\(\int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx\) [563]

Optimal result

Integrand size = 23, antiderivative size = 614 \[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {3 x \sqrt {x^2 \left (2-3 x^2\right )}}{\sqrt {x^2} \left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} x \left (1-\left (1-3 x^2\right )^{2/3}\right ) \sqrt {\frac {\sqrt [3]{2}+\sqrt [3]{2} \left (1-3 x^2\right )^{2/3}+\sqrt [3]{2} \left (1-3 x^2\right )^{4/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1-\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}{2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {2} \sqrt {x^2} \sqrt {x^2 \left (2-3 x^2\right )} \sqrt {\frac {2^{2/3}-2^{2/3} \left (1-3 x^2\right )^{2/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}}}-\frac {x \left (1-\left (1-3 x^2\right )^{2/3}\right ) \sqrt {\frac {\sqrt [3]{2}+\sqrt [3]{2} \left (1-3 x^2\right )^{2/3}+\sqrt [3]{2} \left (1-3 x^2\right )^{4/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1-\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}{2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {x^2} \sqrt {x^2 \left (2-3 x^2\right )} \sqrt {\frac {2^{2/3}-2^{2/3} \left (1-3 x^2\right )^{2/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}}} \] Output:

-3*x*(x^2*(-3*x^2+2))^(1/2)/(x^2)^(1/2)/(2^(2/3)*(1+3^(1/2))-2^(2/3)*(-3*x 
^2+1)^(2/3))+1/4*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*x*(1-(-3*x^2+1)^(2/3))* 
((2^(1/3)+2^(1/3)*(-3*x^2+1)^(2/3)+2^(1/3)*(-3*x^2+1)^(4/3))/(2^(2/3)*(1+3 
^(1/2))-2^(2/3)*(-3*x^2+1)^(2/3))^2)^(1/2)*EllipticE((2^(2/3)*(1-3^(1/2))- 
2^(2/3)*(-3*x^2+1)^(2/3))/(2^(2/3)*(1+3^(1/2))-2^(2/3)*(-3*x^2+1)^(2/3)),I 
*3^(1/2)+2*I)*2^(1/2)/(x^2)^(1/2)/(x^2*(-3*x^2+2))^(1/2)/((2^(2/3)-2^(2/3) 
*(-3*x^2+1)^(2/3))/(2^(2/3)*(1+3^(1/2))-2^(2/3)*(-3*x^2+1)^(2/3))^2)^(1/2) 
-1/3*x*(1-(-3*x^2+1)^(2/3))*((2^(1/3)+2^(1/3)*(-3*x^2+1)^(2/3)+2^(1/3)*(-3 
*x^2+1)^(4/3))/(2^(2/3)*(1+3^(1/2))-2^(2/3)*(-3*x^2+1)^(2/3))^2)^(1/2)*Ell 
ipticF((2^(2/3)*(1-3^(1/2))-2^(2/3)*(-3*x^2+1)^(2/3))/(2^(2/3)*(1+3^(1/2)) 
-2^(2/3)*(-3*x^2+1)^(2/3)),I*3^(1/2)+2*I)*3^(3/4)/(x^2)^(1/2)/(x^2*(-3*x^2 
+2))^(1/2)/((2^(2/3)-2^(2/3)*(-3*x^2+1)^(2/3))/(2^(2/3)*(1+3^(1/2))-2^(2/3 
)*(-3*x^2+1)^(2/3))^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {6 x \sqrt [3]{2-6 x^2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},\frac {1}{2},\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )}{\sqrt {2-3 x^2} \left (6 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},\frac {1}{2},\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )+x^2 \left (3 \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},\frac {3}{2},\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )-4 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},\frac {1}{2},\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )\right )\right )} \] Input:

Integrate[(2 - 6*x^2)^(1/3)/Sqrt[2 - 3*x^2],x]
 

Output:

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

Rubi [C] (verified)

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

Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {333}

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.

\(\displaystyle \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {x \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},\frac {1}{2},\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )}{\sqrt [6]{2}}\)

Input:

Int[(2 - 6*x^2)^(1/3)/Sqrt[2 - 3*x^2],x]
 

Output:

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

Defintions of rubi rules used

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {\left (-6 x^{2}+2\right )^{\frac {1}{3}}}{\sqrt {-3 x^{2}+2}}d x\]

Input:

int((-6*x^2+2)^(1/3)/(-3*x^2+2)^(1/2),x)
 

Output:

int((-6*x^2+2)^(1/3)/(-3*x^2+2)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {{\left (-6 \, x^{2} + 2\right )}^{\frac {1}{3}}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:

integrate((-6*x^2+2)^(1/3)/(-3*x^2+2)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-3*x^2 + 2)*(-6*x^2 + 2)^(1/3)/(3*x^2 - 2), x)
 

Sympy [F]

\[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=\sqrt [3]{2} \int \frac {\sqrt [3]{1 - 3 x^{2}}}{\sqrt {2 - 3 x^{2}}}\, dx \] Input:

integrate((-6*x**2+2)**(1/3)/(-3*x**2+2)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {{\left (-6 \, x^{2} + 2\right )}^{\frac {1}{3}}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:

integrate((-6*x^2+2)^(1/3)/(-3*x^2+2)^(1/2),x, algorithm="maxima")
 

Output:

integrate((-6*x^2 + 2)^(1/3)/sqrt(-3*x^2 + 2), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {{\left (-6 \, x^{2} + 2\right )}^{\frac {1}{3}}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \] Input:

integrate((-6*x^2+2)^(1/3)/(-3*x^2+2)^(1/2),x, algorithm="giac")
 

Output:

integrate((-6*x^2 + 2)^(1/3)/sqrt(-3*x^2 + 2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {{\left (2-6\,x^2\right )}^{1/3}}{\sqrt {2-3\,x^2}} \,d x \] Input:

int((2 - 6*x^2)^(1/3)/(2 - 3*x^2)^(1/2),x)
 

Output:

int((2 - 6*x^2)^(1/3)/(2 - 3*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{2-6 x^2}}{\sqrt {2-3 x^2}} \, dx=-2^{\frac {1}{3}} \left (\int \frac {\left (-3 x^{2}+1\right )^{\frac {1}{3}} \sqrt {-3 x^{2}+2}}{3 x^{2}-2}d x \right ) \] Input:

int((-6*x^2+2)^(1/3)/(-3*x^2+2)^(1/2),x)
 

Output:

 - 2**(1/3)*int((( - 3*x**2 + 1)**(1/3)*sqrt( - 3*x**2 + 2))/(3*x**2 - 2), 
x)