Integrand size = 21, antiderivative size = 650 \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac {3 x}{16 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt {3}}-\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{48\ 2^{2/3} 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {i \log (2+3 i x)}{48 \sqrt [3]{2}}-\frac {i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{48 \sqrt [3]{2}} \]
1/48*I*(-27*x^2+4)^(2/3)/(2+3*I*x)+1/96*I*ln(2+3*I*x)*2^(2/3)-1/96*I*ln(-5 4+81*I*x+27*2^(2/3)*(-27*x^2+4)^(1/3))*2^(2/3)-3/16*x/(-(-27*x^2+4)^(1/3)+ 2^(2/3)*(1-3^(1/2)))-1/144*I*arctan(-1/3*3^(1/2)-1/3*2^(1/3)*(2-3*I*x)/(-2 7*x^2+4)^(1/3)*3^(1/2))*2^(2/3)*3^(1/2)+1/432*2^(5/6)*(2^(2/3)-(-27*x^2+4) ^(1/3))*EllipticF((-(-27*x^2+4)^(1/3)+2^(2/3)*(1+3^(1/2)))/(-(-27*x^2+4)^( 1/3)+2^(2/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((2*2^(1/3)+2^(2/3)*(-27*x^2+4)^( 1/3)+(-27*x^2+4)^(2/3))/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)* 3^(3/4)/x/((-2^(2/3)+(-27*x^2+4)^(1/3))/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^( 1/2)))^2)^(1/2)-1/288*2^(1/3)*(2^(2/3)-(-27*x^2+4)^(1/3))*EllipticE((-(-27 *x^2+4)^(1/3)+2^(2/3)*(1+3^(1/2)))/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)) ),2*I-I*3^(1/2))*((2*2^(1/3)+2^(2/3)*(-27*x^2+4)^(1/3)+(-27*x^2+4)^(2/3))/ (-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2) )*3^(1/4)/x/((-2^(2/3)+(-27*x^2+4)^(1/3))/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3 ^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.20 \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\frac {\sqrt [3]{\frac {2 \sqrt {3}-9 x}{2 i-3 x}} \sqrt [3]{\frac {2 \sqrt {3}+9 x}{-2 i+3 x}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},\frac {1}{3},\frac {8}{3},\frac {2 \left (3 i+\sqrt {3}\right )}{6 i-9 x},\frac {2 \left (-3 i+\sqrt {3}\right )}{-6 i+9 x}\right )}{5\ 3^{2/3} (-2 i+3 x) \sqrt [3]{4-27 x^2}} \]
(((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1 /3)*AppellF1[5/3, 1/3, 1/3, 8/3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))/(-6*I + 9*x)])/(5*3^(2/3)*(-2*I + 3*x)*(4 - 27*x^2)^(1/3))
Time = 0.60 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {498, 25, 719, 233, 501, 833, 760, 2418}
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.
