\(\int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx\) [705]

Optimal result

Integrand size = 19, antiderivative size = 656 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac {3 x}{32 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{4+27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{96\ 2^{2/3} 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {\left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right ),-7+4 \sqrt {3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {\log (2+3 x)}{192 \sqrt [3]{2}}+\frac {\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{192 \sqrt [3]{2}} \]

[Out]

-1/96*(27*x^2+4)^(2/3)/(2+3*x)^2-1/96*(27*x^2+4)^(2/3)/(2+3*x)-1/384*ln(2+3*x)*2^(2/3)+1/384*ln(54-81*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*arctan(-1/3*3^(1/2)-1/3*2^
(1/3)*(2-3*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))*Ellip
ticE((-(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 = 0.30 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {759, 849, 858, 241, 310, 225, 1893, 765} \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx=-\frac {\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {\arctan \left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{96 \sqrt [3]{2} \sqrt {3}}-\frac {3 x}{32 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac {\log (3 x+2)}{192 \sqrt [3]{2}} \]

[In]

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

[Out]

-1/96*(4 + 27*x^2)^(2/3)/(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 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*} \text {integral}& = -\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} \text {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} \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 {\left (\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 )}{32 \sqrt [3]{2} \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 {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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 18.83 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.34 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx=\frac {-12 \left (4+4 x+27 x^2+27 x^3\right )-4 \sqrt [3]{3} (2+3 x)^2 \sqrt [3]{\frac {-2 i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {2 i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {6-2 i \sqrt {3}}{6+9 x},\frac {6+2 i \sqrt {3}}{6+9 x}\right )+\sqrt [3]{6} \sqrt [3]{2 \sqrt {3}-9 i x} (2+3 x)^2 \left (-2 i+3 \sqrt {3} x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {1}{2}+\frac {3}{4} i \sqrt {3} x\right )}{384 (2+3 x)^2 \sqrt [3]{4+27 x^2}} \]

[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))

Maple [F]

\[\int \frac {1}{\left (2+3 x \right )^{3} \left (27 x^{2}+4\right )^{\frac {1}{3}}}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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