\(\int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx\) [2507]

Optimal result

Integrand size = 22, antiderivative size = 671 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {9 (1+x)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{6\ 2^{2/3} \sqrt {3}}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{72 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}-\frac {\left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{36\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}+\frac {\log (2+3 x)}{12\ 2^{2/3}}-\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{12\ 2^{2/3}} \]

[Out]

-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(-108-81*x+27*2^(1/3)*(27*x^2+54*x+28)^(1/3
))*2^(1/3)-9/2*(1+x)/(-2^(1/3)*(108+(54+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+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)*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)*(108+(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)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {758, 12, 857, 633, 241, 310, 225, 1893, 766} \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),-7+4 \sqrt {3}\right )}{36\ 3^{3/4} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{72 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{12\ 2^{2/3}}-\frac {9 (x+1)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}+\frac {\log (3 x+2)}{12\ 2^{2/3}} \]

[In]

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

[Out]

-1/12*(28 + 54*x + 27*x^2)^(2/3)/(2 + 3*x) - (9*(1 + x))/(2*(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (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))]/(6*2^(2/3)*Sqrt[3]) + (S
qrt[2 + Sqrt[3]]*(6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) + (28 + 54*x +
27*x^2)^(2/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcSin[(6*(1 + Sqrt[3]) -
2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]
])/(72*Sqrt[2]*3^(1/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(10
8 + (54 + 54*x)^2)^(1/3))^2)]) - ((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/
3) + (28 + 54*x + 27*x^2)^(2/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(
6*(1 + Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))
], -7 + 4*Sqrt[3]])/(36*3^(3/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^
(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2)]) + Log[2 + 3*x]/(12*2^(2/3)) - Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x
+ 27*x^2)^(1/3)]/(12*2^(2/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[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 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] + Dist[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[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 766

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[-3*c*e^2*(2*c
*d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(ArcTan[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] && NeQ[2*c*d - b*e, 0] && 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 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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 (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {1}{36} \int -\frac {27 x}{(2+3 x) \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 {3}{4} \int \frac {x}{(2+3 x) \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 {1}{4} \int \frac {1}{\sqrt [3]{28+54 x+27 x^2}} \, dx-\frac {1}{2} \int \frac {1}{(2+3 x) \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 {\tan ^{-1}\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 {\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}}+\frac {1}{216} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{108}}} \, dx,x,54+54 x\right ) \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {\tan ^{-1}\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 {\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}}+\frac {\sqrt {(54+54 x)^2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{8 \sqrt {3} (54+54 x)} \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {\tan ^{-1}\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 {\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}}-\frac {\sqrt {(54+54 x)^2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{8 \sqrt {3} (54+54 x)}+\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{8 \sqrt {3} (54+54 x)} \\ & = -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {3 (1+x)}{4 \left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )}+\frac {\tan ^{-1}\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 (1-\sqrt [3]{28+54 x+27 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 (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{24\ 3^{3/4} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}-\frac {\left (1-\sqrt [3]{28+54 x+27 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 (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 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}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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 \[ \int \frac {1}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {-4 \left (28+54 x+27 x^2\right )+4 \sqrt [3]{3} (2+3 x) \sqrt [3]{\frac {9-i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {9+i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},-\frac {3+i \sqrt {3}}{6+9 x},\frac {-3+i \sqrt {3}}{6+9 x}\right )+2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 i+\sqrt {3}-9 i x} (2+3 x) \left (-i+3 \sqrt {3}+3 \sqrt {3} x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {9 i+\sqrt {3}+9 i x}{2 \sqrt {3}}\right )}{48 (2+3 x) \sqrt [3]{28+54 x+27 x^2}} \]

[In]

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

[Out]

(-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*Sqrt[3]*x)*Hypergeometric2F1[1/3, 2
/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(48*(2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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