\(\int \frac {x^2}{(-2+3 x^2) \sqrt [4]{-1+3 x^2}} \, dx\) [1458]

Optimal result

Integrand size = 24, antiderivative size = 226 \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2 x \sqrt [4]{-1+3 x^2}}{3 \left (1+\sqrt {-1+3 x^2}\right )}-\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{-1+3 x^2}}{x}\right )}{3 \sqrt {6}}-\frac {2 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) E\left (2 \arctan \left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{3 \sqrt {3} x}+\frac {\sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{-1+3 x^2}\right ),\frac {1}{2}\right )}{3 \sqrt {3} x} \] Output:

2*x*(3*x^2-1)^(1/4)/(3+3*(3*x^2-1)^(1/2))-1/18*arctan(1/2*6^(1/2)*x/(3*x^2 
-1)^(1/4))*6^(1/2)-1/18*arctanh(1/3*6^(1/2)/x*(3*x^2-1)^(1/4))*6^(1/2)-2/9 
*(x^2/(1+(3*x^2-1)^(1/2))^2)^(1/2)*(1+(3*x^2-1)^(1/2))*EllipticE(sin(2*arc 
tan((3*x^2-1)^(1/4))),1/2*2^(1/2))*3^(1/2)/x+1/9*(x^2/(1+(3*x^2-1)^(1/2))^ 
2)^(1/2)*(1+(3*x^2-1)^(1/2))*InverseJacobiAM(2*arctan((3*x^2-1)^(1/4)),1/2 
*2^(1/2))*3^(1/2)/x
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 5.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.23 \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=-\frac {x^3 \sqrt [4]{1-3 x^2} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )}{6 \sqrt [4]{-1+3 x^2}} \] Input:

Integrate[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]
 

Output:

-1/6*(x^3*(1 - 3*x^2)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2])/ 
(-1 + 3*x^2)^(1/4)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.99, 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.

\(\displaystyle \int \frac {x^2}{\left (3 x^2-2\right ) \sqrt [4]{3 x^2-1}} \, dx\)

\(\Big \downarrow \) 349

\(\displaystyle \int \left (\frac {2}{3 \left (3 x^2-2\right ) \sqrt [4]{3 x^2-1}}+\frac {1}{3 \sqrt [4]{3 x^2-1}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}}+\frac {\sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{3 x^2-1}\right ),\frac {1}{2}\right )}{3 \sqrt {3} x}-\frac {2 \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) E\left (2 \arctan \left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{3 \sqrt {3} x}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}}+\frac {2 \sqrt [4]{3 x^2-1} x}{3 \left (\sqrt {3 x^2-1}+1\right )}\)

Input:

Int[x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(1/4)),x]
 

Output:

(2*x*(-1 + 3*x^2)^(1/4))/(3*(1 + Sqrt[-1 + 3*x^2])) - ArcTan[(Sqrt[3/2]*x) 
/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4 
)]/(3*Sqrt[6]) - (2*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^ 
2])*EllipticE[2*ArcTan[(-1 + 3*x^2)^(1/4)], 1/2])/(3*Sqrt[3]*x) + (Sqrt[x^ 
2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticF[2*ArcTan[(-1 
+ 3*x^2)^(1/4)], 1/2])/(3*Sqrt[3]*x)
 

Defintions of rubi rules used

rule 349

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {x^{2}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}}d x\]

Input:

int(x^2/(3*x^2-2)/(3*x^2-1)^(1/4),x)
 

Output:

int(x^2/(3*x^2-2)/(3*x^2-1)^(1/4),x)
 

Fricas [F]

\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \] Input:

integrate(x^2/(3*x^2-2)/(3*x^2-1)^(1/4),x, algorithm="fricas")
 

Output:

integral((3*x^2 - 1)^(3/4)*x^2/(9*x^4 - 9*x^2 + 2), x)
 

Sympy [F]

\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int \frac {x^{2}}{\left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \] Input:

integrate(x**2/(3*x**2-2)/(3*x**2-1)**(1/4),x)
 

Output:

Integral(x**2/((3*x**2 - 2)*(3*x**2 - 1)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \] Input:

integrate(x^2/(3*x^2-2)/(3*x^2-1)^(1/4),x, algorithm="maxima")
 

Output:

integrate(x^2/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)), x)
 

Giac [F]

\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \] Input:

integrate(x^2/(3*x^2-2)/(3*x^2-1)^(1/4),x, algorithm="giac")
 

Output:

integrate(x^2/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int \frac {x^2}{{\left (3\,x^2-1\right )}^{1/4}\,\left (3\,x^2-2\right )} \,d x \] Input:

int(x^2/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)),x)
 

Output:

int(x^2/((3*x^2 - 1)^(1/4)*(3*x^2 - 2)), x)
 

Reduce [F]

\[ \int \frac {x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int \frac {x^{2}}{3 \left (3 x^{2}-1\right )^{\frac {1}{4}} x^{2}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}d x \] Input:

int(x^2/(3*x^2-2)/(3*x^2-1)^(1/4),x)
 

Output:

int(x**2/(3*(3*x**2 - 1)**(1/4)*x**2 - 2*(3*x**2 - 1)**(1/4)),x)