3.2502 \(\int \frac{1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=744 \[ \frac{\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right ),4 \sqrt{3}-7\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac{\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{300 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{\log (3 x+2)}{600\ 10^{2/3}}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54000 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

[Out]

-(52 - 54*x + 27*x^2)^(2/3)/(600*(2 + 3*x)^2) - (52 - 54*x + 27*x^2)^(2/3)/(1500*(2 + 3*x)) + (9*(1 - x))/(50*
5^(2/3)*(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/
(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(300*Sqrt[3]*10^(2/3)) - (Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x
)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4
*Sqrt[3]])/(54000*Sqrt[2]*3^(1/4)*5^(1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1
- Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + ((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqr
t[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*
x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(27000*3^(3/4)*5^(
1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 +
54*x)^2)^(1/3))^2)]) - Log[2 + 3*x]/(600*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/
(600*10^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 0.674776, antiderivative size = 744, 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, 750} \[ -\frac{\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac{\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{300 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{\log (3 x+2)}{600\ 10^{2/3}}+\frac{\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{54000 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

Antiderivative was successfully verified.

[In]

Int[1/((2 + 3*x)^3*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

-(52 - 54*x + 27*x^2)^(2/3)/(600*(2 + 3*x)^2) - (52 - 54*x + 27*x^2)^(2/3)/(1500*(2 + 3*x)) + (9*(1 - x))/(50*
5^(2/3)*(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/
(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(300*Sqrt[3]*10^(2/3)) - (Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x
)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4
*Sqrt[3]])/(54000*Sqrt[2]*3^(1/4)*5^(1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1
- Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + ((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqr
t[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*
x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(27000*3^(3/4)*5^(
1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 +
54*x)^2)^(1/3))^2)]) - Log[2 + 3*x]/(600*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/
(600*10^(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 750

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] && PosQ[c*e^2*(2*c*d - b*e)]

Rubi steps

\begin{align*} \int \frac{1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\int \frac{-324+54 x}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx}{1800}\\ &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac{\int \frac{22680+9720 x}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx}{1620000}\\ &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac{1}{500} \int \frac{1}{\sqrt [3]{52-54 x+27 x^2}} \, dx+\frac{1}{100} \int \frac{1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt{3} 10^{2/3}}-\frac{\log (2+3 x)}{600\ 10^{2/3}}+\frac{\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1+\frac{x^2}{2700}}} \, dx,x,-54+54 x\right )}{27000\ 5^{2/3}}\\ &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt{3} 10^{2/3}}-\frac{\log (2+3 x)}{600\ 10^{2/3}}+\frac{\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}+\frac{\sqrt{(-54+54 x)^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{200 \sqrt{3} 5^{2/3} (-54+54 x)}\\ &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt{3} 10^{2/3}}-\frac{\log (2+3 x)}{600\ 10^{2/3}}+\frac{\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}-\frac{\sqrt{(-54+54 x)^2} \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{200 \sqrt{3} 5^{2/3} (-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,\frac{\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{100\ 5^{2/3} (-54+54 x)}\\ &=-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac{\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac{9 (1-x)}{50\ 5^{2/3} \left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt{3} 10^{2/3}}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt{\frac{900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{30+30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt{3}\right )}{54000 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac{\left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt{\frac{900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{30+30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt{3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} (1-x) \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt{3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}-\frac{\log (2+3 x)}{600\ 10^{2/3}}+\frac{\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.297046, size = 233, normalized size = 0.31 \[ \frac{-150 \sqrt [3]{3} \sqrt [3]{\frac{9 x-5 i \sqrt{3}-9}{3 x+2}} \sqrt [3]{\frac{9 x+5 i \sqrt{3}-9}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+3^{5/6} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt{3}+9 i} \left (9 x-5 i \sqrt{3}-9\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right )-\frac{90 (2 x+3) \left (27 x^2-54 x+52\right )}{(3 x+2)^2}}{90000 \sqrt [3]{27 x^2-54 x+52}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + 3*x)^3*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

((-90*(3 + 2*x)*(52 - 54*x + 27*x^2))/(2 + 3*x)^2 - 150*3^(1/3)*((-9 - (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*(
(-9 + (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 +
(5*I)*Sqrt[3])/(6 + 9*x)] + 3^(5/6)*10^(2/3)*(9*I + 5*Sqrt[3] - (9*I)*x)^(1/3)*(-9 - (5*I)*Sqrt[3] + 9*x)*Hyp
ergeometric2F1[1/3, 2/3, 5/3, (-9*I + 5*Sqrt[3] + (9*I)*x)/(10*Sqrt[3])])/(90000*(52 - 54*x + 27*x^2)^(1/3))

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Maple [F]  time = 1.249, 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+52}}}}\, 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+52)^(1/3),x)

[Out]

int(1/(2+3*x)^3/(27*x^2-54*x+52)^(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 + 52\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+52)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 - 54*x + 52)^(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 + 52\right )}^{\frac{2}{3}}}{729 \, x^{5} - 540 \, x^{3} + 1080 \, x^{2} + 1440 \, x + 416}, 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+52)^(1/3),x, algorithm="fricas")

[Out]

integral((27*x^2 - 54*x + 52)^(2/3)/(729*x^5 - 540*x^3 + 1080*x^2 + 1440*x + 416), 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 + 52}}\, 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+52)**(1/3),x)

[Out]

Integral(1/((3*x + 2)**3*(27*x**2 - 54*x + 52)**(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 + 52\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+52)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^3), x)