3.711 \(\int \frac{1}{(2+3 i x)^3 \sqrt [3]{4-27 x^2}} \, dx\)

Optimal. Leaf size=676 \[ \frac{\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right ),4 \sqrt{3}-7\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{192 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}-\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac{i \log (2+3 i x)}{192 \sqrt [3]{2}} \]

[Out]

((I/96)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x)^2 + ((I/96)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x) - (3*x)/(32*(2^(2/3)*(
1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))) + ((I/96)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3*I)*x))/(Sqrt[3]*(4 - 27*x^2)
^(1/3))])/(2^(1/3)*Sqrt[3]) - (Sqrt[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]])/(96*2^(2/
3)*3^(3/4)*x*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/
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]*EllipticF[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]])/(144*2^(1/6)*3^(1/4)*x*Sqrt[-((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^
(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)]) + ((I/192)*Log[2 + (3*I)*x])/2^(1/3) - ((I/192)*Log[-54 + (81*I
)*x + 27*2^(2/3)*(4 - 27*x^2)^(1/3)])/2^(1/3)

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Rubi [A]  time = 0.412514, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {745, 835, 844, 235, 304, 219, 1879, 751} \[ -\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{192 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}+\frac{\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac{i \log (2+3 i x)}{192 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((2 + (3*I)*x)^3*(4 - 27*x^2)^(1/3)),x]

[Out]

((I/96)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x)^2 + ((I/96)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x) - (3*x)/(32*(2^(2/3)*(
1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))) + ((I/96)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3*I)*x))/(Sqrt[3]*(4 - 27*x^2)
^(1/3))])/(2^(1/3)*Sqrt[3]) - (Sqrt[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]])/(96*2^(2/
3)*3^(3/4)*x*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/
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]*EllipticF[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]])/(144*2^(1/6)*3^(1/4)*x*Sqrt[-((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^
(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)]) + ((I/192)*Log[2 + (3*I)*x])/2^(1/3) - ((I/192)*Log[-54 + (81*I
)*x + 27*2^(2/3)*(4 - 27*x^2)^(1/3)])/2^(1/3)

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(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] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(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]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[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
] + Simp[(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]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rule 751

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[(6*c^2*e^2)/d^2, 3]}, -Simp
[(Sqrt[3]*c*e*ArcTan[1/Sqrt[3] + (2*c*(d - e*x))/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))])/(d^2*q^2), x] + (-Simp[(3*c
*e*Log[d + e*x])/(2*d^2*q^2), x] + Simp[(3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)])/(2*d^2*q^2), x])] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(2+3 i x)^3 \sqrt [3]{4-27 x^2}} \, dx &=\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}-\frac{3}{32} \int \frac{-4+2 i x}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx\\ &=\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}-\frac{\int \frac{-192-144 i x}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx}{1536}\\ &=\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{1}{32} \int \frac{1}{\sqrt [3]{4-27 x^2}} \, dx+\frac{1}{16} \int \frac{1}{(2+3 i x) \sqrt [3]{4-27 x^2}} \, dx\\ &=\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{192 \sqrt [3]{2}}-\frac{i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{192 \sqrt [3]{2}}-\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{64 \sqrt{3} x}\\ &=\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}+\frac{i \log (2+3 i x)}{192 \sqrt [3]{2}}-\frac{i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{192 \sqrt [3]{2}}+\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{2^{2/3} \left (1+\sqrt{3}\right )-x}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{64 \sqrt{3} x}-\frac{\sqrt{-x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{16\ 2^{5/6} \sqrt{3 \left (2-\sqrt{3}\right )} x}\\ &=\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}-\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \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 (\sin ^{-1}\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 )}{96\ 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}} F\left (\sin ^{-1}\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 )}{144 \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)}{192 \sqrt [3]{2}}-\frac{i \log \left (-54+81 i x+27\ 2^{2/3} \sqrt [3]{4-27 x^2}\right )}{192 \sqrt [3]{2}}\\ \end{align*}

Mathematica [C]  time = 0.100555, size = 134, normalized size = 0.2 \[ -\frac{i \sqrt [3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt [3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left (\frac{8}{3};\frac{1}{3},\frac{1}{3};\frac{11}{3};\frac{2 \left (3 i+\sqrt{3}\right )}{6 i-9 x},\frac{2 \left (-3 i+\sqrt{3}\right )}{9 x-6 i}\right )}{8\ 3^{2/3} (-3 x+2 i)^2 \sqrt [3]{4-27 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + (3*I)*x)^3*(4 - 27*x^2)^(1/3)),x]

[Out]

((-I/8)*((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1/3)*AppellF1[8/3, 1/3, 1/3, 1
1/3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))/(-6*I + 9*x)])/(3^(2/3)*(2*I - 3*x)^2*(4 - 27*x^2)^
(1/3))

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Maple [F]  time = 0.463, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( 2+3\,ix \right ) ^{3}}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*I*x)^3/(-27*x^2+4)^(1/3),x)

[Out]

int(1/(2+3*I*x)^3/(-27*x^2+4)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)^3/(-27*x^2+4)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)^3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)^3/(-27*x^2+4)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{4 - 27 x^{2}} \left (3 i x + 2\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)**3/(-27*x**2+4)**(1/3),x)

[Out]

Integral(1/((4 - 27*x**2)**(1/3)*(3*I*x + 2)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*I*x)^3/(-27*x^2+4)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)^3), x)