Integrand size = 34, antiderivative size = 620 \[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=-\frac {3 \sqrt {x^2} \sqrt {x^2 \left (2-3 x^2\right )}}{x \left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {x^2} \left (1-\left (1-3 x^2\right )^{2/3}\right ) \sqrt {\frac {\sqrt [3]{2}+\sqrt [3]{2} \left (1-3 x^2\right )^{2/3}+\sqrt [3]{2} \left (1-3 x^2\right )^{4/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1-\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}{2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {2} x \sqrt {x^2 \left (2-3 x^2\right )} \sqrt {\frac {2^{2/3}-2^{2/3} \left (1-3 x^2\right )^{2/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}}}-\frac {\sqrt {x^2} \left (1-\left (1-3 x^2\right )^{2/3}\right ) \sqrt {\frac {\sqrt [3]{2}+\sqrt [3]{2} \left (1-3 x^2\right )^{2/3}+\sqrt [3]{2} \left (1-3 x^2\right )^{4/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1-\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}{2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} x \sqrt {x^2 \left (2-3 x^2\right )} \sqrt {\frac {2^{2/3}-2^{2/3} \left (1-3 x^2\right )^{2/3}}{\left (2^{2/3} \left (1+\sqrt {3}\right )-2^{2/3} \left (1-3 x^2\right )^{2/3}\right )^2}}} \] Output:
-3*(x^2)^(1/2)*(x^2*(-3*x^2+2))^(1/2)/x/(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^2)^(1/2)*(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/(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^2)^(1/2)*(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)*EllipticF((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/(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)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 1.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=\frac {6 x \sqrt {x^2} \sqrt [3]{2-6 x^2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},\frac {1}{2},\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )}{\sqrt {2 x^2-3 x^4} \left (6 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},\frac {1}{2},\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )+x^2 \left (3 \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},\frac {3}{2},\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )-4 \operatorname {AppellF1}\left (\frac {3}{2},\frac {2}{3},\frac {1}{2},\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )\right )\right )} \] Input:
Integrate[(Sqrt[x^2]*(2 - 6*x^2)^(1/3))/Sqrt[2*x^2 - 3*x^4],x]
Output:
(6*x*Sqrt[x^2]*(2 - 6*x^2)^(1/3)*AppellF1[1/2, -1/3, 1/2, 3/2, 3*x^2, (3*x ^2)/2])/(Sqrt[2*x^2 - 3*x^4]*(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*AppellF 1[3/2, 2/3, 1/2, 5/2, 3*x^2, (3*x^2)/2])))
Time = 0.44 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.48, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {34, 27, 1940, 1118, 27, 266, 807, 832, 759, 2416}
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.
\(\displaystyle \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx\) |
\(\Big \downarrow \) 34 |
\(\displaystyle \frac {\sqrt {x^2} \int \frac {\sqrt [3]{2} x \sqrt [3]{1-3 x^2}}{\sqrt {2 x^2-3 x^4}}dx}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt [3]{2} \sqrt {x^2} \int \frac {x \sqrt [3]{1-3 x^2}}{\sqrt {2 x^2-3 x^4}}dx}{x}\) |
\(\Big \downarrow \) 1940 |
\(\displaystyle \frac {\sqrt {x^2} \int \frac {\sqrt [3]{1-3 x^2}}{\sqrt {2 x^2-3 x^4}}dx^2}{2^{2/3} x}\) |
\(\Big \downarrow \) 1118 |
\(\displaystyle -\frac {\sqrt {x^2} \int \frac {\sqrt {3} \sqrt [3]{1-3 x^2}}{\sqrt {1-x^4}}d\left (1-3 x^2\right )}{3\ 2^{2/3} x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {x^2} \int \frac {\sqrt [3]{1-3 x^2}}{\sqrt {1-x^4}}d\left (1-3 x^2\right )}{2^{2/3} \sqrt {3} x}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {\sqrt {3} \sqrt {x^2} \int \frac {x^6}{\sqrt {1-x^{12}}}d\sqrt [3]{1-3 x^2}}{2^{2/3} x}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {\sqrt {3} \sqrt {x^2} \int \frac {x^4}{\sqrt {1-x^6}}dx^4}{2\ 2^{2/3} x}\) |
\(\Big \downarrow \) 832 |
\(\displaystyle -\frac {\sqrt {3} \sqrt {x^2} \left (\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^6}}dx^4-\int \frac {3 x^2-\sqrt {3}}{\sqrt {1-x^6}}dx^4\right )}{2\ 2^{2/3} x}\) |
\(\Big \downarrow \) 759 |
