Integrand size = 24, antiderivative size = 389 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {2 \left (23 c d^2-2 b d e-19 a e^2\right ) x}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac {2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}+\frac {2 \left (16 c d^2+26 b d e+247 a e^2\right ) x}{1215 d^4 e^2 \sqrt {d+e x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2+26 b d e+247 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{1215 \sqrt [4]{3} d^4 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:
2/21*(a*e^2-b*d*e+c*d^2)*x/d/e^2/(e*x^3+d)^(7/2)-2/315*(-19*a*e^2-2*b*d*e+ 23*c*d^2)*x/d^2/e^2/(e*x^3+d)^(5/2)+2/2835*(247*a*e^2+26*b*d*e+16*c*d^2)*x /d^3/e^2/(e*x^3+d)^(3/2)+2/1215*(247*a*e^2+26*b*d*e+16*c*d^2)*x/d^4/e^2/(e *x^3+d)^(1/2)+2/3645*(1/2*6^(1/2)+1/2*2^(1/2))*(247*a*e^2+26*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)*3^(3/4)/d^4/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 = 10.23 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.51 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\frac {2 x \left (c d^2 \left (-56 d^3-189 d^2 e x^3+384 d e^2 x^6+112 e^3 x^9\right )+e \left (b d \left (-91 d^3+756 d^2 e x^3+624 d e^2 x^6+182 e^3 x^9\right )+a e \left (3388 d^3+7182 d^2 e x^3+5928 d e^2 x^6+1729 e^3 x^9\right )\right )\right )+7 \left (16 c d^2+13 e (2 b d+19 a e)\right ) x \left (d+e x^3\right )^3 \sqrt {1+\frac {e x^3}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {e x^3}{d}\right )}{8505 d^4 e^2 \left (d+e x^3\right )^{7/2}} \] Input:
Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]
Output:
(2*x*(c*d^2*(-56*d^3 - 189*d^2*e*x^3 + 384*d*e^2*x^6 + 112*e^3*x^9) + e*(b *d*(-91*d^3 + 756*d^2*e*x^3 + 624*d*e^2*x^6 + 182*e^3*x^9) + a*e*(3388*d^3 + 7182*d^2*e*x^3 + 5928*d*e^2*x^6 + 1729*e^3*x^9))) + 7*(16*c*d^2 + 13*e* (2*b*d + 19*a*e))*x*(d + e*x^3)^3*Sqrt[1 + (e*x^3)/d]*Hypergeometric2F1[1/ 3, 1/2, 4/3, -((e*x^3)/d)])/(8505*d^4*e^2*(d + e*x^3)^(7/2))
Time = 0.44 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1739, 27, 910, 749, 749, 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.
| \(\displaystyle \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1739 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {2 \int \frac {-21 c d e x^3+2 c d^2-e (2 b d+19 a e)}{2 \left (e x^3+d\right )^{7/2}}dx}{21 d e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {\int \frac {-21 c d e x^3+2 c d^2-19 a e^2-2 b d e}{\left (e x^3+d\right )^{7/2}}dx}{21 d e^2}\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {\frac {2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{15 d \left (d+e x^3\right )^{5/2}}-\frac {\left (247 a e^2+26 b d e+16 c d^2\right ) \int \frac {1}{\left (e x^3+d\right )^{5/2}}dx}{15 d}}{21 d e^2}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {\frac {2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{15 d \left (d+e x^3\right )^{5/2}}-\frac {\left (247 a e^2+26 b d e+16 c d^2\right ) \left (\frac {7 \int \frac {1}{\left (e x^3+d\right )^{3/2}}dx}{9 d}+\frac {2 x}{9 d \left (d+e x^3\right )^{3/2}}\right )}{15 d}}{21 d e^2}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {\frac {2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{15 d \left (d+e x^3\right )^{5/2}}-\frac {\left (247 a e^2+26 b d e+16 c d^2\right ) \left (\frac {7 \left (\frac {\int \frac {1}{\sqrt {e x^3+d}}dx}{3 d}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{9 d}+\frac {2 x}{9 d \left (d+e x^3\right )^{3/2}}\right )}{15 d}}{21 d e^2}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}-\frac {\frac {2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{15 d \left (d+e x^3\right )^{5/2}}-\frac {\left (247 a e^2+26 b d e+16 c d^2\right ) \left (\frac {7 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{9 d}+\frac {2 x}{9 d \left (d+e x^3\right )^{3/2}}\right )}{15 d}}{21 d e^2}\) |
