \, {\left (16 \, c d^{4} - 46 \, b d^{3} e + 391 \, a d^{2} e^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) + {\left (935 \, c e^{4} x^{10} + 55 \, {\left (26 \, c d e^{3} + 23 \, b e^{4}\right )} x^{7} + 5 \, {\left (27 \, c d^{2} e^{2} + 460 \, b d e^{3} + 391 \, a e^{4}\right )} x^{4} - {\left (216 \, c d^{3} e - 621 \, b d^{2} e^{2} - 5474 \, a d e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{21505 \, e^{3}} \] Input:
integrate((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
2/21505*(27*(16*c*d^4 - 46*b*d^3*e + 391*a*d^2*e^2)*sqrt(e)*weierstrassPIn verse(0, -4*d/e, x) + (935*c*e^4*x^10 + 55*(26*c*d*e^3 + 23*b*e^4)*x^7 + 5 *(27*c*d^2*e^2 + 460*b*d*e^3 + 391*a*e^4)*x^4 - (216*c*d^3*e - 621*b*d^2*e ^2 - 5474*a*d*e^3)*x)*sqrt(e*x^3 + d))/e^3
Time = 2.51 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.72 \[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {a d^{\frac {3}{2}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a \sqrt {d} e x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b d^{\frac {3}{2}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b \sqrt {d} e x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {c d^{\frac {3}{2}} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {c \sqrt {d} e x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \] Input:
integrate((e*x**3+d)**(3/2)*(c*x**6+b*x**3+a),x)
Output:
a*d**(3/2)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/ d)/(3*gamma(4/3)) + a*sqrt(d)*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)) + b*d**(3/2)*x**4*gamma(4/3)*hyp er((-1/2, 4/3), (7/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + b*sqrt( d)*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)) + c*d**(3/2)*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + c*sqrt(d)*e*x**10*gamma(10/3)* hyper((-1/2, 10/3), (13/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(13/3))
\[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
integrate((c*x^6 + b*x^3 + a)*(e*x^3 + d)^(3/2), x)
\[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
integrate((c*x^6 + b*x^3 + a)*(e*x^3 + d)^(3/2), x)
Timed out. \[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\int {\left (e\,x^3+d\right )}^{3/2}\,\left (c\,x^6+b\,x^3+a\right ) \,d x \] Input:
int((d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6),x)
Output:
int((d + e*x^3)^(3/2)*(a + b*x^3 + c*x^6), x)
\[ \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx=\frac {10948 \sqrt {e \,x^{3}+d}\, a d \,e^{2} x +3910 \sqrt {e \,x^{3}+d}\, a \,e^{3} x^{4}+1242 \sqrt {e \,x^{3}+d}\, b \,d^{2} e x +4600 \sqrt {e \,x^{3}+d}\, b d \,e^{2} x^{4}+2530 \sqrt {e \,x^{3}+d}\, b \,e^{3} x^{7}-432 \sqrt {e \,x^{3}+d}\, c \,d^{3} x +270 \sqrt {e \,x^{3}+d}\, c \,d^{2} e \,x^{4}+2860 \sqrt {e \,x^{3}+d}\, c d \,e^{2} x^{7}+1870 \sqrt {e \,x^{3}+d}\, c \,e^{3} x^{10}+10557 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) a \,d^{2} e^{2}-1242 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) b \,d^{3} e +432 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) c \,d^{4}}{21505 e^{2}} \] Input:
int((e*x^3+d)^(3/2)*(c*x^6+b*x^3+a),x)
Output:
(10948*sqrt(d + e*x**3)*a*d*e**2*x + 3910*sqrt(d + e*x**3)*a*e**3*x**4 + 1 242*sqrt(d + e*x**3)*b*d**2*e*x + 4600*sqrt(d + e*x**3)*b*d*e**2*x**4 + 25 30*sqrt(d + e*x**3)*b*e**3*x**7 - 432*sqrt(d + e*x**3)*c*d**3*x + 270*sqrt (d + e*x**3)*c*d**2*e*x**4 + 2860*sqrt(d + e*x**3)*c*d*e**2*x**7 + 1870*sq rt(d + e*x**3)*c*e**3*x**10 + 10557*int(sqrt(d + e*x**3)/(d + e*x**3),x)*a *d**2*e**2 - 1242*int(sqrt(d + e*x**3)/(d + e*x**3),x)*b*d**3*e + 432*int( sqrt(d + e*x**3)/(d + e*x**3),x)*c*d**4)/(21505*e**2)