Integrand size = 24, antiderivative size = 246 \[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=-\frac {\left (-1+3 x^2\right )^{3/4}}{2 x}+\frac {3 x \sqrt [4]{-1+3 x^2}}{2 \left (1+\sqrt {-1+3 x^2}\right )}-\frac {1}{4} \sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{4} \sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) E\left (2 \arctan \left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{2 x}+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{-1+3 x^2}\right ),\frac {1}{2}\right )}{4 x} \]
-1/2*(3*x^2-1)^(3/4)/x-1/8*arctan(1/2*x*6^(1/2)/(3*x^2-1)^(1/4))*6^(1/2)-1 /8*arctanh(1/2*x*6^(1/2)/(3*x^2-1)^(1/4))*6^(1/2)+3/2*x*(3*x^2-1)^(1/4)/(1 +(3*x^2-1)^(1/2))-1/2*(cos(2*arctan((3*x^2-1)^(1/4)))^2)^(1/2)/cos(2*arcta n((3*x^2-1)^(1/4)))*EllipticE(sin(2*arctan((3*x^2-1)^(1/4))),1/2*2^(1/2))* (1+(3*x^2-1)^(1/2))*(x^2/(1+(3*x^2-1)^(1/2))^2)^(1/2)/x*3^(1/2)+1/4*(cos(2 *arctan((3*x^2-1)^(1/4)))^2)^(1/2)/cos(2*arctan((3*x^2-1)^(1/4)))*Elliptic F(sin(2*arctan((3*x^2-1)^(1/4))),1/2*2^(1/2))*(1+(3*x^2-1)^(1/2))*(x^2/(1+ (3*x^2-1)^(1/2))^2)^(1/2)/x*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {4-12 x^2-3 x^4 \sqrt [4]{1-3 x^2} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )}{8 x \sqrt [4]{-1+3 x^2}} \]
(4 - 12*x^2 - 3*x^4*(1 - 3*x^2)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3 *x^2)/2])/(8*x*(-1 + 3*x^2)^(1/4))
Time = 0.35 (sec) , antiderivative size = 246, 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.
\(\displaystyle \int \frac {1}{x^2 \left (3 x^2-2\right ) \sqrt [4]{3 x^2-1}} \, dx\) |
\(\Big \downarrow \) 349 |
\(\displaystyle \int \left (\frac {3}{2 \left (3 x^2-2\right ) \sqrt [4]{3 x^2-1}}-\frac {1}{2 x^2 \sqrt [4]{3 x^2-1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{4} \sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{3 x^2-1}\right ),\frac {1}{2}\right )}{4 x}-\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) E\left (2 \arctan \left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{2 x}-\frac {1}{4} \sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {3 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt {3 x^2-1}+1\right )}-\frac {\left (3 x^2-1\right )^{3/4}}{2 x}\) |
-1/2*(-1 + 3*x^2)^(3/4)/x + (3*x*(-1 + 3*x^2)^(1/4))/(2*(1 + Sqrt[-1 + 3*x ^2])) - (Sqrt[3/2]*ArcTan[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/4 - (Sqrt[3/2 ]*ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/4 - (Sqrt[3]*Sqrt[x^2/(1 + Sq rt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticE[2*ArcTan[(-1 + 3*x^2)^ (1/4)], 1/2])/(2*x) + (Sqrt[3]*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqr t[-1 + 3*x^2])*EllipticF[2*ArcTan[(-1 + 3*x^2)^(1/4)], 1/2])/(4*x)
3.11.51.3.1 Defintions of rubi rules used
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])
\[\int \frac {1}{x^{2} \left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}}d x\]
\[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int \frac {1}{x^{2} \cdot \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \]
\[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int \frac {1}{x^2\,{\left (3\,x^2-1\right )}^{1/4}\,\left (3\,x^2-2\right )} \,d x \]