\(\int \frac {(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx\) [706]

Optimal result

Integrand size = 21, antiderivative size = 564 \[ \int \frac {(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx=-\frac {4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac {1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac {16 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}+\frac {32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4-27 x^2}+\left (4-27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]

[Out]

-4/35*(7*I-4*x)*(-27*x^2+4)^(2/3)-1/30*I*(2+3*I*x)^2*(-27*x^2+4)^(2/3)-96/7*x/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3
^(1/2)))+32/189*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)-16/63*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)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {757, 794, 241, 310, 225, 1893} \[ \int \frac {(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx=\frac {32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right ),-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac {16 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt {\frac {\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac {1}{30} i \left (4-27 x^2\right )^{2/3} (2+3 i x)^2-\frac {4}{35} (-4 x+7 i) \left (4-27 x^2\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )} \]

[In]

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

[Out]

(-4*(7*I - 4*x)*(4 - 27*x^2)^(2/3))/35 - (I/30)*(2 + (3*I)*x)^2*(4 - 27*x^2)^(2/3) - (96*x)/(7*(2^(2/3)*(1 - S
qrt[3]) - (4 - 27*x^2)^(1/3))) - (16*2^(1/3)*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[A
rcSin[(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]])/(21*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)]) + (
32*2^(5/6)*(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]])/(63*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)])

Rule 225

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]*Sq
rt[(-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]

Rule 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 757

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)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*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] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1893

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]/(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] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac {1}{90} \int \frac {(2+3 i x) (-288-864 i x)}{\sqrt [3]{4-27 x^2}} \, dx \\ & = -\frac {4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac {1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}+\frac {32}{7} \int \frac {1}{\sqrt [3]{4-27 x^2}} \, dx \\ & = -\frac {4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac {1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac {\left (16 \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 \sqrt {3} x} \\ & = -\frac {4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac {1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}+\frac {\left (16 \sqrt {-x^2}\right ) \text {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 )}{7 \sqrt {3} x}-\frac {\left (16\ 2^{2/3} \left (1+\sqrt {3}\right ) \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4-27 x^2}\right )}{7 \sqrt {3} x} \\ & = -\frac {4}{35} (7 i-4 x) \left (4-27 x^2\right )^{2/3}-\frac {1}{30} i (2+3 i x)^2 \left (4-27 x^2\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4-27 x^2}\right )}-\frac {16 \sqrt [3]{2} \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 )}{21\ 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 {32\ 2^{5/6} \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 )}{63 \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}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.11 \[ \int \frac {(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx=\left (4-27 x^2\right )^{2/3} \left (-\frac {14 i}{15}+\frac {6 x}{7}+\frac {3 i x^2}{10}\right )+\frac {16}{7} \sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {27 x^2}{4}\right ) \]

[In]

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

[Out]

(4 - 27*x^2)^(2/3)*((-14*I)/15 + (6*x)/7 + ((3*I)/10)*x^2) + (16*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (2
7*x^2)/4])/7

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 2.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.09

method	result	size
risch	\(-\frac {i \left (63 x^{2}-180 i x -196\right ) \left (27 x^{2}-4\right )}{210 \left (-27 x^{2}+4\right )^{\frac {1}{3}}}+\frac {16 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^{2}}{4}\right )}{7}\)	\(49\)
meijerg	\(4 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {27 x^{2}}{4}\right )+9 i 2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;\frac {27 x^{2}}{4}\right )-9 \,2^{\frac {1}{3}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {3}{2};\frac {5}{2};\frac {27 x^{2}}{4}\right )-\frac {27 i 2^{\frac {1}{3}} x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},2;3;\frac {27 x^{2}}{4}\right )}{8}\)	\(78\)

[In]

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

[Out]

-1/210*I*(-180*I*x+63*x^2-196)*(27*x^2-4)/(-27*x^2+4)^(1/3)+16/7*2^(1/3)*x*hypergeom([1/3,1/2],[3/2],27/4*x^2)

Fricas [F]

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

[In]

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

[Out]

1/630*(630*x*integral(128/63*(-27*x^2 + 4)^(2/3)/(27*x^4 - 4*x^2), x) + (189*I*x^3 + 540*x^2 - 588*I*x - 320)*
(-27*x^2 + 4)^(2/3))/x

Sympy [A] (verification not implemented)

Time = 3.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.26 \[ \int \frac {(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx=- 9 \cdot \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )} + 4 \cdot \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{2 i \pi }}{4}} \right )} - i \left (4 - 27 x^{2}\right )^{\frac {2}{3}} - 27 i \left (\begin {cases} \frac {x^{2} \left (27 x^{2} - 4\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}}}{90} + \frac {\left (27 x^{2} - 4\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}}}{405} & \text {for}\: \left |{x^{2}}\right | > \frac {4}{27} \\- \frac {x^{2} \left (4 - 27 x^{2}\right )^{\frac {2}{3}}}{90} - \frac {\left (4 - 27 x^{2}\right )^{\frac {2}{3}}}{405} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

-9*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(2*I*pi)/4) + 4*2**(1/3)*x*hyper((1/3, 1/2), (3/2,
), 27*x**2*exp_polar(2*I*pi)/4) - I*(4 - 27*x**2)**(2/3) - 27*I*Piecewise((x**2*(27*x**2 - 4)**(2/3)*exp(-I*pi
/3)/90 + (27*x**2 - 4)**(2/3)*exp(-I*pi/3)/405, Abs(x**2) > 4/27), (-x**2*(4 - 27*x**2)**(2/3)/90 - (4 - 27*x*
*2)**(2/3)/405, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 i x)^3}{\sqrt [3]{4-27 x^2}} \, dx=\int \frac {{\left (2+x\,3{}\mathrm {i}\right )}^3}{{\left (4-27\,x^2\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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