\(\int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx\) [2502]

Optimal result

Integrand size = 22, antiderivative size = 744 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac {9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{54000 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {\left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}-\frac {\log (2+3 x)}{600\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}} \]

[Out]

-1/600*(27*x^2-54*x+52)^(2/3)/(2+3*x)^2-1/1500*(27*x^2-54*x+52)^(2/3)/(2+3*x)-1/6000*ln(2+3*x)*10^(1/3)+1/6000
*ln(216-81*x-27*10^(1/3)*(27*x^2-54*x+52)^(1/3))*10^(1/3)+9/250*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2)^(
1/3)+30-30*3^(1/2))+1/9000*arctan(-1/3*3^(1/2)-1/15*2^(2/3)*(8-3*x)*5^(2/3)/(27*x^2-54*x+52)^(1/3)*3^(1/2))*10
^(1/3)*3^(1/2)+1/405000*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^
2)^(1/3)+30+30*3^(1/2))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2)),2*I-I*3^(1/2))*((900+30*10^(1/3)*(
2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2
))^2)^(1/2)*3^(1/4)/(1-x)/((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30
*3^(1/2))^2)^(1/2)-1/810000*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54
*x)^2)^(1/3)+30+30*3^(1/2))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2)),2*I-I*3^(1/2))*((900+30*10^(1/
3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^
(1/2))^2)^(1/2)*(1/2+1/2*3^(1/2))*3^(3/4)/(1-x)/((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-5
4+54*x)^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 744, 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, 848, 857, 633, 241, 310, 225, 1893, 764} \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right ),-7+4 \sqrt {3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{54000 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}+\frac {9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac {\log (3 x+2)}{600\ 10^{2/3}} \]

[In]

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

[Out]

-1/600*(52 - 54*x + 27*x^2)^(2/3)/(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 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 764

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)]

Rule 848

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 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 (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 {\text {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} \text {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} \text {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 (\left (1+\sqrt {3}\right ) \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\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 {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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 15.99 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.31 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {-\frac {90 (3+2 x) \left (52-54 x+27 x^2\right )}{(2+3 x)^2}-150 \sqrt [3]{3} \sqrt [3]{\frac {-9-5 i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {-9+5 i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {15-5 i \sqrt {3}}{6+9 x},\frac {15+5 i \sqrt {3}}{6+9 x}\right )+3^{5/6} 10^{2/3} \sqrt [3]{9 i+5 \sqrt {3}-9 i x} \left (-9-5 i \sqrt {3}+9 x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {-9 i+5 \sqrt {3}+9 i x}{10 \sqrt {3}}\right )}{90000 \sqrt [3]{52-54 x+27 x^2}} \]

[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))

Maple [F]

\[\int \frac {1}{\left (2+3 x \right )^{3} \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{{\left (3\,x+2\right )}^3\,{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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