Integrand size = 17, antiderivative size = 211 \[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\frac {1944 e x \sqrt {9+x^4}}{13 \left (3+x^2\right )}+\frac {108 x \left (195 d+77 e x^2\right ) \sqrt {9+x^4}}{1001}+\frac {10 x \left (117 d+77 e x^2\right ) \left (9+x^4\right )^{3/2}}{1001}+\frac {1}{143} x \left (13 d+11 e x^2\right ) \left (9+x^4\right )^{5/2}-\frac {1944 \sqrt {3} e \left (3+x^2\right ) \sqrt {\frac {9+x^4}{\left (3+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right )|\frac {1}{2}\right )}{13 \sqrt {9+x^4}}+\frac {972 \sqrt {3} (65 d+77 e) \left (3+x^2\right ) \sqrt {\frac {9+x^4}{\left (3+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{1001 \sqrt {9+x^4}} \] Output:
1944*e*x*(x^4+9)^(1/2)/(13*x^2+39)+108/1001*x*(77*e*x^2+195*d)*(x^4+9)^(1/ 2)+10/1001*x*(77*e*x^2+117*d)*(x^4+9)^(3/2)+1/143*x*(11*e*x^2+13*d)*(x^4+9 )^(5/2)-1944/13*3^(1/2)*e*(x^2+3)*((x^4+9)/(x^2+3)^2)^(1/2)*EllipticE(sin( 2*arctan(1/3*x*3^(1/2))),1/2*2^(1/2))/(x^4+9)^(1/2)+972/1001*3^(1/2)*(65*d +77*e)*(x^2+3)*((x^4+9)/(x^2+3)^2)^(1/2)*InverseJacobiAM(2*arctan(1/3*x*3^ (1/2)),1/2*2^(1/2))/(x^4+9)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.64 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.21 \[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=243 d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},-\frac {x^4}{9}\right )+81 e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},-\frac {x^4}{9}\right ) \] Input:
Integrate[(d + e*x^2)*(9 + x^4)^(5/2),x]
Output:
243*d*x*Hypergeometric2F1[-5/2, 1/4, 5/4, -1/9*x^4] + 81*e*x^3*Hypergeomet ric2F1[-5/2, 3/4, 7/4, -1/9*x^4]
Time = 0.62 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {1491, 27, 1491, 27, 1491, 27, 1512, 27, 761, 1510}
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.
| \(\displaystyle \int \left (x^4+9\right )^{5/2} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {5}{143} \int 18 \left (11 e x^2+13 d\right ) \left (x^4+9\right )^{3/2}dx+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \int \left (11 e x^2+13 d\right ) \left (x^4+9\right )^{3/2}dx+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {90}{143} \left (\frac {1}{21} \int 18 \left (77 e x^2+117 d\right ) \sqrt {x^4+9}dx+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \int \left (77 e x^2+117 d\right ) \sqrt {x^4+9}dx+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {1}{15} \int \frac {54 \left (77 e x^2+195 d\right )}{\sqrt {x^4+9}}dx+\frac {1}{5} x \sqrt {x^4+9} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \int \frac {77 e x^2+195 d}{\sqrt {x^4+9}}dx+\frac {1}{5} x \sqrt {x^4+9} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (3 (65 d+77 e) \int \frac {1}{\sqrt {x^4+9}}dx-231 e \int \frac {3-x^2}{3 \sqrt {x^4+9}}dx\right )+\frac {1}{5} x \sqrt {x^4+9} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (3 (65 d+77 e) \int \frac {1}{\sqrt {x^4+9}}dx-77 e \int \frac {3-x^2}{\sqrt {x^4+9}}dx\right )+\frac {1}{5} x \sqrt {x^4+9} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (\frac {\sqrt {3} \left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} (65 d+77 e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {x^4+9}}-77 e \int \frac {3-x^2}{\sqrt {x^4+9}}dx\right )+\frac {1}{5} x \sqrt {x^4+9} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (\frac {\sqrt {3} \left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} (65 d+77 e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {x^4+9}}-77 e \left (\frac {\sqrt {3} \left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+9}}-\frac {x \sqrt {x^4+9}}{x^2+3}\right )\right )+\frac {1}{5} x \sqrt {x^4+9} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (x^4+9\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (x^4+9\right )^{5/2} \left (13 d+11 e x^2\right )\) |
