3.8.11 \(\int \frac {1}{(2+3 i x)^3 \sqrt [3]{4-27 x^2}} \, dx\) [711]

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

[Out]

1/96*I*(-27*x^2+4)^(2/3)/(2+3*I*x)^2+1/96*I*(-27*x^2+4)^(2/3)/(2+3*I*x)+1/384*I*ln(2+3*I*x)*2^(2/3)-1/384*I*ln
(-54+81*I*x+27*2^(2/3)*(-27*x^2+4)^(1/3))*2^(2/3)-3/32*x/(-(-27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))-1/576*I*arct
an(-1/3*3^(1/2)-1/3*2^(1/3)*(2-3*I*x)/(-27*x^2+4)^(1/3)*3^(1/2))*2^(2/3)*3^(1/2)+1/864*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)-1/576*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)

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Rubi [A]
time = 0.31, antiderivative size = 676, normalized size of antiderivative = 1.00, 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 = {759, 849, 858, 241, 310, 225, 1893, 765} \begin {gather*} \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 (\text {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 )}{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 (\text {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 )}{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 \text {ArcTan}\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 {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 \log (2+3 i x)}{192 \sqrt [3]{2}} \end {gather*}

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 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 759

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 765

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]

Rule 849

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 858

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 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*} \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} \text {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} \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 )}{64 \sqrt {3} x}-\frac {\sqrt {-x^2} \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 7.51, size = 134, normalized size = 0.20 \begin {gather*} -\frac {i \sqrt [3]{\frac {2 \sqrt {3}-9 x}{2 i-3 x}} \sqrt [3]{\frac {2 \sqrt {3}+9 x}{-2 i+3 x}} 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 )}{-6 i+9 x}\right )}{8\ 3^{2/3} (2 i-3 x)^2 \sqrt [3]{4-27 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/8*I)*((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,
11/3, (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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (3 i x +2\right )^{3} \left (-27 x^{2}+4\right )^{\frac {1}{3}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} i \int \frac {1}{27 x^{3} \sqrt [3]{4 - 27 x^{2}} - 54 i x^{2} \sqrt [3]{4 - 27 x^{2}} - 36 x \sqrt [3]{4 - 27 x^{2}} + 8 i \sqrt [3]{4 - 27 x^{2}}}\, dx \end {gather*}

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]

I*Integral(1/(27*x**3*(4 - 27*x**2)**(1/3) - 54*I*x**2*(4 - 27*x**2)**(1/3) - 36*x*(4 - 27*x**2)**(1/3) + 8*I*
(4 - 27*x**2)**(1/3)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (2+x\,3{}\mathrm {i}\right )}^3\,{\left (4-27\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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