\(\displaystyle -\frac {\sqrt {3} \sqrt {x^2} \left (-\int \frac {3 x^2-\sqrt {3}}{\sqrt {1-x^6}}dx^4-\frac {2 \sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt {\frac {x^4-3 x^2+2}{\left (3 x^2+\sqrt {3}\right )^2}} x^2 \operatorname {EllipticF}\left (\arcsin \left (\frac {3 x^2-\sqrt {3}}{3 x^2+\sqrt {3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {x^2}{\left (3 x^2+\sqrt {3}\right )^2}} \sqrt {1-x^6}}\right )}{2\ 2^{2/3} x}\) |
\(\Big \downarrow \) 2416 |
\(\displaystyle -\frac {\sqrt {3} \sqrt {x^2} \left (-\frac {2 \sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt {\frac {x^4-3 x^2+2}{\left (3 x^2+\sqrt {3}\right )^2}} x^2 \operatorname {EllipticF}\left (\arcsin \left (\frac {3 x^2-\sqrt {3}}{3 x^2+\sqrt {3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {x^2}{\left (3 x^2+\sqrt {3}\right )^2}} \sqrt {1-x^6}}-\frac {3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {x^4-3 x^2+2}{\left (3 x^2+\sqrt {3}\right )^2}} x^2 E\left (\arcsin \left (\frac {3 x^2-\sqrt {3}}{3 x^2+\sqrt {3}}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x^2}{\left (3 x^2+\sqrt {3}\right )^2}} \sqrt {1-x^6}}+\frac {2 \sqrt {1-x^6}}{3 x^2+\sqrt {3}}\right )}{2\ 2^{2/3} x}\) |
Input:
Int[(Sqrt[x^2]*(2 - 6*x^2)^(1/3))/Sqrt[2*x^2 - 3*x^4],x]
Output:
-1/2*(Sqrt[3]*Sqrt[x^2]*((2*Sqrt[1 - x^6])/(Sqrt[3] + 3*x^2) - (3^(3/4)*Sq rt[2 - Sqrt[3]]*x^2*Sqrt[(2 - 3*x^2 + x^4)/(Sqrt[3] + 3*x^2)^2]*EllipticE[ ArcSin[(-Sqrt[3] + 3*x^2)/(Sqrt[3] + 3*x^2)], -7 - 4*Sqrt[3]])/(Sqrt[x^2/( Sqrt[3] + 3*x^2)^2]*Sqrt[1 - x^6]) - (2*3^(1/4)*(1 - Sqrt[3])*Sqrt[2 + Sqr t[3]]*x^2*Sqrt[(2 - 3*x^2 + x^4)/(Sqrt[3] + 3*x^2)^2]*EllipticF[ArcSin[(-S qrt[3] + 3*x^2)/(Sqrt[3] + 3*x^2)], -7 - 4*Sqrt[3]])/(Sqrt[x^2/(Sqrt[3] + 3*x^2)^2]*Sqrt[1 - x^6])))/(2^(2/3)*x)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F racPart[p]/x^(m*FracPart[p])) Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x ] && !IntegerQ[p]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[1/e Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_) ^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1) *(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && I ntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
\[\int \frac {\sqrt {x^{2}}\, \left (-6 x^{2}+2\right )^{\frac {1}{3}}}{\sqrt {-3 x^{4}+2 x^{2}}}d x\]
Input:
int((x^2)^(1/2)*(-6*x^2+2)^(1/3)/(-3*x^4+2*x^2)^(1/2),x)
Output:
int((x^2)^(1/2)*(-6*x^2+2)^(1/3)/(-3*x^4+2*x^2)^(1/2),x)
\[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=\int { \frac {\sqrt {x^{2}} {\left (-6 \, x^{2} + 2\right )}^{\frac {1}{3}}}{\sqrt {-3 \, x^{4} + 2 \, x^{2}}} \,d x } \] Input:
integrate((x^2)^(1/2)*(-6*x^2+2)^(1/3)/(-3*x^4+2*x^2)^(1/2),x, algorithm=" fricas")
Output:
integral(-sqrt(-3*x^4 + 2*x^2)*sqrt(x^2)*(-6*x^2 + 2)^(1/3)/(3*x^4 - 2*x^2 ), sqrt(x^2))
\[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=\sqrt [3]{2} \int \frac {\sqrt [3]{1 - 3 x^{2}} \sqrt {x^{2}}}{\sqrt {- 3 x^{4} + 2 x^{2}}}\, dx \] Input:
integrate((x**2)**(1/2)*(-6*x**2+2)**(1/3)/(-3*x**4+2*x**2)**(1/2),x)
Output:
2**(1/3)*Integral((1 - 3*x**2)**(1/3)*sqrt(x**2)/sqrt(-3*x**4 + 2*x**2), x )
\[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=\int { \frac {\sqrt {x^{2}} {\left (-6 \, x^{2} + 2\right )}^{\frac {1}{3}}}{\sqrt {-3 \, x^{4} + 2 \, x^{2}}} \,d x } \] Input:
integrate((x^2)^(1/2)*(-6*x^2+2)^(1/3)/(-3*x^4+2*x^2)^(1/2),x, algorithm=" maxima")
Output:
integrate(sqrt(x^2)*(-6*x^2 + 2)^(1/3)/sqrt(-3*x^4 + 2*x^2), x)
\[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=\int { \frac {\sqrt {x^{2}} {\left (-6 \, x^{2} + 2\right )}^{\frac {1}{3}}}{\sqrt {-3 \, x^{4} + 2 \, x^{2}}} \,d x } \] Input:
integrate((x^2)^(1/2)*(-6*x^2+2)^(1/3)/(-3*x^4+2*x^2)^(1/2),x, algorithm=" giac")
Output:
integrate(sqrt(x^2)*(-6*x^2 + 2)^(1/3)/sqrt(-3*x^4 + 2*x^2), x)
Timed out. \[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=\int \frac {{\left (2-6\,x^2\right )}^{1/3}\,\sqrt {x^2}}{\sqrt {2\,x^2-3\,x^4}} \,d x \] Input:
int(((2 - 6*x^2)^(1/3)*(x^2)^(1/2))/(2*x^2 - 3*x^4)^(1/2),x)
Output:
int(((2 - 6*x^2)^(1/3)*(x^2)^(1/2))/(2*x^2 - 3*x^4)^(1/2), x)
\[ \int \frac {\sqrt {x^2} \sqrt [3]{2-6 x^2}}{\sqrt {2 x^2-3 x^4}} \, dx=-2^{\frac {1}{3}} \left (\int \frac {\left (-3 x^{2}+1\right )^{\frac {1}{3}} \sqrt {-3 x^{2}+2}}{3 x^{2}-2}d x \right ) \] Input:
int((x^2)^(1/2)*(-6*x^2+2)^(1/3)/(-3*x^4+2*x^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)