Input:
Int[(d + e*x^2)*(9 + x^4)^(5/2),x]
Output:
(x*(13*d + 11*e*x^2)*(9 + x^4)^(5/2))/143 + (90*((x*(117*d + 77*e*x^2)*(9 + x^4)^(3/2))/63 + (6*((x*(195*d + 77*e*x^2)*Sqrt[9 + x^4])/5 + (18*(-77*e *(-((x*Sqrt[9 + x^4])/(3 + x^2)) + (Sqrt[3]*(3 + x^2)*Sqrt[(9 + x^4)/(3 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[3]], 1/2])/Sqrt[9 + x^4]) + (Sqrt[3]*(65 *d + 77*e)*(3 + x^2)*Sqrt[(9 + x^4)/(3 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt [3]], 1/2])/(2*Sqrt[9 + x^4])))/5))/7))/143
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_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*( d*(4*p + 3) + e*(4*p + 1)*x^2)*((a + c*x^4)^p/((4*p + 1)*(4*p + 3))), x] + Simp[2*(p/((4*p + 1)*(4*p + 3))) Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 1))*x^2, x]*(a + c*x^4)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c *d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Time = 1.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.16
| method | result | size |
| meijerg | \(81 e \,x^{3} \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {x^{4}}{9}\right )+243 d x \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -\frac {x^{4}}{9}\right )\) | \(34\) |
| risch | \(\frac {x \left (77 e \,x^{10}+91 d \,x^{8}+2156 e \,x^{6}+2808 d \,x^{4}+21483 e \,x^{2}+38961 d \right ) \sqrt {x^{4}+9}}{1001}+\frac {3240 d \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{77 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}+\frac {648 i e \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{13 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\) | \(189\) |
| default | \(d \left (\frac {x^{9} \sqrt {x^{4}+9}}{11}+\frac {216 x^{5} \sqrt {x^{4}+9}}{77}+\frac {2997 x \sqrt {x^{4}+9}}{77}+\frac {3240 \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{77 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\right )+e \left (\frac {x^{11} \sqrt {x^{4}+9}}{13}+\frac {28 x^{7} \sqrt {x^{4}+9}}{13}+\frac {279 x^{3} \sqrt {x^{4}+9}}{13}+\frac {648 i \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{13 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\right )\) | \(219\) |
| elliptic | \(\frac {e \,x^{11} \sqrt {x^{4}+9}}{13}+\frac {d \,x^{9} \sqrt {x^{4}+9}}{11}+\frac {28 e \,x^{7} \sqrt {x^{4}+9}}{13}+\frac {216 d \,x^{5} \sqrt {x^{4}+9}}{77}+\frac {279 e \,x^{3} \sqrt {x^{4}+9}}{13}+\frac {2997 d x \sqrt {x^{4}+9}}{77}+\frac {3240 d \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{77 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}+\frac {648 i e \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{13 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\) | \(221\) |
Input:
int((e*x^2+d)*(x^4+9)^(5/2),x,method=_RETURNVERBOSE)
Output:
81*e*x^3*hypergeom([-5/2,3/4],[7/4],-1/9*x^4)+243*d*x*hypergeom([-5/2,1/4] ,[5/4],-1/9*x^4)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.45 \[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\frac {449064 i \, \sqrt {3 i} e x E(\arcsin \left (\frac {\sqrt {3 i}}{x}\right )\,|\,-1) + 1944 i \, \sqrt {3 i} {\left (65 \, d - 231 \, e\right )} x F(\arcsin \left (\frac {\sqrt {3 i}}{x}\right )\,|\,-1) + {\left (77 \, e x^{12} + 91 \, d x^{10} + 2156 \, e x^{8} + 2808 \, d x^{6} + 21483 \, e x^{4} + 38961 \, d x^{2} + 149688 \, e\right )} \sqrt {x^{4} + 9}}{1001 \, x} \] Input:
integrate((e*x^2+d)*(x^4+9)^(5/2),x, algorithm="fricas")
Output:
1/1001*(449064*I*sqrt(3*I)*e*x*elliptic_e(arcsin(sqrt(3*I)/x), -1) + 1944* I*sqrt(3*I)*(65*d - 231*e)*x*elliptic_f(arcsin(sqrt(3*I)/x), -1) + (77*e*x ^12 + 91*d*x^10 + 2156*e*x^8 + 2808*d*x^6 + 21483*e*x^4 + 38961*d*x^2 + 14 9688*e)*sqrt(x^4 + 9))/x
Result contains complex when optimal does not.