Input:
Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]
Output:
(2*(c*d^2 - b*d*e + a*e^2)*x)/(21*d*e^2*(d + e*x^3)^(7/2)) - ((2*(23*c*d^2 - 2*b*d*e - 19*a*e^2)*x)/(15*d*(d + e*x^3)^(5/2)) - ((16*c*d^2 + 26*b*d*e + 247*a*e^2)*((2*x)/(9*d*(d + e*x^3)^(3/2)) + (7*((2*x)/(3*d*Sqrt[d + e*x ^3]) + (2*Sqrt[2 + Sqrt[3]]*(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[ 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]])/(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)))/ (15*d))/(21*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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[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[(-(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 = 0.67 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.24
| method | result | size |
| elliptic | \(\frac {2 x \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{21 d \,e^{6} \left (x^{3}+\frac {d}{e}\right )^{4}}+\frac {2 x \left (19 a \,e^{2}+2 b d e -23 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{315 d^{2} e^{5} \left (x^{3}+\frac {d}{e}\right )^{3}}+\frac {2 x \left (247 a \,e^{2}+26 b d e +16 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{2835 d^{3} e^{4} \left (x^{3}+\frac {d}{e}\right )^{2}}+\frac {2 x \left (247 a \,e^{2}+26 b d e +16 c \,d^{2}\right )}{1215 e^{2} d^{4} \sqrt {\left (x^{3}+\frac {d}{e}\right ) e}}-\frac {2 i \left (247 a \,e^{2}+26 b d e +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3645 d^{4} e^{3} \sqrt {e \,x^{3}+d}}\) | \(484\) |
| default | \(\text {Expression too large to display}\) | \(1182\) |
Input:
int((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
2/21*x/d/e^6*(a*e^2-b*d*e+c*d^2)*(e*x^3+d)^(1/2)/(x^3+d/e)^4+2/315*x/d^2*( 19*a*e^2+2*b*d*e-23*c*d^2)/e^5*(e*x^3+d)^(1/2)/(x^3+d/e)^3+2/2835*x/d^3*(2 47*a*e^2+26*b*d*e+16*c*d^2)/e^4*(e*x^3+d)^(1/2)/(x^3+d/e)^2+2/1215/e^2*x/d ^4*(247*a*e^2+26*b*d*e+16*c*d^2)/((x^3+d/e)*e)^(1/2)-2/3645*I*(247*a*e^2+2 6*b*d*e+16*c*d^2)/d^4/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 = 0.08 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\frac {2 \, {\left (7 \, {\left ({\left (16 \, c d^{2} e^{4} + 26 \, b d e^{5} + 247 \, a e^{6}\right )} x^{12} + 4 \, {\left (16 \, c d^{3} e^{3} + 26 \, b d^{2} e^{4} + 247 \, a d e^{5}\right )} x^{9} + 16 \, c d^{6} + 26 \, b d^{5} e + 247 \, a d^{4} e^{2} + 6 \, {\left (16 \, c d^{4} e^{2} + 26 \, b d^{3} e^{3} + 247 \, a d^{2} e^{4}\right )} x^{6} + 4 \, {\left (16 \, c d^{5} e + 26 \, b d^{4} e^{2} + 247 \, a d^{3} e^{3}\right )} x^{3}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) + {\left (7 \, {\left (16 \, c d^{2} e^{4} + 26 \, b d e^{5} + 247 \, a e^{6}\right )} x^{10} + 24 \, {\left (16 \, c d^{3} e^{3} + 26 \, b d^{2} e^{4} + 247 \, a d e^{5}\right )} x^{7} - 189 \, {\left (c d^{4} e^{2} - 4 \, b d^{3} e^{3} - 38 \, a d^{2} e^{4}\right )} x^{4} - 7 \, {\left (8 \, c d^{5} e + 13 \, b d^{4} e^{2} - 484 \, a d^{3} e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{8505 \, {\left (d^{4} e^{7} x^{12} + 4 \, d^{5} e^{6} x^{9} + 6 \, d^{6} e^{5} x^{6} + 4 \, d^{7} e^{4} x^{3} + d^{8} e^{3}\right )}} \] Input:
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="fricas")
Output:
2/8505*(7*((16*c*d^2*e^4 + 26*b*d*e^5 + 247*a*e^6)*x^12 + 4*(16*c*d^3*e^3 + 26*b*d^2*e^4 + 247*a*d*e^5)*x^9 + 16*c*d^6 + 26*b*d^5*e + 247*a*d^4*e^2 + 6*(16*c*d^4*e^2 + 26*b*d^3*e^3 + 247*a*d^2*e^4)*x^6 + 4*(16*c*d^5*e + 26 *b*d^4*e^2 + 247*a*d^3*e^3)*x^3)*sqrt(e)*weierstrassPInverse(0, -4*d/e, x) + (7*(16*c*d^2*e^4 + 26*b*d*e^5 + 247*a*e^6)*x^10 + 24*(16*c*d^3*e^3 + 26 *b*d^2*e^4 + 247*a*d*e^5)*x^7 - 189*(c*d^4*e^2 - 4*b*d^3*e^3 - 38*a*d^2*e^ 4)*x^4 - 7*(8*c*d^5*e + 13*b*d^4*e^2 - 484*a*d^3*e^3)*x)*sqrt(e*x^3 + d))/ (d^4*e^7*x^12 + 4*d^5*e^6*x^9 + 6*d^6*e^5*x^6 + 4*d^7*e^4*x^3 + d^8*e^3)
Timed out. \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(9/2),x)
Output:
Timed out
\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="maxima")
Output:
integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2), x)
\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x, algorithm="giac")
Output:
integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2), x)
Timed out. \[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx=\int \frac {c\,x^6+b\,x^3+a}{{\left (e\,x^3+d\right )}^{9/2}} \,d x \] Input:
int((a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x)
Output:
int((a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2), x)
\[ \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx =\text {Too large to display} \] Input:
int((c*x^6+b*x^3+a)/(e*x^3+d)^(9/2),x)
Output:
( - 26*sqrt(d + e*x**3)*b*e*x - 16*sqrt(d + e*x**3)*c*d*x - 38*sqrt(d + e* x**3)*c*e*x**4 + 247*int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3* e**2*x**6 + 10*d**2*e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*a*d**4*e** 2 + 988*int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e**2*x**6 + 1 0*d**2*e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*a*d**3*e**3*x**3 + 1482 *int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e**2*x**6 + 10*d**2* e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*a*d**2*e**4*x**6 + 988*int(sqr t(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e**2*x**6 + 10*d**2*e**3*x** 9 + 5*d*e**4*x**12 + e**5*x**15),x)*a*d*e**5*x**9 + 247*int(sqrt(d + e*x** 3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e**2*x**6 + 10*d**2*e**3*x**9 + 5*d*e** 4*x**12 + e**5*x**15),x)*a*e**6*x**12 + 26*int(sqrt(d + e*x**3)/(d**5 + 5* d**4*e*x**3 + 10*d**3*e**2*x**6 + 10*d**2*e**3*x**9 + 5*d*e**4*x**12 + e** 5*x**15),x)*b*d**5*e + 104*int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10 *d**3*e**2*x**6 + 10*d**2*e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*b*d* *4*e**2*x**3 + 156*int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e* *2*x**6 + 10*d**2*e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*b*d**3*e**3* x**6 + 104*int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e**2*x**6 + 10*d**2*e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*b*d**2*e**4*x**9 + 2 6*int(sqrt(d + e*x**3)/(d**5 + 5*d**4*e*x**3 + 10*d**3*e**2*x**6 + 10*d**2 *e**3*x**9 + 5*d*e**4*x**12 + e**5*x**15),x)*b*d*e**5*x**12 + 16*int(sq...