Integrand size = 19, antiderivative size = 125 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{5/2} \, dx=\frac {108 x \left (195 d+77 e x^2\right ) \sqrt {9-x^4}}{1001}+\frac {10 x \left (117 d+77 e x^2\right ) \left (9-x^4\right )^{3/2}}{1001}+\frac {1}{143} x \left (13 d+11 e x^2\right ) \left (9-x^4\right )^{5/2}+\frac {1944}{13} \sqrt {3} e E\left (\left .\arcsin \left (\frac {x}{\sqrt {3}}\right )\right |-1\right )+\frac {1944 \sqrt {3} (65 d-77 e) \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3}}\right ),-1\right )}{1001} \] Output:
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*Ellipt icE(1/3*x*3^(1/2),I)+1944/1001*3^(1/2)*(65*d-77*e)*EllipticF(1/3*x*3^(1/2) ,I)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.74 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.36 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{5/2} \, dx=243 d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},\frac {x^4}{9}\right )+81 e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},\frac {x^4}{9}\right ) \] Input:
Integrate[(d + e*x^2)*(9 - x^4)^(5/2),x]
Output:
243*d*x*Hypergeometric2F1[-5/2, 1/4, 5/4, x^4/9] + 81*e*x^3*Hypergeometric 2F1[-5/2, 3/4, 7/4, x^4/9]
Time = 0.58 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1491, 27, 1491, 27, 1491, 27, 1495, 399, 284, 327, 762}
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.
| \(\displaystyle \int \left (9-x^4\right )^{5/2} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {5}{143} \int 18 \left (11 e x^2+13 d\right ) \left (9-x^4\right )^{3/2}dx+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \int \left (11 e x^2+13 d\right ) \left (9-x^4\right )^{3/2}dx+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {90}{143} \left (\frac {1}{21} \int 18 \left (77 e x^2+117 d\right ) \sqrt {9-x^4}dx+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \int \left (77 e x^2+117 d\right ) \sqrt {9-x^4}dx+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {1}{15} \int \frac {54 \left (77 e x^2+195 d\right )}{\sqrt {9-x^4}}dx+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \int \frac {77 e x^2+195 d}{\sqrt {9-x^4}}dx+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 1495 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \int \frac {77 e x^2+195 d}{\sqrt {3-x^2} \sqrt {x^2+3}}dx+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (3 (65 d-77 e) \int \frac {1}{\sqrt {3-x^2} \sqrt {x^2+3}}dx+77 e \int \frac {\sqrt {x^2+3}}{\sqrt {3-x^2}}dx\right )+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 284 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (3 (65 d-77 e) \int \frac {1}{\sqrt {9-x^4}}dx+77 e \int \frac {\sqrt {x^2+3}}{\sqrt {3-x^2}}dx\right )+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (3 (65 d-77 e) \int \frac {1}{\sqrt {9-x^4}}dx+77 \sqrt {3} e E\left (\left .\arcsin \left (\frac {x}{\sqrt {3}}\right )\right |-1\right )\right )+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {90}{143} \left (\frac {6}{7} \left (\frac {18}{5} \left (\sqrt {3} (65 d-77 e) \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3}}\right ),-1\right )+77 \sqrt {3} e E\left (\left .\arcsin \left (\frac {x}{\sqrt {3}}\right )\right |-1\right )\right )+\frac {1}{5} x \sqrt {9-x^4} \left (195 d+77 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (117 d+77 e x^2\right )\right )+\frac {1}{143} x \left (9-x^4\right )^{5/2} \left (13 d+11 e x^2\right )\) |
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*Sq rt[3]*e*EllipticE[ArcSin[x/Sqrt[3]], -1] + Sqrt[3]*(65*d - 77*e)*EllipticF [ArcSin[x/Sqrt[3]], -1]))/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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> I nt[(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0]))
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
