Integrand size = 19, antiderivative size = 99 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=\frac {6}{35} x \left (15 d+7 e x^2\right ) \sqrt {9-x^4}+\frac {1}{63} x \left (9 d+7 e x^2\right ) \left (9-x^4\right )^{3/2}+\frac {108}{5} \sqrt {3} e E\left (\left .\arcsin \left (\frac {x}{\sqrt {3}}\right )\right |-1\right )+\frac {108}{35} \sqrt {3} (5 d-7 e) \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3}}\right ),-1\right ) \] Output:
6/35*x*(7*e*x^2+15*d)*(-x^4+9)^(1/2)+1/63*x*(7*e*x^2+9*d)*(-x^4+9)^(3/2)+1 08/5*3^(1/2)*e*EllipticE(1/3*x*3^(1/2),I)+108/35*3^(1/2)*(5*d-7*e)*Ellipti cF(1/3*x*3^(1/2),I)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.45 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=27 d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {x^4}{9}\right )+9 e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {x^4}{9}\right ) \] Input:
Integrate[(d + e*x^2)*(9 - x^4)^(3/2),x]
Output:
27*d*x*Hypergeometric2F1[-3/2, 1/4, 5/4, x^4/9] + 9*e*x^3*Hypergeometric2F 1[-3/2, 3/4, 7/4, x^4/9]
Time = 0.46 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {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.
| \(\displaystyle \int \left (9-x^4\right )^{3/2} \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {1}{21} \int 18 \left (7 e x^2+9 d\right ) \sqrt {9-x^4}dx+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} \int \left (7 e x^2+9 d\right ) \sqrt {9-x^4}dx+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {6}{7} \left (\frac {1}{15} \int \frac {54 \left (7 e x^2+15 d\right )}{\sqrt {9-x^4}}dx+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} \left (\frac {18}{5} \int \frac {7 e x^2+15 d}{\sqrt {9-x^4}}dx+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 1495 |
| \(\displaystyle \frac {6}{7} \left (\frac {18}{5} \int \frac {7 e x^2+15 d}{\sqrt {3-x^2} \sqrt {x^2+3}}dx+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {6}{7} \left (\frac {18}{5} \left (3 (5 d-7 e) \int \frac {1}{\sqrt {3-x^2} \sqrt {x^2+3}}dx+7 e \int \frac {\sqrt {x^2+3}}{\sqrt {3-x^2}}dx\right )+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 284 |
| \(\displaystyle \frac {6}{7} \left (\frac {18}{5} \left (3 (5 d-7 e) \int \frac {1}{\sqrt {9-x^4}}dx+7 e \int \frac {\sqrt {x^2+3}}{\sqrt {3-x^2}}dx\right )+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {6}{7} \left (\frac {18}{5} \left (3 (5 d-7 e) \int \frac {1}{\sqrt {9-x^4}}dx+7 \sqrt {3} e E\left (\left .\arcsin \left (\frac {x}{\sqrt {3}}\right )\right |-1\right )\right )+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {6}{7} \left (\frac {18}{5} \left (\sqrt {3} (5 d-7 e) \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3}}\right ),-1\right )+7 \sqrt {3} e E\left (\left .\arcsin \left (\frac {x}{\sqrt {3}}\right )\right |-1\right )\right )+\frac {1}{5} x \sqrt {9-x^4} \left (15 d+7 e x^2\right )\right )+\frac {1}{63} x \left (9-x^4\right )^{3/2} \left (9 d+7 e x^2\right )\) |
Input:
Int[(d + e*x^2)*(9 - x^4)^(3/2),x]
Output:
(x*(9*d + 7*e*x^2)*(9 - x^4)^(3/2))/63 + (6*((x*(15*d + 7*e*x^2)*Sqrt[9 - x^4])/5 + (18*(7*Sqrt[3]*e*EllipticE[ArcSin[x/Sqrt[3]], -1] + Sqrt[3]*(5*d - 7*e)*EllipticF[ArcSin[x/Sqrt[3]], -1]))/5))/7
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 = 1.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.34
| method | result | size |
| meijerg | \(9 e \,x^{3} \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], \frac {x^{4}}{9}\right )+27 d x \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], \frac {x^{4}}{9}\right )\) | \(34\) |
| risch | \(\frac {x \left (35 e \,x^{6}+45 d \,x^{4}-693 e \,x^{2}-1215 d \right ) \left (x^{4}-9\right )}{315 \sqrt {-x^{4}+9}}+\frac {36 d \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )}{7 \sqrt {-x^{4}+9}}-\frac {36 e \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3}\, x}{3}, i\right )\right )}{5 \sqrt {-x^{4}+9}}\) | \(137\) |
