\(\int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx\) [714]

Optimal result

Integrand size = 22, antiderivative size = 152 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{6\ 2^{2/3} \sqrt {3}}+\frac {1}{4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right )+\frac {\log (2+3 x)}{12\ 2^{2/3}}-\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}} \] Output:

-1/12*(27*x^2+54*x+28)^(2/3)/(2+3*x)+1/36*arctan(1/3*3^(1/2)+1/3*2^(2/3)*( 
4+3*x)*3^(1/2)/(27*x^2+54*x+28)^(1/3))*2^(1/3)*3^(1/2)+1/4*(1+x)*hypergeom 
([1/3, 1/2],[3/2],-27*(1+x)^2)+1/24*ln(2+3*x)*2^(1/3)-1/24*ln(-108-81*x+27 
*2^(1/3)*(27*x^2+54*x+28)^(1/3))*2^(1/3)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 11.67 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {-4 \left (28+54 x+27 x^2\right )+4 \sqrt [3]{3} (2+3 x) \sqrt [3]{\frac {9-i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {9+i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {3+i \sqrt {3}}{6+9 x},\frac {-3+i \sqrt {3}}{6+9 x}\right )+2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i+\sqrt {3}-9 i x} (2+3 x) \left (-i+3 \sqrt {3}+3 \sqrt {3} x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {9 i+\sqrt {3}+9 i x}{2 \sqrt {3}}\right )}{48 (2+3 x) \sqrt [3]{28+54 x+27 x^2}} \] Input:

Integrate[1/((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(1/3)),x]
 

Output:

(-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))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(481\) vs. \(2(152)=304\).

Time = 0.59 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.16, 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

\(\displaystyle \frac {3}{4} \left (\frac {\sqrt {(54 x+54)^2} \left (-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}-\frac {\sqrt {(54 x+54)^2}}{3 \sqrt {3} \left (-54 x-\sqrt {3}-53\right )}\right )}{6 \sqrt {3} (54 x+54)}-\frac {2}{3} \left (-\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac {\log (3 x+2)}{6\ 2^{2/3}}\right )\right )-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}\)

Input:

Int[1/((2 + 3*x)^2*(28 + 54*x + 27*x^2)^(1/3)),x]
 

Output:

-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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 233

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]
 

rule 760

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]
 

rule 833

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]
 

rule 1090

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]
 

rule 1167

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])
 

rule 1176

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)]
 

rule 1269

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]
 

rule 2418

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]
 

Maple [F]

\[\int \frac {1}{\left (3 x +2\right )^{2} \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]

Input:

int(1/(3*x+2)^2/(27*x^2+54*x+28)^(1/3),x)
 

Output:

int(1/(3*x+2)^2/(27*x^2+54*x+28)^(1/3),x)
 

Fricas [F]

\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \] Input:

integrate(1/(2+3*x)^2/(27*x^2+54*x+28)^(1/3),x, algorithm="fricas")
 

Output:

integral((27*x^2 + 54*x + 28)^(2/3)/(243*x^4 + 810*x^3 + 1008*x^2 + 552*x 
+ 112), x)
 

Sympy [F]

\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right )^{2} \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \] Input:

integrate(1/(2+3*x)**2/(27*x**2+54*x+28)**(1/3),x)
 

Output:

Integral(1/((3*x + 2)**2*(27*x**2 + 54*x + 28)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \] Input:

integrate(1/(2+3*x)^2/(27*x^2+54*x+28)^(1/3),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)^2), x)
 

Giac [F]

\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{2}} \,d x } \] Input:

integrate(1/(2+3*x)^2/(27*x^2+54*x+28)^(1/3),x, algorithm="giac")
 

Output:

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{{\left (3\,x+2\right )}^2\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \] Input:

int(1/((3*x + 2)^2*(54*x + 27*x^2 + 28)^(1/3)),x)
 

Output:

int(1/((3*x + 2)^2*(54*x + 27*x^2 + 28)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{9 \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}} x^{2}+12 \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}} x +4 \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \] Input:

int(1/(2+3*x)^2/(27*x^2+54*x+28)^(1/3),x)
 

Output:

int(1/(9*(27*x**2 + 54*x + 28)**(1/3)*x**2 + 12*(27*x**2 + 54*x + 28)**(1/ 
3)*x + 4*(27*x**2 + 54*x + 28)**(1/3)),x)