Integrand size = 26, antiderivative size = 31 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=-\frac {1}{3} \sqrt {2} E\left (\arcsin (x)\left |\frac {3}{2}\right .\right )+\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right ) \] Output:
-1/3*2^(1/2)*EllipticE(x,1/2*6^(1/2))+1/3*EllipticF(x,1/2*6^(1/2))*2^(1/2)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {-E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{\sqrt {3}} \] Input:
Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]
Output:
(-EllipticE[ArcSin[Sqrt[3/2]*x], 2/3] + EllipticF[ArcSin[Sqrt[3/2]*x], 2/3 ])/Sqrt[3]
Time = 0.17 (sec) , antiderivative size = 31, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx\) |
\(\Big \downarrow \) 389 |
\(\displaystyle \frac {2}{3} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}}dx-\frac {1}{3} \int \frac {\sqrt {2-3 x^2}}{\sqrt {1-x^2}}dx\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )-\frac {1}{3} \int \frac {\sqrt {2-3 x^2}}{\sqrt {1-x^2}}dx\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )-\frac {1}{3} \sqrt {2} E\left (\arcsin (x)\left |\frac {3}{2}\right .\right )\) |
Input:
Int[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]),x]
Output:
-1/3*(Sqrt[2]*EllipticE[ArcSin[x], 3/2]) + (Sqrt[2]*EllipticF[ArcSin[x], 3 /2])/3
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 = 2.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\operatorname {EllipticF}\left (x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {\sqrt {6}}{2}\right )\right )}{3}\) | \(23\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{4}-5 x^{2}+2}}\) | \(67\) |
Input:
int(x^2/(-3*x^2+2)^(1/2)/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*2^(1/2)*(EllipticF(x,1/2*6^(1/2))-EllipticE(x,1/2*6^(1/2)))
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {\sqrt {3} x E(\arcsin \left (\frac {1}{x}\right )\,|\,\frac {2}{3}) - \sqrt {3} x F(\arcsin \left (\frac {1}{x}\right )\,|\,\frac {2}{3}) + \sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{3 \, x} \] Input:
integrate(x^2/(-3*x^2+2)^(1/2)/(-x^2+1)^(1/2),x, algorithm="fricas")
Output:
1/3*(sqrt(3)*x*elliptic_e(arcsin(1/x), 2/3) - sqrt(3)*x*elliptic_f(arcsin( 1/x), 2/3) + sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2))/x
\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int \frac {x^{2}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - 3 x^{2}}}\, dx \] Input:
integrate(x**2/(-3*x**2+2)**(1/2)/(-x**2+1)**(1/2),x)
Output:
Integral(x**2/(sqrt(-(x - 1)*(x + 1))*sqrt(2 - 3*x**2)), x)
\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(1/2)/(-x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)), x)
\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(1/2)/(-x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(x^2/(sqrt(-x^2 + 1)*sqrt(-3*x^2 + 2)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int \frac {x^2}{\sqrt {1-x^2}\,\sqrt {2-3\,x^2}} \,d x \] Input:
int(x^2/((1 - x^2)^(1/2)*(2 - 3*x^2)^(1/2)),x)
Output:
int(x^2/((1 - x^2)^(1/2)*(2 - 3*x^2)^(1/2)), x)
\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int \frac {\sqrt {-3 x^{2}+2}\, \sqrt {-x^{2}+1}\, x^{2}}{3 x^{4}-5 x^{2}+2}d x \] Input:
int(x^2/(-3*x^2+2)^(1/2)/(-x^2+1)^(1/2),x)
Output:
int((sqrt( - 3*x**2 + 2)*sqrt( - x**2 + 1)*x**2)/(3*x**4 - 5*x**2 + 2),x)