### 3.2508 $$\int \frac{1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx$$

Optimal. Leaf size=696 $\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),4 \sqrt{3}-7\right )}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac{\left (27 x^2+54 x+28\right )^{2/3}}{24 (3 x+2)^2}+\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{24\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{12\ 2^{2/3} \sqrt{3}}-\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{9 (x+1)}{2 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}-\frac{\log (3 x+2)}{24\ 2^{2/3}}$

[Out]

-(28 + 54*x + 27*x^2)^(2/3)/(24*(2 + 3*x)^2) + (28 + 54*x + 27*x^2)^(2/3)/(12*(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 + 5
4*x + 27*x^2)^(1/3))]/(12*2^(2/3)*Sqrt[3]) - (Sqrt[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)*(108 + (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)*(10
8 + (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)*(10
8 + (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]/(24*2^(2
/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x + 27*x^2)^(1/3)]/(24*2^(2/3))

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Rubi [A]  time = 0.595902, antiderivative size = 696, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.409, Rules used = {744, 834, 843, 619, 235, 304, 219, 1879, 752} $\frac{\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac{\left (27 x^2+54 x+28\right )^{2/3}}{24 (3 x+2)^2}+\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{24\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{12\ 2^{2/3} \sqrt{3}}+\frac{\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{36\ 3^{3/4} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac{\sqrt{2+\sqrt{3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt{\frac{\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\sin ^{-1}\left (\frac{6 \left (1+\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac{9 (x+1)}{2 \left (6 \left (1-\sqrt{3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}-\frac{\log (3 x+2)}{24\ 2^{2/3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(28 + 54*x + 27*x^2)^(2/3)/(24*(2 + 3*x)^2) + (28 + 54*x + 27*x^2)^(2/3)/(12*(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 + 5
4*x + 27*x^2)^(1/3))]/(12*2^(2/3)*Sqrt[3]) - (Sqrt[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)*(108 + (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)*(10
8 + (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)*(10
8 + (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]/(24*2^(2
/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x + 27*x^2)^(1/3)]/(24*2^(2/3))

Rule 744

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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(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)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 843

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 619

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 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 752

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)]

Rubi steps

\begin{align*} \int \frac{1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx &=-\frac{\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}-\frac{1}{72} \int \frac{108+54 x}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\frac{\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac{\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac{\int \frac{-648-1944 x}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx}{2592}\\ &=-\frac{\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\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}{4} \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}}{24 (2+3 x)^2}+\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 )}{12\ 2^{2/3} \sqrt{3}}-\frac{\log (2+3 x)}{24\ 2^{2/3}}+\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}-\frac{1}{216} \operatorname{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}}{24 (2+3 x)^2}+\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 )}{12\ 2^{2/3} \sqrt{3}}-\frac{\log (2+3 x)}{24\ 2^{2/3}}+\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}-\frac{\sqrt{(54+54 x)^2} \operatorname{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}}{24 (2+3 x)^2}+\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 )}{12\ 2^{2/3} \sqrt{3}}-\frac{\log (2+3 x)}{24\ 2^{2/3}}+\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}+\frac{\sqrt{(54+54 x)^2} \operatorname{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 (\sqrt{\frac{1}{6} \left (2+\sqrt{3}\right )} \sqrt{(54+54 x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{4 (54+54 x)}\\ &=-\frac{\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\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 )}{12\ 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)}{24\ 2^{2/3}}+\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.234778, size = 233, normalized size = 0.33 $\frac{-54 \sqrt [3]{3} \sqrt [3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt [3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )+9 i 2^{2/3} 3^{5/6} \sqrt [3]{-9 i x+\sqrt{3}-9 i} \left (9 i x+\sqrt{3}+9 i\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+\sqrt{3}+9 i}{2 \sqrt{3}}\right )+\frac{162 (2 x+1) \left (27 x^2+54 x+28\right )}{(3 x+2)^2}}{1296 \sqrt [3]{27 x^2+54 x+28}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((162*(1 + 2*x)*(28 + 54*x + 27*x^2))/(2 + 3*x)^2 - 54*3^(1/3)*((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)] + (9*I)*2^(2/3)*3^(5/6)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(9*I + Sqrt[3] + (9*I)*x)*Hypergeometric2F1
[1/3, 2/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(1296*(28 + 54*x + 27*x^2)^(1/3))

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Maple [F]  time = 1.77, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( 2+3\,x \right ) ^{3}}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(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} + 54 \, x + 28\right )}^{\frac{2}{3}}}{729 \, x^{5} + 2916 \, x^{4} + 4644 \, x^{3} + 3672 \, x^{2} + 1440 \, x + 224}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((27*x^2 + 54*x + 28)^(2/3)/(729*x^5 + 2916*x^4 + 4644*x^3 + 3672*x^2 + 1440*x + 224), 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} + 54 x + 28}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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