| default | \(d \left (-\frac {x^{5} \sqrt {-x^{4}+9}}{7}+\frac {27 x \sqrt {-x^{4}+9}}{7}+\frac {36 \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )}{7 \sqrt {-x^{4}+9}}\right )+e \left (-\frac {x^{7} \sqrt {-x^{4}+9}}{9}+\frac {11 x^{3} \sqrt {-x^{4}+9}}{5}-\frac {36 \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3}\, x}{3}, i\right )\right )}{5 \sqrt {-x^{4}+9}}\right )\) | \(156\) |
| elliptic | \(-\frac {e \,x^{7} \sqrt {-x^{4}+9}}{9}-\frac {d \,x^{5} \sqrt {-x^{4}+9}}{7}+\frac {11 e \,x^{3} \sqrt {-x^{4}+9}}{5}+\frac {27 d x \sqrt {-x^{4}+9}}{7}+\frac {36 d \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )}{7 \sqrt {-x^{4}+9}}-\frac {36 e \sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3}\, x}{3}, i\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3}\, x}{3}, i\right )\right )}{5 \sqrt {-x^{4}+9}}\) | \(156\) |
Input:
int((e*x^2+d)*(-x^4+9)^(3/2),x,method=_RETURNVERBOSE)
Output:
9*e*x^3*hypergeom([-3/2,3/4],[7/4],1/9*x^4)+27*d*x*hypergeom([-3/2,1/4],[5 /4],1/9*x^4)
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=\frac {-20412 i \, \sqrt {3} e x E(\arcsin \left (\frac {\sqrt {3}}{x}\right )\,|\,-1) + 972 i \, \sqrt {3} {\left (5 \, d + 21 \, e\right )} x F(\arcsin \left (\frac {\sqrt {3}}{x}\right )\,|\,-1) - {\left (35 \, e x^{8} + 45 \, d x^{6} - 693 \, e x^{4} - 1215 \, d x^{2} + 6804 \, e\right )} \sqrt {-x^{4} + 9}}{315 \, x} \] Input:
integrate((e*x^2+d)*(-x^4+9)^(3/2),x, algorithm="fricas")
Output:
1/315*(-20412*I*sqrt(3)*e*x*elliptic_e(arcsin(sqrt(3)/x), -1) + 972*I*sqrt (3)*(5*d + 21*e)*x*elliptic_f(arcsin(sqrt(3)/x), -1) - (35*e*x^8 + 45*d*x^ 6 - 693*e*x^4 - 1215*d*x^2 + 6804*e)*sqrt(-x^4 + 9))/x
Time = 1.46 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.55 \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=- \frac {3 d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {27 d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {3 e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {27 e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{9}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)*(-x**4+9)**(3/2),x)
Output:
-3*d*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), x**4*exp_polar(2*I*pi)/9)/ (4*gamma(9/4)) + 27*d*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), x**4*exp_pol ar(2*I*pi)/9)/(4*gamma(5/4)) - 3*e*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/ 4,), x**4*exp_polar(2*I*pi)/9)/(4*gamma(11/4)) + 27*e*x**3*gamma(3/4)*hype r((-1/2, 3/4), (7/4,), x**4*exp_polar(2*I*pi)/9)/(4*gamma(7/4))
\[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 9\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-x^4+9)^(3/2),x, algorithm="maxima")
Output:
integrate((-x^4 + 9)^(3/2)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 9\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-x^4+9)^(3/2),x, algorithm="giac")
Output:
integrate((-x^4 + 9)^(3/2)*(e*x^2 + d), x)
Timed out. \[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=\int {\left (9-x^4\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \] Input:
int((9 - x^4)^(3/2)*(d + e*x^2),x)
Output:
int((9 - x^4)^(3/2)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \left (9-x^4\right )^{3/2} \, dx=-\frac {\sqrt {-x^{4}+9}\, d \,x^{5}}{7}+\frac {27 \sqrt {-x^{4}+9}\, d x}{7}-\frac {\sqrt {-x^{4}+9}\, e \,x^{7}}{9}+\frac {11 \sqrt {-x^{4}+9}\, e \,x^{3}}{5}-\frac {324 \left (\int \frac {\sqrt {-x^{4}+9}}{x^{4}-9}d x \right ) d}{7}-\frac {108 \left (\int \frac {\sqrt {-x^{4}+9}\, x^{2}}{x^{4}-9}d x \right ) e}{5} \] Input:
int((e*x^2+d)*(-x^4+9)^(3/2),x)
Output:
( - 45*sqrt( - x**4 + 9)*d*x**5 + 1215*sqrt( - x**4 + 9)*d*x - 35*sqrt( - x**4 + 9)*e*x**7 + 693*sqrt( - x**4 + 9)*e*x**3 - 14580*int(sqrt( - x**4 + 9)/(x**4 - 9),x)*d - 6804*int((sqrt( - x**4 + 9)*x**2)/(x**4 - 9),x)*e)/3 15