Integrand size = 23, antiderivative size = 66 \[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\frac {x \sqrt {1+4 x^2}}{\sqrt {2+3 x^2}}-\frac {E\left (\arctan \left (\sqrt {\frac {3}{2}} x\right )|-\frac {5}{3}\right )}{\sqrt {3}}+\frac {\operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {3}{2}} x\right ),-\frac {5}{3}\right )}{\sqrt {3}} \] Output:
x*(4*x^2+1)^(1/2)/(3*x^2+2)^(1/2)-1/3*EllipticE(x*6^(1/2)/(6*x^2+4)^(1/2), 1/3*I*15^(1/2))*3^(1/2)+1/3*InverseJacobiAM(arctan(1/2*x*6^(1/2)),1/3*I*15 ^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=-\frac {i E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )}{\sqrt {3}} \] Input:
Integrate[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]
Output:
((-I)*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 8/3])/Sqrt[3]
Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(66)=132\).
Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.29, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.
\(\displaystyle \int \frac {\sqrt {4 x^2+1}}{\sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 324 |
\(\displaystyle \int \frac {1}{\sqrt {3 x^2+2} \sqrt {4 x^2+1}}dx+4 \int \frac {x^2}{\sqrt {3 x^2+2} \sqrt {4 x^2+1}}dx\) |
\(\Big \downarrow \) 320 |
\(\displaystyle 4 \int \frac {x^2}{\sqrt {3 x^2+2} \sqrt {4 x^2+1}}dx+\frac {\sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan (2 x),\frac {5}{8}\right )}{2 \sqrt {2} \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle 4 \left (\frac {x \sqrt {3 x^2+2}}{3 \sqrt {4 x^2+1}}-\frac {1}{3} \int \frac {\sqrt {3 x^2+2}}{\left (4 x^2+1\right )^{3/2}}dx\right )+\frac {\sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan (2 x),\frac {5}{8}\right )}{2 \sqrt {2} \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {\sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan (2 x),\frac {5}{8}\right )}{2 \sqrt {2} \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}}+4 \left (\frac {x \sqrt {3 x^2+2}}{3 \sqrt {4 x^2+1}}-\frac {\sqrt {3 x^2+2} E\left (\arctan (2 x)\left |\frac {5}{8}\right .\right )}{3 \sqrt {2} \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}}\right )\) |
Input:
Int[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]
Output:
4*((x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + 4*x^2]) - (Sqrt[2 + 3*x^2]*EllipticE[Ar cTan[2*x], 5/8])/(3*Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + 4*x^2)]*Sqrt[1 + 4*x^2]) ) + (Sqrt[2 + 3*x^2]*EllipticF[ArcTan[2*x], 5/8])/(2*Sqrt[2]*Sqrt[(2 + 3*x ^2)/(1 + 4*x^2)]*Sqrt[1 + 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]
Time = 0.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.30
method | result | size |
default | \(-\frac {i \operatorname {EllipticE}\left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right ) \sqrt {3}}{3}\) | \(20\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}+2\right ) \left (4 x^{2}+1\right )}\, \left (-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {4 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )}{6 \sqrt {12 x^{4}+11 x^{2}+2}}+\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {4 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )\right )}{6 \sqrt {12 x^{4}+11 x^{2}+2}}\right )}{\sqrt {3 x^{2}+2}\, \sqrt {4 x^{2}+1}}\) | \(156\) |
Input:
int((4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/3*I*EllipticE(1/2*I*x*6^(1/2),2/3*6^(1/2))*3^(1/2)
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\frac {-i \, \sqrt {3} x E(\arcsin \left (\frac {i}{2 \, x}\right )\,|\,\frac {8}{3}) + 5 i \, \sqrt {3} x F(\arcsin \left (\frac {i}{2 \, x}\right )\,|\,\frac {8}{3}) + 4 \, \sqrt {4 \, x^{2} + 1} \sqrt {3 \, x^{2} + 2}}{12 \, x} \] Input:
integrate((4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/12*(-I*sqrt(3)*x*elliptic_e(arcsin(1/2*I/x), 8/3) + 5*I*sqrt(3)*x*ellipt ic_f(arcsin(1/2*I/x), 8/3) + 4*sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2))/x
\[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\int \frac {\sqrt {4 x^{2} + 1}}{\sqrt {3 x^{2} + 2}}\, dx \] Input:
integrate((4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
Output:
Integral(sqrt(4*x**2 + 1)/sqrt(3*x**2 + 2), x)
\[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\int { \frac {\sqrt {4 \, x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}} \,d x } \] Input:
integrate((4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(4*x^2 + 1)/sqrt(3*x^2 + 2), x)
\[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\int { \frac {\sqrt {4 \, x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}} \,d x } \] Input:
integrate((4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(4*x^2 + 1)/sqrt(3*x^2 + 2), x)
Timed out. \[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\int \frac {\sqrt {4\,x^2+1}}{\sqrt {3\,x^2+2}} \,d x \] Input:
int((4*x^2 + 1)^(1/2)/(3*x^2 + 2)^(1/2),x)
Output:
int((4*x^2 + 1)^(1/2)/(3*x^2 + 2)^(1/2), x)
\[ \int \frac {\sqrt {1+4 x^2}}{\sqrt {2+3 x^2}} \, dx=\int \frac {\sqrt {3 x^{2}+2}\, \sqrt {4 x^{2}+1}}{3 x^{2}+2}d x \] Input:
int((4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x)
Output:
int((sqrt(3*x**2 + 2)*sqrt(4*x**2 + 1))/(3*x**2 + 2),x)