Integrand size = 24, antiderivative size = 42 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \] Output:
1/3*EllipticE(1/2*x*6^(1/2),1/3*I*6^(1/2))*3^(1/2)-1/3*EllipticF(1/2*x*6^( 1/2),1/3*I*6^(1/2))*3^(1/2)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )-\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \] 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 = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {x^2+1}} \, dx\) |
\(\Big \downarrow \) 389 |
\(\displaystyle \int \frac {\sqrt {x^2+1}}{\sqrt {2-3 x^2}}dx-\int \frac {1}{\sqrt {2-3 x^2} \sqrt {x^2+1}}dx\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \int \frac {\sqrt {x^2+1}}{\sqrt {2-3 x^2}}dx-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}}\) |
Input:
Int[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]
Output:
EllipticE[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3] - EllipticF[ArcSin[Sqrt[3/2]* x], -2/3]/Sqrt[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 = 2.61 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\sqrt {3}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{3}\) | \(35\) |
elliptic | \(-\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{6 \sqrt {-3 x^{2}+2}\, \sqrt {-3 x^{4}-x^{2}+2}}\) | \(83\) |
Input:
int(x^2/(-3*x^2+2)^(1/2)/(x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/3*3^(1/2)*(EllipticF(1/2*6^(1/2)*x,1/3*I*6^(1/2))-EllipticE(1/2*6^(1/2) *x,1/3*I*6^(1/2)))
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.48 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=-\frac {2 \, \sqrt {\frac {2}{3}} \sqrt {-3} x E(\arcsin \left (\frac {\sqrt {\frac {2}{3}}}{x}\right )\,|\,-\frac {3}{2}) - 2 \, \sqrt {\frac {2}{3}} \sqrt {-3} x F(\arcsin \left (\frac {\sqrt {\frac {2}{3}}}{x}\right )\,|\,-\frac {3}{2}) + 3 \, \sqrt {x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{9 \, x} \] Input:
integrate(x^2/(-3*x^2+2)^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas")
Output:
-1/9*(2*sqrt(2/3)*sqrt(-3)*x*elliptic_e(arcsin(sqrt(2/3)/x), -3/2) - 2*sqr t(2/3)*sqrt(-3)*x*elliptic_f(arcsin(sqrt(2/3)/x), -3/2) + 3*sqrt(x^2 + 1)* sqrt(-3*x^2 + 2))/x
\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\int \frac {x^{2}}{\sqrt {2 - 3 x^{2}} \sqrt {x^{2} + 1}}\, dx \] Input:
integrate(x**2/(-3*x**2+2)**(1/2)/(x**2+1)**(1/2),x)
Output:
Integral(x**2/(sqrt(2 - 3*x**2)*sqrt(x**2 + 1)), 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. \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\int \frac {x^2}{\sqrt {x^2+1}\,\sqrt {2-3\,x^2}} \,d x \] Input:
int(x^2/((x^2 + 1)^(1/2)*(2 - 3*x^2)^(1/2)),x)
Output:
int(x^2/((x^2 + 1)^(1/2)*(2 - 3*x^2)^(1/2)), x)
\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=-\left (\int \frac {\sqrt {-3 x^{2}+2}\, \sqrt {x^{2}+1}\, x^{2}}{3 x^{4}+x^{2}-2}d x \right ) \] 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 + x**2 - 2),x)