Time = 1.92 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.05 \[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\frac {3 d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {27 d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {243 d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 e x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} + \frac {27 e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {243 e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)*(x**4+9)**(5/2),x)
Output:
3*d*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), x**4*exp_polar(I*pi)/9)/(4 *gamma(13/4)) + 27*d*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), x**4*exp_p olar(I*pi)/9)/(2*gamma(9/4)) + 243*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4, ), x**4*exp_polar(I*pi)/9)/(4*gamma(5/4)) + 3*e*x**11*gamma(11/4)*hyper((- 1/2, 11/4), (15/4,), x**4*exp_polar(I*pi)/9)/(4*gamma(15/4)) + 27*e*x**7*g amma(7/4)*hyper((-1/2, 7/4), (11/4,), x**4*exp_polar(I*pi)/9)/(2*gamma(11/ 4)) + 243*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), x**4*exp_polar(I*pi )/9)/(4*gamma(7/4))
\[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\int { {\left (x^{4} + 9\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(x^4+9)^(5/2),x, algorithm="maxima")
Output:
integrate((x^4 + 9)^(5/2)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\int { {\left (x^{4} + 9\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(x^4+9)^(5/2),x, algorithm="giac")
Output:
integrate((x^4 + 9)^(5/2)*(e*x^2 + d), x)
Timed out. \[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\int {\left (x^4+9\right )}^{5/2}\,\left (e\,x^2+d\right ) \,d x \] Input:
int((x^4 + 9)^(5/2)*(d + e*x^2),x)
Output:
int((x^4 + 9)^(5/2)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \left (9+x^4\right )^{5/2} \, dx=\frac {\sqrt {x^{4}+9}\, d \,x^{9}}{11}+\frac {216 \sqrt {x^{4}+9}\, d \,x^{5}}{77}+\frac {2997 \sqrt {x^{4}+9}\, d x}{77}+\frac {\sqrt {x^{4}+9}\, e \,x^{11}}{13}+\frac {28 \sqrt {x^{4}+9}\, e \,x^{7}}{13}+\frac {279 \sqrt {x^{4}+9}\, e \,x^{3}}{13}+\frac {29160 \left (\int \frac {\sqrt {x^{4}+9}}{x^{4}+9}d x \right ) d}{77}+\frac {1944 \left (\int \frac {\sqrt {x^{4}+9}\, x^{2}}{x^{4}+9}d x \right ) e}{13} \] Input:
int((e*x^2+d)*(x^4+9)^(5/2),x)
Output:
(91*sqrt(x**4 + 9)*d*x**9 + 2808*sqrt(x**4 + 9)*d*x**5 + 38961*sqrt(x**4 + 9)*d*x + 77*sqrt(x**4 + 9)*e*x**11 + 2156*sqrt(x**4 + 9)*e*x**7 + 21483*s qrt(x**4 + 9)*e*x**3 + 379080*int(sqrt(x**4 + 9)/(x**4 + 9),x)*d + 149688* int((sqrt(x**4 + 9)*x**2)/(x**4 + 9),x)*e)/1001