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[(-a)*c, 2]}, Simp[Sqrt[-c] Int[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 1.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27
| 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 | \(-\frac {x \left (77 e \,x^{10}+91 d \,x^{8}-2156 e \,x^{6}-2808 d \,x^{4}+21483 e \,x^{2}+38961 d \right ) \left (x^{4}-9\right )}{1001 \sqrt {-x^{4}+9}}+\frac {3240 d \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )}{77 \sqrt {-x^{4}+9}}-\frac {648 e \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3}\, x}{3}, i\right )\right )}{13 \sqrt {-x^{4}+9}}\) | \(149\) |
| default | \(d \left (\frac {x^{9} \sqrt {-x^{4}+9}}{11}-\frac {216 x^{5} \sqrt {-x^{4}+9}}{77}+\frac {2997 x \sqrt {-x^{4}+9}}{77}+\frac {3240 \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )}{77 \sqrt {-x^{4}+9}}\right )+e \left (\frac {x^{11} \sqrt {-x^{4}+9}}{13}-\frac {28 x^{7} \sqrt {-x^{4}+9}}{13}+\frac {279 x^{3} \sqrt {-x^{4}+9}}{13}-\frac {648 \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3}\, x}{3}, i\right )\right )}{13 \sqrt {-x^{4}+9}}\right )\) | \(184\) |
| elliptic | \(\frac {e \,x^{11} \sqrt {-x^{4}+9}}{13}+\frac {d \,x^{9} \sqrt {-x^{4}+9}}{11}-\frac {28 e \,x^{7} \sqrt {-x^{4}+9}}{13}-\frac {216 d \,x^{5} \sqrt {-x^{4}+9}}{77}+\frac {279 e \,x^{3} \sqrt {-x^{4}+9}}{13}+\frac {2997 d x \sqrt {-x^{4}+9}}{77}+\frac {3240 d \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )}{77 \sqrt {-x^{4}+9}}-\frac {648 e \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3}\, x}{3}, i\right )\right )}{13 \sqrt {-x^{4}+9}}\) | \(186\) |
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)
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{5/2} \, dx=\frac {-449064 i \, \sqrt {3} e x E(\arcsin \left (\frac {\sqrt {3}}{x}\right )\,|\,-1) + 1944 i \, \sqrt {3} {\left (65 \, d + 231 \, e\right )} x F(\arcsin \left (\frac {\sqrt {3}}{x}\right )\,|\,-1) + {\left (77 \, e x^{12} + 91 \, d x^{10} - 2156 \, e x^{8} - 2808 \, d x^{6} + 21483 \, e x^{4} + 38961 \, d x^{2} - 149688 \, e\right )} \sqrt {-x^{4} + 9}}{1001 \, x} \] Input:
integrate((e*x^2+d)*(-x^4+9)^(5/2),x, algorithm="fricas")
Output:
1/1001*(-449064*I*sqrt(3)*e*x*elliptic_e(arcsin(sqrt(3)/x), -1) + 1944*I*s qrt(3)*(65*d + 231*e)*x*elliptic_f(arcsin(sqrt(3)/x), -1) + (77*e*x^12 + 9 1*d*x^10 - 2156*e*x^8 - 2808*d*x^6 + 21483*e*x^4 + 38961*d*x^2 - 149688*e) *sqrt(-x^4 + 9))/x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (114) = 228\).
Time = 2.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.85 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{5/2} \, dx=\frac {3 d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} - \frac {27 d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {243 d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 e x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} - \frac {27 e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {243 e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] 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(2*I*pi)/9)/ (4*gamma(13/4)) - 27*d*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), x**4*exp _polar(2*I*pi)/9)/(2*gamma(9/4)) + 243*d*x*gamma(1/4)*hyper((-1/2, 1/4), ( 5/4,), x**4*exp_polar(2*I*pi)/9)/(4*gamma(5/4)) + 3*e*x**11*gamma(11/4)*hy per((-1/2, 11/4), (15/4,), x**4*exp_polar(2*I*pi)/9)/(4*gamma(15/4)) - 27* e*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), x**4*exp_polar(2*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(2*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. \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{5/2} \, dx=\int {\left (9-x^4\right )}^{5/2}\,\left (e\,x^2+d\right ) \,d x \] Input:
int((9 - x^4)^(5/2)*(d + e*x^2),x)
Output:
int((9 - x^4)^(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*sqrt( - 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