| \(\displaystyle \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle \frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac {3}{16} \int -\frac {i x+2}{(3 i x+2) \sqrt [3]{4-27 x^2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{16} \int \frac {i x+2}{(3 i x+2) \sqrt [3]{4-27 x^2}}dx+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {3}{16} \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{4-27 x^2}}dx+\frac {4}{3} \int \frac {1}{(3 i x+2) \sqrt [3]{4-27 x^2}}dx\right )+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {3}{16} \left (-\frac {\sqrt {-x^2} \int \frac {\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}}{6 \sqrt {3} x}+\frac {4}{3} \int \frac {1}{(3 i x+2) \sqrt [3]{4-27 x^2}}dx\right )+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
| \(\Big \downarrow \) 501 |
| \(\displaystyle \frac {3}{16} \left (-\frac {\sqrt {-x^2} \int \frac {\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}}{6 \sqrt {3} x}+\frac {4}{3} \left (\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt {3}}-\frac {i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac {i \log (2+3 i x)}{12 \sqrt [3]{2}}\right )\right )+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {3}{16} \left (-\frac {\sqrt {-x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}\right )}{6 \sqrt {3} x}+\frac {4}{3} \left (\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt {3}}-\frac {i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac {i \log (2+3 i x)}{12 \sqrt [3]{2}}\right )\right )+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {3}{16} \left (-\frac {\sqrt {-x^2} \left (-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{3 \sqrt {3} \sqrt {-x^2}}d\sqrt [3]{4-27 x^2}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}\right )}{6 \sqrt {3} x}+\frac {4}{3} \left (\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt {3}}-\frac {i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac {i \log (2+3 i x)}{12 \sqrt [3]{2}}\right )\right )+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3}{16} \left (-\frac {\sqrt {-x^2} \left (-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac {6 \sqrt {3} \sqrt {-x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )}{6 \sqrt {3} x}+\frac {4}{3} \left (\frac {i \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 i x)}{\sqrt {3} \sqrt [3]{4-27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt {3}}-\frac {i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{12 \sqrt [3]{2}}+\frac {i \log (2+3 i x)}{12 \sqrt [3]{2}}\right )\right )+\frac {i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}\) |
((I/48)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x) + (3*(-1/6*(Sqrt[-x^2]*((-6*Sqrt [3]*Sqrt[-x^2])/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3)) + (2^(1/3)*Sq rt[2 + Sqrt[3]]*(2^(2/3) - (4 - 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*( 4 - 27*x^2)^(1/3) + (4 - 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x ^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3) )/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(1/ 4)*Sqrt[-x^2]*Sqrt[-((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)]) - (2*2^(1/3)*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*( 2^(2/3) - (4 - 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 - 27*x^2)^(1/3) + (4 - 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2]*Ell ipticF[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - S qrt[3]) - (4 - 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*Sqrt[-x^2]*Sqr t[-((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^( 1/3))^2)])))/(Sqrt[3]*x) + (4*(((I/6)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3* I)*x))/(Sqrt[3]*(4 - 27*x^2)^(1/3))])/(2^(1/3)*Sqrt[3]) + ((I/12)*Log[2 + (3*I)*x])/2^(1/3) - ((I/12)*Log[-54 + (81*I)*x + 27*2^(2/3)*(4 - 27*x^2)^( 1/3)])/2^(1/3)))/3))/16
3.8.10.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(1/3)), x_Symbol] :> With[ {q = Rt[6*b^2*(d^2/c^2), 3]}, Simp[(-Sqrt[3])*b*d*(ArcTan[1/Sqrt[3] + 2*b*( (c - d*x)/(Sqrt[3]*c*q*(a + b*x^2)^(1/3)))]/(c^2*q^2)), x] + (-Simp[3*b*d*( Log[c + d*x]/(2*c^2*q^2)), x] + Simp[3*b*d*(Log[b*c - b*d*x - c*q*(a + b*x^ 2)^(1/3)]/(2*c^2*q^2)), x])] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 - 3*a*d ^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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 ] && NegQ[a]
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] && NegQ[a]
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 - S qrt[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] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \frac {1}{\left (3 i x +2\right )^{2} \left (-27 x^{2}+4\right )^{\frac {1}{3}}}d x\]
Timed out. \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=- \int \frac {1}{9 x^{2} \sqrt [3]{4 - 27 x^{2}} - 12 i x \sqrt [3]{4 - 27 x^{2}} - 4 \sqrt [3]{4 - 27 x^{2}}}\, dx \]
-Integral(1/(9*x**2*(4 - 27*x**2)**(1/3) - 12*I*x*(4 - 27*x**2)**(1/3) - 4 *(4 - 27*x**2)**(1/3)), x)
\[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\int { \frac {1}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 i \, x + 2\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\int { \frac {1}{{\left (-27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 i \, x + 2\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx=\int \frac {1}{{\left (2+x\,3{}\mathrm {i}\right )}^2\,{\left (4-27\,x^2\right )}^{1/3}} \,d x \]