\(\int \frac {x^6}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\) [1485]

Optimal result

Integrand size = 24, antiderivative size = 182 \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {80}{567} x \sqrt [4]{2-3 x^2}+\frac {2}{63} x^3 \sqrt [4]{2-3 x^2}+\frac {8\ 2^{3/4} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{2-3 x^2}}{2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {160\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{567 \sqrt {3}} \] Output:

80/567*x*(-3*x^2+2)^(1/4)+2/63*x^3*(-3*x^2+2)^(1/4)+8/81*2^(3/4)*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)-8 
/81*2^(3/4)*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)-160/1701*2^(3/4)*InverseJacobiAM(1/2*arcsin(1/2*x*6^(1/2) 
),2^(1/2))*3^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 6.42 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{567} x \left (31 \sqrt [4]{2} x^2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+\frac {80-102 x^2-27 x^4+\frac {1280 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{\left (-4+3 x^2\right ) \left (4 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )\right )}}{\left (2-3 x^2\right )^{3/4}}\right ) \] Input:

Integrate[x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
 

Output:

(2*x*(31*2^(1/4)*x^2*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4] + (8 
0 - 102*x^2 - 27*x^4 + (1280*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2) 
/4])/((-4 + 3*x^2)*(4*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x 
^2*(2*AppellF1[3/2, 3/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + 3*AppellF1[3/2, 7 
/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4]))))/(2 - 3*x^2)^(3/4)))/567
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 182, 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 = {352, 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.

\(\displaystyle \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

\(\Big \downarrow \) 352

\(\displaystyle \int \left (-\frac {4 x^2}{9 \left (2-3 x^2\right )^{3/4}}-\frac {16}{27 \left (2-3 x^2\right )^{3/4}}+\frac {64}{27 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}-\frac {x^4}{3 \left (2-3 x^2\right )^{3/4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {160\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{567 \sqrt {3}}+\frac {8\ 2^{3/4} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}+\frac {80}{567} \sqrt [4]{2-3 x^2} x+\frac {2}{63} \sqrt [4]{2-3 x^2} x^3\)

Input:

Int[x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
 

Output:

(80*x*(2 - 3*x^2)^(1/4))/567 + (2*x^3*(2 - 3*x^2)^(1/4))/63 + (8*2^(3/4)*A 
rcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/ 
(27*Sqrt[3]) - (8*2^(3/4)*ArcTanh[(2^(3/4) + 2^(1/4)*Sqrt[2 - 3*x^2])/(Sqr 
t[3]*x*(2 - 3*x^2)^(1/4))])/(27*Sqrt[3]) - (160*2^(3/4)*EllipticF[ArcSin[S 
qrt[3/2]*x]/2, 2])/(567*Sqrt[3])
 

Defintions of rubi rules used

rule 352

Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol 
] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(3/4)*(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]

\[\int \frac {x^{6}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}d x\]

Input:

int(x^6/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
 

Output:

int(x^6/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
 

Fricas [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(x^6/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")
 

Output:

integral((-3*x^2 + 2)^(1/4)*x^6/(9*x^4 - 18*x^2 + 8), x)
 

Sympy [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{6}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(x**6/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
 

Output:

-Integral(x**6/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(x^6/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")
 

Output:

-integrate(x^6/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)), x)
 

Giac [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(x^6/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")
 

Output:

integrate(-x^6/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\int \frac {x^6}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \] Input:

int(-x^6/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)),x)
 

Output:

-int(x^6/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)), x)
 

Reduce [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\left (\int \frac {x^{6}}{3 \left (-3 x^{2}+2\right )^{\frac {3}{4}} x^{2}-4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}d x \right ) \] Input:

int(x^6/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
 

Output:

 - int(x**6/(3*( - 3*x**2 + 2)**(3/4)*x**2 - 4*( - 3*x**2 + 2)**(3/4)),x)