Integrand size = 17, antiderivative size = 158 \[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x^2\right )}{18 \sqrt {9+x^4}}-\frac {e x \sqrt {9+x^4}}{18 \left (3+x^2\right )}+\frac {e \left (3+x^2\right ) \sqrt {\frac {9+x^4}{\left (3+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {9+x^4}}+\frac {(d-3 e) \left (3+x^2\right ) \sqrt {\frac {9+x^4}{\left (3+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{36 \sqrt {3} \sqrt {9+x^4}} \] Output:
1/18*x*(e*x^2+d)/(x^4+9)^(1/2)-e*x*(x^4+9)^(1/2)/(18*x^2+54)+1/18*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)+1/108*(d-3*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))*3^(1/2)/(x^4+9)^(1/ 2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.39 \[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\frac {1}{162} x \left (\frac {9 d}{\sqrt {9+x^4}}+3 d \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {x^4}{9}\right )+2 e x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {x^4}{9}\right )\right ) \] Input:
Integrate[(d + e*x^2)/(9 + x^4)^(3/2),x]
Output:
(x*((9*d)/Sqrt[9 + x^4] + 3*d*Hypergeometric2F1[1/4, 1/2, 5/4, -1/9*x^4] + 2*e*x^2*Hypergeometric2F1[3/4, 3/2, 7/4, -1/9*x^4]))/162
Time = 0.43 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1493, 25, 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.
| \(\displaystyle \int \frac {d+e x^2}{\left (x^4+9\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1493 |
| \(\displaystyle \frac {x \left (d+e x^2\right )}{18 \sqrt {x^4+9}}-\frac {1}{18} \int -\frac {d-e x^2}{\sqrt {x^4+9}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{18} \int \frac {d-e x^2}{\sqrt {x^4+9}}dx+\frac {x \left (d+e x^2\right )}{18 \sqrt {x^4+9}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {1}{18} \left ((d-3 e) \int \frac {1}{\sqrt {x^4+9}}dx+3 e \int \frac {3-x^2}{3 \sqrt {x^4+9}}dx\right )+\frac {x \left (d+e x^2\right )}{18 \sqrt {x^4+9}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left ((d-3 e) \int \frac {1}{\sqrt {x^4+9}}dx+e \int \frac {3-x^2}{\sqrt {x^4+9}}dx\right )+\frac {x \left (d+e x^2\right )}{18 \sqrt {x^4+9}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {1}{18} \left (e \int \frac {3-x^2}{\sqrt {x^4+9}}dx+\frac {\left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} (d-3 e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {x^4+9}}\right )+\frac {x \left (d+e x^2\right )}{18 \sqrt {x^4+9}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {1}{18} \left (\frac {\left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} (d-3 e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {x^4+9}}+e \left (\frac {\sqrt {3} \left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+9}}-\frac {x \sqrt {x^4+9}}{x^2+3}\right )\right )+\frac {x \left (d+e x^2\right )}{18 \sqrt {x^4+9}}\) |
Input:
Int[(d + e*x^2)/(9 + x^4)^(3/2),x]
Output:
(x*(d + e*x^2))/(18*Sqrt[9 + x^4]) + (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]) + ((d - 3*e)*(3 + x^2)*Sqrt[(9 + x^4)/(3 + x^2)^2 ]*EllipticF[2*ArcTan[x/Sqrt[3]], 1/2])/(2*Sqrt[3]*Sqrt[9 + x^4]))/18
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 + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) ) Int[Simp[d*(4*p + 5) + e*(4*p + 7)*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] && LtQ[p, -1] && Integer Q[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 = 1.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.22
| method | result | size |
| meijerg | \(\frac {e \,x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {7}{4}\right ], -\frac {x^{4}}{9}\right )}{81}+\frac {d x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], -\frac {x^{4}}{9}\right )}{27}\) | \(34\) |
| risch | \(\frac {x \left (e \,x^{2}+d \right )}{18 \sqrt {x^{4}+9}}-\frac {i e \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{54 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}+\frac {d \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{162 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\) | \(162\) |
| elliptic | \(-\frac {2 \left (-\frac {1}{36} e \,x^{3}-\frac {1}{36} d x \right )}{\sqrt {x^{4}+9}}-\frac {i e \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{54 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}+\frac {d \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{162 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\) | \(165\) |
| default | \(d \left (\frac {x}{18 \sqrt {x^{4}+9}}+\frac {\sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{162 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\right )+e \left (\frac {x^{3}}{18 \sqrt {x^{4}+9}}-\frac {i \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{54 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\right )\) | \(171\) |
Input:
int((e*x^2+d)/(x^4+9)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/81*e*x^3*hypergeom([3/4,3/2],[7/4],-1/9*x^4)+1/27*d*x*hypergeom([1/4,3/2 ],[5/4],-1/9*x^4)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.48 \[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {1}{3} i} {\left (i \, e x^{4} + 9 i \, e\right )} E(\arcsin \left (\sqrt {\frac {1}{3} i} x\right )\,|\,-1) + \sqrt {\frac {1}{3} i} {\left (-i \, {\left (d + e\right )} x^{4} - 9 i \, d - 9 i \, e\right )} F(\arcsin \left (\sqrt {\frac {1}{3} i} x\right )\,|\,-1) + {\left (e x^{3} + d x\right )} \sqrt {x^{4} + 9}}{18 \, {\left (x^{4} + 9\right )}} \] Input:
integrate((e*x^2+d)/(x^4+9)^(3/2),x, algorithm="fricas")
Output:
1/18*(sqrt(1/3*I)*(I*e*x^4 + 9*I*e)*elliptic_e(arcsin(sqrt(1/3*I)*x), -1) + sqrt(1/3*I)*(-I*(d + e)*x^4 - 9*I*d - 9*I*e)*elliptic_f(arcsin(sqrt(1/3* I)*x), -1) + (e*x^3 + d*x)*sqrt(x^4 + 9))/(x^4 + 9)
Result contains complex when optimal does not.
Time = 1.68 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.41 \[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{108 \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{108 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)/(x**4+9)**(3/2),x)
Output:
d*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(I*pi)/9)/(108*gamm a(5/4)) + e*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/4,), x**4*exp_polar(I*pi) /9)/(108*gamma(7/4))
\[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (x^{4} + 9\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(x^4+9)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)/(x^4 + 9)^(3/2), x)
\[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (x^{4} + 9\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(x^4+9)^(3/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)/(x^4 + 9)^(3/2), x)
Timed out. \[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (x^4+9\right )}^{3/2}} \,d x \] Input:
int((d + e*x^2)/(x^4 + 9)^(3/2),x)
Output:
int((d + e*x^2)/(x^4 + 9)^(3/2), x)
\[ \int \frac {d+e x^2}{\left (9+x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {x^{4}+9}}{x^{8}+18 x^{4}+81}d x \right ) d +\left (\int \frac {\sqrt {x^{4}+9}\, x^{2}}{x^{8}+18 x^{4}+81}d x \right ) e \] Input:
int((e*x^2+d)/(x^4+9)^(3/2),x)
Output:
int(sqrt(x**4 + 9)/(x**8 + 18*x**4 + 81),x)*d + int((sqrt(x**4 + 9)*x**2)/ (x**8 + 18*x**4 + 81),x)*e