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

Optimal. Leaf size=656 \[ -\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right ),4 \sqrt{3}-7\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{96 \sqrt [3]{2} \sqrt{3}}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{192 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.411907, antiderivative size = 656, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {745, 835, 844, 235, 304, 219, 1879, 751} \[ -\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{96 \sqrt [3]{2} \sqrt{3}}-\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{192 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4 + 27*x^2)^(2/3)/(96*(2 + 3*x)^2) - (4 + 27*x^2)^(2/3)/(96*(2 + 3*x)) - (3*x)/(32*(2^(2/3)*(1 - Sqrt[3]) -
(4 + 27*x^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3))]/(96*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)]) - Log[2 + 3*x]/(192*2^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^2)^(1/3)]/(192*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 x)^3 \sqrt [3]{4+27 x^2}} \, dx &=-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac{3}{32} \int \frac{-4+2 x}{(2+3 x)^2 \sqrt [3]{4+27 x^2}} \, dx\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}+\frac{\int \frac{192+144 x}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx}{1536}\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}+\frac{1}{32} \int \frac{1}{\sqrt [3]{4+27 x^2}} \, dx+\frac{1}{16} \int \frac{1}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx\\ &=-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}-\frac{\log (2+3 x)}{192 \sqrt [3]{2}}+\frac{\log \left (54-81 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{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}-\frac{\log (2+3 x)}{192 \sqrt [3]{2}}+\frac{\log \left (54-81 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{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac{\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4+27 x^2}\right )}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 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{\log (2+3 x)}{192 \sqrt [3]{2}}+\frac{\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{192 \sqrt [3]{2}}\\ \end{align*}

Mathematica [C]  time = 0.271006, size = 223, normalized size = 0.34 \[ \frac{-4 \sqrt [3]{3} \sqrt [3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt [3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} (3 x+2)^2 F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+\sqrt [3]{6} \sqrt [3]{2 \sqrt{3}-9 i x} \left (3 \sqrt{3} x-2 i\right ) (3 x+2)^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right )-12 \left (27 x^3+27 x^2+4 x+4\right )}{384 (3 x+2)^2 \sqrt [3]{27 x^2+4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-12*(4 + 4*x + 27*x^2 + 27*x^3) - 4*3^(1/3)*(2 + 3*x)^2*(((-2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*(((2*I)*Sqrt
[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6
+ 9*x)] + 6^(1/3)*(2*Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)^2*(-2*I + 3*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/
3, 1/2 + ((3*I)/4)*Sqrt[3]*x])/(384*(2 + 3*x)^2*(4 + 27*x^2)^(1/3))

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Maple [F]  time = 0.375, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( 2+3\,x \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*x)^3/(27*x^2+4)^(1/3),x)

[Out]

int(1/(2+3*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 \, x + 2\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{729 \, x^{5} + 1458 \, x^{4} + 1080 \, x^{3} + 432 \, x^{2} + 144 \, x + 32}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((27*x^2 + 4)^(2/3)/(729*x^5 + 1458*x^4 + 1080*x^3 + 432*x^2 + 144*x + 32), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((3*x + 2)**3*(27*x**2 + 4)**(1/3)), 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 \, x + 2\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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