Integrand size = 23, antiderivative size = 182 \[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\frac {x \sqrt {d^2+e^2 x^4}}{d+e x^2}-\frac {\sqrt {d} \left (d+e x^2\right ) \sqrt {\frac {d^2+e^2 x^4}{\left (d+e x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|\frac {1}{2}\right )}{\sqrt {e} \sqrt {d^2+e^2 x^4}}+\frac {\sqrt {d} \left (d+e x^2\right ) \sqrt {\frac {d^2+e^2 x^4}{\left (d+e x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),\frac {1}{2}\right )}{\sqrt {e} \sqrt {d^2+e^2 x^4}} \] Output:
x*(e^2*x^4+d^2)^(1/2)/(e*x^2+d)-d^(1/2)*(e*x^2+d)*((e^2*x^4+d^2)/(e*x^2+d) ^2)^(1/2)*EllipticE(sin(2*arctan(e^(1/2)*x/d^(1/2))),1/2*2^(1/2))/e^(1/2)/ (e^2*x^4+d^2)^(1/2)+d^(1/2)*(e*x^2+d)*((e^2*x^4+d^2)/(e*x^2+d)^2)^(1/2)*In verseJacobiAM(2*arctan(e^(1/2)*x/d^(1/2)),1/2*2^(1/2))/e^(1/2)/(e^2*x^4+d^ 2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.48 \[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\frac {\sqrt {1+\frac {e^2 x^4}{d^2}} \left (3 d x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {e^2 x^4}{d^2}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {e^2 x^4}{d^2}\right )\right )}{3 \sqrt {d^2+e^2 x^4}} \] Input:
Integrate[(d + e*x^2)/Sqrt[d^2 + e^2*x^4],x]
Output:
(Sqrt[1 + (e^2*x^4)/d^2]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, -((e^2*x^ 4)/d^2)] + e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -((e^2*x^4)/d^2)]))/(3*S qrt[d^2 + e^2*x^4])
Time = 0.45 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.
\(\displaystyle \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx\) |
\(\Big \downarrow \) 1512 |
\(\displaystyle 2 d \int \frac {1}{\sqrt {e^2 x^4+d^2}}dx-d \int \frac {d-e x^2}{d \sqrt {e^2 x^4+d^2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 d \int \frac {1}{\sqrt {e^2 x^4+d^2}}dx-\int \frac {d-e x^2}{\sqrt {e^2 x^4+d^2}}dx\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {\sqrt {d} \left (d+e x^2\right ) \sqrt {\frac {d^2+e^2 x^4}{\left (d+e x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),\frac {1}{2}\right )}{\sqrt {e} \sqrt {d^2+e^2 x^4}}-\int \frac {d-e x^2}{\sqrt {e^2 x^4+d^2}}dx\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {\sqrt {d} \left (d+e x^2\right ) \sqrt {\frac {d^2+e^2 x^4}{\left (d+e x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),\frac {1}{2}\right )}{\sqrt {e} \sqrt {d^2+e^2 x^4}}-\frac {\sqrt {d} \left (d+e x^2\right ) \sqrt {\frac {d^2+e^2 x^4}{\left (d+e x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|\frac {1}{2}\right )}{\sqrt {e} \sqrt {d^2+e^2 x^4}}+\frac {x \sqrt {d^2+e^2 x^4}}{d+e x^2}\) |
Input:
Int[(d + e*x^2)/Sqrt[d^2 + e^2*x^4],x]
Output:
(x*Sqrt[d^2 + e^2*x^4])/(d + e*x^2) - (Sqrt[d]*(d + e*x^2)*Sqrt[(d^2 + e^2 *x^4)/(d + e*x^2)^2]*EllipticE[2*ArcTan[(Sqrt[e]*x)/Sqrt[d]], 1/2])/(Sqrt[ e]*Sqrt[d^2 + e^2*x^4]) + (Sqrt[d]*(d + e*x^2)*Sqrt[(d^2 + e^2*x^4)/(d + e *x^2)^2]*EllipticF[2*ArcTan[(Sqrt[e]*x)/Sqrt[d]], 1/2])/(Sqrt[e]*Sqrt[d^2 + e^2*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]
