Integrand size = 22, antiderivative size = 671 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {9 (1+x)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{6\ 2^{2/3} \sqrt {3}}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{72 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}-\frac {\left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{36\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}+\frac {\log (2+3 x)}{12\ 2^{2/3}}-\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}} \]
-1/12*(27*x^2+54*x+28)^(2/3)/(2+3*x)+1/24*ln(2+3*x)*2^(1/3)-1/24*ln(-108-8 1*x+27*2^(1/3)*(27*x^2+54*x+28)^(1/3))*2^(1/3)-9/2*(1+x)/(-2^(1/3)*(108+(5 4+54*x)^2)^(1/3)+6-6*3^(1/2))+1/36*arctan(1/3*3^(1/2)+1/3*2^(2/3)*(4+3*x)/ (27*x^2+54*x+28)^(1/3)*3^(1/2))*2^(1/3)*3^(1/2)-1/108*(6-2^(1/3)*(108+(54+ 54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6+6*3^(1/2))/( -2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2)),2*I-I*3^(1/2))*((1+(27*x^2+5 4*x+28)^(1/3)+(27*x^2+54*x+28)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6- 6*3^(1/2))^2)^(1/2)*3^(1/4)/(1+x)/((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(- 2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))^2)^(1/2)+1/432*(6-2^(1/3)*(10 8+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6+6*3^(1 /2))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2)),2*I-I*3^(1/2))*((1+(27 *x^2+54*x+28)^(1/3)+(27*x^2+54*x+28)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1 /3)+6-6*3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(3/4)/(1+x)*2^(1/2)/ ((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6- 6*3^(1/2))^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.20 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {-4 \left (28+54 x+27 x^2\right )+4 \sqrt [3]{3} (2+3 x) \sqrt [3]{\frac {9-i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {9+i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {3+i \sqrt {3}}{6+9 x},\frac {-3+i \sqrt {3}}{6+9 x}\right )+2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i+\sqrt {3}-9 i x} (2+3 x) \left (-i+3 \sqrt {3}+3 \sqrt {3} x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {9 i+\sqrt {3}+9 i x}{2 \sqrt {3}}\right )}{48 (2+3 x) \sqrt [3]{28+54 x+27 x^2}} \]
(-4*(28 + 54*x + 27*x^2) + 4*3^(1/3)*(2 + 3*x)*((9 - I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((9 + I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/ 3, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + 2^(2/3 )*3^(1/3)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)*(-I + 3*Sqrt[3] + 3*S qrt[3]*x)*Hypergeometric2F1[1/3, 2/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sq rt[3])])/(48*(2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3))
Time = 0.58 (sec) , antiderivative size = 481, normalized size of antiderivative = 0.72, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1167, 27, 1269, 1090, 233, 833, 760, 1176, 2418}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(3 x+2)^2 \sqrt [3]{27 x^2+54 x+28}} \, dx\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle -\frac {1}{36} \int -\frac {27 x}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} \int \frac {x}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{27 x^2+54 x+28}}dx-\frac {2}{3} \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{162} \int \frac {1}{\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}d(54 x+54)-\frac {2}{3} \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {3}{4} \left (\frac {\sqrt {(54 x+54)^2} \int \frac {6 \sqrt {3} \sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{6 \sqrt {3} (54 x+54)}-\frac {2}{3} \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {3}{4} \left (\frac {\sqrt {(54 x+54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {6 \sqrt {3}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}\right )}{6 \sqrt {3} (54 x+54)}-\frac {2}{3} \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {3}{4} \left (\frac {\sqrt {(54 x+54)^2} \left (-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}\right )}{6 \sqrt {3} (54 x+54)}-\frac {2}{3} \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+54 x+28}}dx\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 1176 |
| \(\displaystyle \frac {3}{4} \left (\frac {\sqrt {(54 x+54)^2} \left (-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}\right )}{6 \sqrt {3} (54 x+54)}-\frac {2}{3} \left (-\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac {\log (3 x+2)}{6\ 2^{2/3}}\right )\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3}{4} \left (\frac {\sqrt {(54 x+54)^2} \left (-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}-\frac {\sqrt {(54 x+54)^2}}{3 \sqrt {3} \left (-54 x-\sqrt {3}-53\right )}\right )}{6 \sqrt {3} (54 x+54)}-\frac {2}{3} \left (-\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac {\log (3 x+2)}{6\ 2^{2/3}}\right )\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\) |
-1/12*(28 + 54*x + 27*x^2)^(2/3)/(2 + 3*x) + (3*((Sqrt[(54 + 54*x)^2]*(-1/ 3*Sqrt[(54 + 54*x)^2]/(Sqrt[3]*(-53 - Sqrt[3] - 54*x)) + (6*3^(3/4)*Sqrt[2 + Sqrt[3]]*(-53 - 54*x)*Sqrt[(55 + 54*x + (1 + (54 + 54*x)^2/108)^(2/3))/ (-53 - Sqrt[3] - 54*x)^2]*EllipticE[ArcSin[(-53 + Sqrt[3] - 54*x)/(-53 - S qrt[3] - 54*x)], -7 + 4*Sqrt[3]])/(Sqrt[-((-53 - 54*x)/(-53 - Sqrt[3] - 54 *x)^2)]*Sqrt[(54 + 54*x)^2]) - (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3]) *(-53 - 54*x)*Sqrt[(55 + 54*x + (1 + (54 + 54*x)^2/108)^(2/3))/(-53 - Sqrt [3] - 54*x)^2]*EllipticF[ArcSin[(-53 + Sqrt[3] - 54*x)/(-53 - Sqrt[3] - 54 *x)], -7 + 4*Sqrt[3]])/(Sqrt[-((-53 - 54*x)/(-53 - Sqrt[3] - 54*x)^2)]*Sqr t[(54 + 54*x)^2])))/(6*Sqrt[3]*(54 + 54*x)) - (2*(-1/3*ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(2^(2/3)*Sqrt[3] ) - Log[2 + 3*x]/(6*2^(2/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x + 2 7*x^2)^(1/3)]/(6*2^(2/3))))/3))/4
3.26.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[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]
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]*Sqrt[(- 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]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[ (d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Sy mbol] :> With[{q = Rt[-3*c*e^2*(2*c*d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(Arc Tan[1/Sqrt[3] - 2*((c*d - b*e - c*e*x)/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3))) ]/q^2), x] + (-Simp[3*c*e*(Log[d + e*x]/(2*q^2)), x] + Simp[3*c*e*(Log[c*d - b*e - c*e*x + q*(a + b*x + c*x^2)^(1/3)]/(2*q^2)), x])] /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2, 0] && NegQ[c*e^ 2*(2*c*d - b*e)]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] + S imp[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 - S qrt[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*Sqrt[3])*a*d^3, 0]
\[\int \frac {1}{\left (2+3 x \right )^{2} \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]
\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right )^{2} \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]
\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{{\left (3\,x+2\right )}^2\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \]