Integrand size = 26, antiderivative size = 35 \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=\frac {E\left (\arcsin (2 x)\left |-\frac {3}{8}\right .\right )}{3 \sqrt {2}}-\frac {\operatorname {EllipticF}\left (\arcsin (2 x),-\frac {3}{8}\right )}{3 \sqrt {2}} \] Output:
1/6*2^(1/2)*EllipticE(2*x,1/4*I*6^(1/2))-1/6*EllipticF(2*x,1/4*I*6^(1/2))* 2^(1/2)
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=\frac {E\left (\arcsin (2 x)\left |-\frac {3}{8}\right .\right )-\operatorname {EllipticF}\left (\arcsin (2 x),-\frac {3}{8}\right )}{3 \sqrt {2}} \] Input:
Integrate[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 + 3*x^2]),x]
Output:
(EllipticE[ArcSin[2*x], -3/8] - EllipticF[ArcSin[2*x], -3/8])/(3*Sqrt[2])
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {389, 321, 327}
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.
\(\displaystyle \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 389 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {3 x^2+2}}{\sqrt {1-4 x^2}}dx-\frac {2}{3} \int \frac {1}{\sqrt {1-4 x^2} \sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {3 x^2+2}}{\sqrt {1-4 x^2}}dx-\frac {\operatorname {EllipticF}\left (\arcsin (2 x),-\frac {3}{8}\right )}{3 \sqrt {2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {E\left (\arcsin (2 x)\left |-\frac {3}{8}\right .\right )}{3 \sqrt {2}}-\frac {\operatorname {EllipticF}\left (\arcsin (2 x),-\frac {3}{8}\right )}{3 \sqrt {2}}\) |
Input:
Int[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 + 3*x^2]),x]
Output:
EllipticE[ArcSin[2*x], -3/8]/(3*Sqrt[2]) - EllipticF[ArcSin[2*x], -3/8]/(3 *Sqrt[2])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[1/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b Int [1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N eQ[b*c - a*d, 0] && !SimplerSqrtQ[-b/a, -d/c]
Time = 2.99 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\left (\operatorname {EllipticF}\left (2 x , \frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (2 x , \frac {i \sqrt {6}}{4}\right )\right ) \sqrt {2}}{6}\) | \(29\) |
elliptic | \(-\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (4 x^{2}-1\right )}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (2 x , \frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (2 x , \frac {i \sqrt {6}}{4}\right )\right )}{6 \sqrt {3 x^{2}+2}\, \sqrt {-12 x^{4}-5 x^{2}+2}}\) | \(76\) |
Input:
int(x^2/(-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(EllipticF(2*x,1/4*I*6^(1/2))-EllipticE(2*x,1/4*I*6^(1/2)))*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=-\frac {\sqrt {-3} x E(\arcsin \left (\frac {1}{2 \, x}\right )\,|\,-\frac {8}{3}) - \sqrt {-3} x F(\arcsin \left (\frac {1}{2 \, x}\right )\,|\,-\frac {8}{3}) + 4 \, \sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}}{48 \, x} \] Input:
integrate(x^2/(-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
-1/48*(sqrt(-3)*x*elliptic_e(arcsin(1/2/x), -8/3) - sqrt(-3)*x*elliptic_f( arcsin(1/2/x), -8/3) + 4*sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1))/x
\[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^{2}}{\sqrt {- \left (2 x - 1\right ) \left (2 x + 1\right )} \sqrt {3 x^{2} + 2}}\, dx \] Input:
integrate(x**2/(-4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
Output:
Integral(x**2/(sqrt(-(2*x - 1)*(2*x + 1))*sqrt(3*x**2 + 2)), x)
\[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}} \,d x } \] Input:
integrate(x^2/(-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)), x)
\[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}} \,d x } \] Input:
integrate(x^2/(-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^2}{\sqrt {3\,x^2+2}\,\sqrt {1-4\,x^2}} \,d x \] Input:
int(x^2/((3*x^2 + 2)^(1/2)*(1 - 4*x^2)^(1/2)),x)
Output:
int(x^2/((3*x^2 + 2)^(1/2)*(1 - 4*x^2)^(1/2)), x)
\[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx=-\left (\int \frac {\sqrt {3 x^{2}+2}\, \sqrt {-4 x^{2}+1}\, x^{2}}{12 x^{4}+5 x^{2}-2}d x \right ) \] Input:
int(x^2/(-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)*x**2)/(12*x**4 + 5*x**2 - 2), x)