Integrand size = 24, antiderivative size = 148 \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\sqrt [4]{2} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{2-3 x^2}}{2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}} \] Output:
1/9*2^(1/4)*arctan(1/3*(2^(3/4)-2^(1/4)*(-3*x^2+2)^(1/2))*3^(1/2)/x/(-3*x^ 2+2)^(1/4))*3^(1/2)+1/9*2^(1/4)*arctanh(3^(1/2)*x*(-3*x^2+2)^(1/4)/(2^(3/4 )+2^(1/4)*(-3*x^2+2)^(1/2)))*3^(1/2)-2/9*2^(1/4)*EllipticE(sin(1/2*arcsin( 1/2*x*6^(1/2))),2^(1/2))*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 5.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.25 \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{12 \sqrt [4]{2}} \] Input:
Integrate[x^2/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]
Output:
(x^3*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(12*2^(1/4))
Time = 0.24 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {349, 2009}
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.
\(\displaystyle \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\) |
\(\Big \downarrow \) 349 |
\(\displaystyle \int \left (\frac {4}{3 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}-\frac {1}{3 \sqrt [4]{2-3 x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}\) |
Input:
Int[x^2/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]
Output:
(2^(1/4)*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x^2])/(Sqrt[3]*x*(2 - 3*x^2) ^(1/4))])/(3*Sqrt[3]) + (2^(1/4)*ArcTanh[(2^(3/4) + 2^(1/4)*Sqrt[2 - 3*x^2 ])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(3*Sqrt[3]) - (2*2^(1/4)*EllipticE[ArcS in[Sqrt[3/2]*x]/2, 2])/(3*Sqrt[3])
Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol ] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(1/4)*(c + d*x^2)), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a] || In tegerQ[m/2])
\[\int \frac {x^{2}}{\left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (-3 x^{2}+4\right )}d x\]
Input:
int(x^2/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)
Output:
int(x^2/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)
\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(1/4)/(-3*x^2+4),x, algorithm="fricas")
Output:
integral((-3*x^2 + 2)^(3/4)*x^2/(9*x^4 - 18*x^2 + 8), x)
\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{2}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \] Input:
integrate(x**2/(-3*x**2+2)**(1/4)/(-3*x**2+4),x)
Output:
-Integral(x**2/(3*x**2*(2 - 3*x**2)**(1/4) - 4*(2 - 3*x**2)**(1/4)), x)
\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(1/4)/(-3*x^2+4),x, algorithm="maxima")
Output:
-integrate(x^2/((3*x^2 - 4)*(-3*x^2 + 2)^(1/4)), x)
\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(1/4)/(-3*x^2+4),x, algorithm="giac")
Output:
integrate(-x^2/((3*x^2 - 4)*(-3*x^2 + 2)^(1/4)), x)
Timed out. \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\int \frac {x^2}{{\left (2-3\,x^2\right )}^{1/4}\,\left (3\,x^2-4\right )} \,d x \] Input:
int(-x^2/((2 - 3*x^2)^(1/4)*(3*x^2 - 4)),x)
Output:
-int(x^2/((2 - 3*x^2)^(1/4)*(3*x^2 - 4)), x)
\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\left (\int \frac {x^{2}}{3 \left (-3 x^{2}+2\right )^{\frac {1}{4}} x^{2}-4 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}d x \right ) \] Input:
int(x^2/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)
Output:
- int(x**2/(3*( - 3*x**2 + 2)**(1/4)*x**2 - 4*( - 3*x**2 + 2)**(1/4)),x)