Integrand size = 21, antiderivative size = 143 \[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=\frac {x \sqrt {4+x^2}}{\sqrt {2+3 x^2}}-\frac {2 \sqrt {4+x^2} E\left (\arctan \left (\sqrt {\frac {3}{2}} x\right )|\frac {5}{6}\right )}{\sqrt {3} \sqrt {\frac {4+x^2}{2+3 x^2}} \sqrt {2+3 x^2}}+\frac {2 \sqrt {4+x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {5}{6}\right )}{\sqrt {3} \sqrt {\frac {4+x^2}{2+3 x^2}} \sqrt {2+3 x^2}} \] Output:
x*(x^2+4)^(1/2)/(3*x^2+2)^(1/2)-2/3*(x^2+4)^(1/2)*EllipticE(x*6^(1/2)/(6*x ^2+4)^(1/2),1/6*30^(1/2))*3^(1/2)/((x^2+4)/(3*x^2+2))^(1/2)/(3*x^2+2)^(1/2 )+2/3*(x^2+4)^(1/2)*InverseJacobiAM(arctan(1/2*x*6^(1/2)),1/6*30^(1/2))*3^ (1/2)/((x^2+4)/(3*x^2+2))^(1/2)/(3*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=-\frac {2 i E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {1}{6}\right )}{\sqrt {3}} \] Input:
Integrate[Sqrt[4 + x^2]/Sqrt[2 + 3*x^2],x]
Output:
((-2*I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 1/6])/Sqrt[3]
Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {324, 320, 388, 313}
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.
\(\displaystyle \int \frac {\sqrt {x^2+4}}{\sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 324 |
\(\displaystyle 4 \int \frac {1}{\sqrt {x^2+4} \sqrt {3 x^2+2}}dx+\int \frac {x^2}{\sqrt {x^2+4} \sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \int \frac {x^2}{\sqrt {x^2+4} \sqrt {3 x^2+2}}dx+\frac {2 \sqrt {2} \sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{2}\right ),-5\right )}{\sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle -\frac {4}{3} \int \frac {\sqrt {3 x^2+2}}{\left (x^2+4\right )^{3/2}}dx+\frac {2 \sqrt {2} \sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{2}\right ),-5\right )}{\sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}}+\frac {\sqrt {3 x^2+2} x}{3 \sqrt {x^2+4}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {2 \sqrt {2} \sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {x}{2}\right ),-5\right )}{\sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}}-\frac {\sqrt {2} \sqrt {3 x^2+2} E\left (\left .\arctan \left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}}+\frac {\sqrt {3 x^2+2} x}{3 \sqrt {x^2+4}}\) |
Input:
Int[Sqrt[4 + x^2]/Sqrt[2 + 3*x^2],x]
Output:
(x*Sqrt[2 + 3*x^2])/(3*Sqrt[4 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE [ArcTan[x/2], -5])/(3*Sqrt[4 + x^2]*Sqrt[(2 + 3*x^2)/(4 + x^2)]) + (2*Sqrt [2]*Sqrt[2 + 3*x^2]*EllipticF[ArcTan[x/2], -5])/(Sqrt[4 + x^2]*Sqrt[(2 + 3 *x^2)/(4 + x^2)])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.18
method | result | size |
default | \(-\frac {i \left (5 \operatorname {EllipticF}\left (\frac {i x}{2}, \sqrt {6}\right )+\operatorname {EllipticE}\left (\frac {i x}{2}, \sqrt {6}\right )\right ) \sqrt {2}}{3}\) | \(26\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}+2\right ) \left (x^{2}+4\right )}\, \left (-\frac {2 i \sqrt {x^{2}+4}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (\frac {i x}{2}, \sqrt {6}\right )}{\sqrt {3 x^{4}+14 x^{2}+8}}+\frac {i \sqrt {x^{2}+4}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {i x}{2}, \sqrt {6}\right )-\operatorname {EllipticE}\left (\frac {i x}{2}, \sqrt {6}\right )\right )}{3 \sqrt {3 x^{4}+14 x^{2}+8}}\right )}{\sqrt {3 x^{2}+2}\, \sqrt {x^{2}+4}}\) | \(127\) |
Input:
int((x^2+4)^(1/2)/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/3*I*(5*EllipticF(1/2*I*x,6^(1/2))+EllipticE(1/2*I*x,6^(1/2)))*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=-\frac {2 \, \sqrt {3} \sqrt {-\frac {2}{3}} x E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,6) - 20 \, \sqrt {3} \sqrt {-\frac {2}{3}} x F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,6) - 3 \, \sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 4}}{9 \, x} \] Input:
integrate((x^2+4)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
-1/9*(2*sqrt(3)*sqrt(-2/3)*x*elliptic_e(arcsin(sqrt(-2/3)/x), 6) - 20*sqrt (3)*sqrt(-2/3)*x*elliptic_f(arcsin(sqrt(-2/3)/x), 6) - 3*sqrt(3*x^2 + 2)*s qrt(x^2 + 4))/x
\[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=\int \frac {\sqrt {x^{2} + 4}}{\sqrt {3 x^{2} + 2}}\, dx \] Input:
integrate((x**2+4)**(1/2)/(3*x**2+2)**(1/2),x)
Output:
Integral(sqrt(x**2 + 4)/sqrt(3*x**2 + 2), x)
\[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=\int { \frac {\sqrt {x^{2} + 4}}{\sqrt {3 \, x^{2} + 2}} \,d x } \] Input:
integrate((x^2+4)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x^2 + 4)/sqrt(3*x^2 + 2), x)
\[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=\int { \frac {\sqrt {x^{2} + 4}}{\sqrt {3 \, x^{2} + 2}} \,d x } \] Input:
integrate((x^2+4)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x^2 + 4)/sqrt(3*x^2 + 2), x)
Timed out. \[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=\int \frac {\sqrt {x^2+4}}{\sqrt {3\,x^2+2}} \,d x \] Input:
int((x^2 + 4)^(1/2)/(3*x^2 + 2)^(1/2),x)
Output:
int((x^2 + 4)^(1/2)/(3*x^2 + 2)^(1/2), x)
\[ \int \frac {\sqrt {4+x^2}}{\sqrt {2+3 x^2}} \, dx=\int \frac {\sqrt {3 x^{2}+2}\, \sqrt {x^{2}+4}}{3 x^{2}+2}d x \] Input:
int((x^2+4)^(1/2)/(3*x^2+2)^(1/2),x)
Output:
int((sqrt(3*x**2 + 2)*sqrt(x**2 + 4))/(3*x**2 + 2),x)