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {d \sqrt {1-\frac {i e \,x^{2}}{d}}\, \sqrt {1+\frac {i e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i e}{d}}, i\right )}{\sqrt {\frac {i e}{d}}\, \sqrt {e^{2} x^{4}+d^{2}}}+\frac {i d \sqrt {1-\frac {i e \,x^{2}}{d}}\, \sqrt {1+\frac {i e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i e}{d}}, i\right )\right )}{\sqrt {\frac {i e}{d}}\, \sqrt {e^{2} x^{4}+d^{2}}}\) | \(153\) |
elliptic | \(\frac {d \sqrt {1-\frac {i e \,x^{2}}{d}}\, \sqrt {1+\frac {i e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i e}{d}}, i\right )}{\sqrt {\frac {i e}{d}}\, \sqrt {e^{2} x^{4}+d^{2}}}+\frac {i d \sqrt {1-\frac {i e \,x^{2}}{d}}\, \sqrt {1+\frac {i e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i e}{d}}, i\right )\right )}{\sqrt {\frac {i e}{d}}\, \sqrt {e^{2} x^{4}+d^{2}}}\) | \(153\) |
Input:
int((e*x^2+d)/(e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
d/(I*e/d)^(1/2)*(1-I*e*x^2/d)^(1/2)*(1+I*e*x^2/d)^(1/2)/(e^2*x^4+d^2)^(1/2 )*EllipticF(x*(I*e/d)^(1/2),I)+I*d/(I*e/d)^(1/2)*(1-I*e*x^2/d)^(1/2)*(1+I* e*x^2/d)^(1/2)/(e^2*x^4+d^2)^(1/2)*(EllipticF(x*(I*e/d)^(1/2),I)-EllipticE (x*(I*e/d)^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.53 \[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\frac {d e x \left (-\frac {d^{2}}{e^{2}}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (d e - e^{2}\right )} x \left (-\frac {d^{2}}{e^{2}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {e^{2} x^{4} + d^{2}} d}{d e x} \] Input:
integrate((e*x^2+d)/(e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
Output:
(d*e*x*(-d^2/e^2)^(3/4)*elliptic_e(arcsin((-d^2/e^2)^(1/4)/x), -1) - (d*e - e^2)*x*(-d^2/e^2)^(3/4)*elliptic_f(arcsin((-d^2/e^2)^(1/4)/x), -1) + sqr t(e^2*x^4 + d^2)*d)/(d*e*x)
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.41 \[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{i \pi }}{d^{2}}} \right )}}{4 d \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((e*x**2+d)/(e**2*x**4+d**2)**(1/2),x)
Output:
x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), e**2*x**4*exp_polar(I*pi)/d**2)/(4* gamma(5/4)) + e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), e**2*x**4*exp_po lar(I*pi)/d**2)/(4*d*gamma(7/4))
\[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {e^{2} x^{4} + d^{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(e^2*x^4+d^2)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)/sqrt(e^2*x^4 + d^2), x)
\[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\int { \frac {e x^{2} + d}{\sqrt {e^{2} x^{4} + d^{2}}} \,d x } \] Input:
integrate((e*x^2+d)/(e^2*x^4+d^2)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)/sqrt(e^2*x^4 + d^2), x)
Timed out. \[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\int \frac {e\,x^2+d}{\sqrt {d^2+e^2\,x^4}} \,d x \] Input:
int((d + e*x^2)/(d^2 + e^2*x^4)^(1/2),x)
Output:
int((d + e*x^2)/(d^2 + e^2*x^4)^(1/2), x)
\[ \int \frac {d+e x^2}{\sqrt {d^2+e^2 x^4}} \, dx=\left (\int \frac {\sqrt {e^{2} x^{4}+d^{2}}}{e^{2} x^{4}+d^{2}}d x \right ) d +\left (\int \frac {\sqrt {e^{2} x^{4}+d^{2}}\, x^{2}}{e^{2} x^{4}+d^{2}}d x \right ) e \] Input:
int((e*x^2+d)/(e^2*x^4+d^2)^(1/2),x)
Output:
int(sqrt(d**2 + e**2*x**4)/(d**2 + e**2*x**4),x)*d + int((sqrt(d**2 + e**2 *x**4)*x**2)/(d**2 + e**2*x**4),x)*e