Integrand size = 17, antiderivative size = 132 \[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\frac {e x \sqrt {9+x^4}}{3+x^2}-\frac {\sqrt {3} e \left (3+x^2\right ) \sqrt {\frac {9+x^4}{\left (3+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right )|\frac {1}{2}\right )}{\sqrt {9+x^4}}+\frac {(d+3 e) \left (3+x^2\right ) \sqrt {\frac {9+x^4}{\left (3+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {9+x^4}} \] Output:
e*x*(x^4+9)^(1/2)/(x^2+3)-3^(1/2)*e*(x^2+3)*((x^4+9)/(x^2+3)^2)^(1/2)*Elli pticE(sin(2*arctan(1/3*x*3^(1/2))),1/2*2^(1/2))/(x^4+9)^(1/2)+1/6*(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 = 10.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.37 \[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\frac {1}{3} d x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {x^4}{9}\right )+\frac {1}{9} e x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {x^4}{9}\right ) \] Input:
Integrate[(d + e*x^2)/Sqrt[9 + x^4],x]
Output:
(d*x*Hypergeometric2F1[1/4, 1/2, 5/4, -1/9*x^4])/3 + (e*x^3*Hypergeometric 2F1[1/2, 3/4, 7/4, -1/9*x^4])/9
Time = 0.38 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {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.
\(\displaystyle \int \frac {d+e x^2}{\sqrt {x^4+9}} \, dx\) |
\(\Big \downarrow \) 1512 |
\(\displaystyle (d+3 e) \int \frac {1}{\sqrt {x^4+9}}dx-3 e \int \frac {3-x^2}{3 \sqrt {x^4+9}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle (d+3 e) \int \frac {1}{\sqrt {x^4+9}}dx-e \int \frac {3-x^2}{\sqrt {x^4+9}}dx\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {\left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} (d+3 e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {x^4+9}}-e \int \frac {3-x^2}{\sqrt {x^4+9}}dx\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {\left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} (d+3 e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right ),\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {x^4+9}}-e \left (\frac {\sqrt {3} \left (x^2+3\right ) \sqrt {\frac {x^4+9}{\left (x^2+3\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {3}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+9}}-\frac {x \sqrt {x^4+9}}{x^2+3}\right )\) |
Input:
Int[(d + e*x^2)/Sqrt[9 + x^4],x]
Output:
-(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])
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)/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 = 0.63 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.26
method | result | size |
meijerg | \(\frac {e \,x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {x^{4}}{9}\right )}{9}+\frac {d x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -\frac {x^{4}}{9}\right )}{3}\) | \(34\) |
default | \(\frac {d \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{9 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}+\frac {i e \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{3 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\) | \(145\) |
elliptic | \(\frac {d \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )}{9 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}+\frac {i e \sqrt {-3 i x^{2}+9}\, \sqrt {3 i x^{2}+9}\, \left (\operatorname {EllipticF}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )-\operatorname {EllipticE}\left (x \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ), i\right )\right )}{3 \left (\frac {\sqrt {6}}{6}+\frac {i \sqrt {6}}{6}\right ) \sqrt {x^{4}+9}}\) | \(145\) |
Input:
int((e*x^2+d)/(x^4+9)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/9*e*x^3*hypergeom([1/2,3/4],[7/4],-1/9*x^4)+1/3*d*x*hypergeom([1/4,1/2], [5/4],-1/9*x^4)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.41 \[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\frac {9 i \, \sqrt {3 i} e x E(\arcsin \left (\frac {\sqrt {3 i}}{x}\right )\,|\,-1) + i \, \sqrt {3 i} {\left (d - 9 \, e\right )} x F(\arcsin \left (\frac {\sqrt {3 i}}{x}\right )\,|\,-1) + 3 \, \sqrt {x^{4} + 9} e}{3 \, x} \] Input:
integrate((e*x^2+d)/(x^4+9)^(1/2),x, algorithm="fricas")
Output:
1/3*(9*I*sqrt(3*I)*e*x*elliptic_e(arcsin(sqrt(3*I)/x), -1) + I*sqrt(3*I)*( d - 9*e)*x*elliptic_f(arcsin(sqrt(3*I)/x), -1) + 3*sqrt(x^4 + 9)*e)/x
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.49 \[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{12 \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{9}} \right )}}{12 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)/(x**4+9)**(1/2),x)
Output:
d*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**4*exp_polar(I*pi)/9)/(12*gamma (5/4)) + e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4*exp_polar(I*pi)/ 9)/(12*gamma(7/4))
\[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {x^{4} + 9}} \,d x } \] Input:
integrate((e*x^2+d)/(x^4+9)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)/sqrt(x^4 + 9), x)
\[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {x^{4} + 9}} \,d x } \] Input:
integrate((e*x^2+d)/(x^4+9)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)/sqrt(x^4 + 9), x)
Timed out. \[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\int \frac {e\,x^2+d}{\sqrt {x^4+9}} \,d x \] Input:
int((d + e*x^2)/(x^4 + 9)^(1/2),x)
Output:
int((d + e*x^2)/(x^4 + 9)^(1/2), x)
\[ \int \frac {d+e x^2}{\sqrt {9+x^4}} \, dx=\left (\int \frac {\sqrt {x^{4}+9}}{x^{4}+9}d x \right ) d +\left (\int \frac {\sqrt {x^{4}+9}\, x^{2}}{x^{4}+9}d x \right ) e \] Input:
int((e*x^2+d)/(x^4+9)^(1/2),x)
Output:
int(sqrt(x**4 + 9)/(x**4 + 9),x)*d + int((sqrt(x**4 + 9)*x**2)/(x**4 + 9), x)*e