\(\int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx\) [708]

Optimal result

Integrand size = 22, antiderivative size = 193 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {(1-x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (1-x)^2\right )}{500\ 5^{2/3}}-\frac {\log (2+3 x)}{600\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}} \] Output:

-1/600*(27*x^2-54*x+52)^(2/3)/(2+3*x)^2-(27*x^2-54*x+52)^(2/3)/(3000+4500* 
x)+1/9000*arctan(-1/3*3^(1/2)-1/15*2^(2/3)*(8-3*x)*3^(1/2)*5^(2/3)/(27*x^2 
-54*x+52)^(1/3))*3^(1/2)*10^(1/3)-1/2500*5^(1/3)*(1-x)*hypergeom([1/3, 1/2 
],[3/2],-27/25*(1-x)^2)-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)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 18.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {-\frac {90 (3+2 x) \left (52-54 x+27 x^2\right )}{(2+3 x)^2}-150 \sqrt [3]{3} \sqrt [3]{\frac {-9-5 i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {-9+5 i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {15-5 i \sqrt {3}}{6+9 x},\frac {15+5 i \sqrt {3}}{6+9 x}\right )+3^{5/6} 10^{2/3} \sqrt [3]{9 i+5 \sqrt {3}-9 i x} \left (-9-5 i \sqrt {3}+9 x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {-9 i+5 \sqrt {3}+9 i x}{10 \sqrt {3}}\right )}{90000 \sqrt [3]{52-54 x+27 x^2}} \] Input:

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

Output:

((-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)*Hypergeometric2F1[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))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(519\) vs. \(2(193)=386\).

Time = 0.66 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.69, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1167, 27, 1237, 27, 1269, 1090, 233, 833, 760, 1175, 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.

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

\(\Big \downarrow \) 1167

\(\displaystyle -\frac {\int -\frac {54 (6-x)}{(3 x+2)^2 \sqrt [3]{27 x^2-54 x+52}}dx}{1800}-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{100} \int \frac {6-x}{(3 x+2)^2 \sqrt [3]{27 x^2-54 x+52}}dx-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {3}{100} \left (-\frac {1}{900} \int -\frac {60 (3 x+7)}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \int \frac {3 x+7}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (\int \frac {1}{\sqrt [3]{27 x^2-54 x+52}}dx+5 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (5 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx+\frac {\int \frac {1}{\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}d(54 x-54)}{54\ 5^{2/3}}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (5 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx+\frac {\sqrt [3]{5} \sqrt {(54 x-54)^2} \int \frac {30 \sqrt {3} \sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{2 \sqrt {3} (54 x-54)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (5 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx+\frac {\sqrt [3]{5} \sqrt {(54 x-54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {30 \sqrt {3}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}\right )}{2 \sqrt {3} (54 x-54)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (\frac {\sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}\right )}{2 \sqrt {3} (54 x-54)}+5 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}}dx\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 1175

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (\frac {\sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}\right )}{2 \sqrt {3} (54 x-54)}+5 \left (-\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}}\right )\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3}{100} \left (\frac {1}{15} \left (\frac {\sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}+\frac {30\ 3^{3/4} \sqrt {2+\sqrt {3}} (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}-\frac {\sqrt {(54 x-54)^2}}{15 \sqrt {3} \left (-54 x-\sqrt {3}+55\right )}\right )}{2 \sqrt {3} (54 x-54)}+5 \left (-\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}}\right )\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{45 (3 x+2)}\right )-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}\)

Input:

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

Output:

-1/600*(52 - 54*x + 27*x^2)^(2/3)/(2 + 3*x)^2 + (3*(-1/45*(52 - 54*x + 27* 
x^2)^(2/3)/(2 + 3*x) + ((5^(1/3)*Sqrt[(-54 + 54*x)^2]*(-1/15*Sqrt[(-54 + 5 
4*x)^2]/(Sqrt[3]*(55 - Sqrt[3] - 54*x)) + (30*3^(3/4)*Sqrt[2 + Sqrt[3]]*(5 
5 - 54*x)*Sqrt[(-53 + 54*x + (1 + (-54 + 54*x)^2/2700)^(2/3))/(55 - Sqrt[3 
] - 54*x)^2]*EllipticE[ArcSin[(55 + Sqrt[3] - 54*x)/(55 - Sqrt[3] - 54*x)] 
, -7 + 4*Sqrt[3]])/(Sqrt[-((55 - 54*x)/(55 - Sqrt[3] - 54*x)^2)]*Sqrt[(-54 
 + 54*x)^2]) - (60*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(55 - 54*x)*Sqr 
t[(-53 + 54*x + (1 + (-54 + 54*x)^2/2700)^(2/3))/(55 - Sqrt[3] - 54*x)^2]* 
EllipticF[ArcSin[(55 + Sqrt[3] - 54*x)/(55 - Sqrt[3] - 54*x)], -7 + 4*Sqrt 
[3]])/(Sqrt[-((55 - 54*x)/(55 - Sqrt[3] - 54*x)^2)]*Sqrt[(-54 + 54*x)^2])) 
)/(2*Sqrt[3]*(-54 + 54*x)) + 5*(-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))]/(Sqrt[3]*10^(2/3)) - Log[2 
 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^( 
1/3)]/(6*10^(2/3))))/15))/100
 

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 1175

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*(ArcT 
an[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] && PosQ[c*e^2 
*(2*c*d - b*e)]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] + Simp[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] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{27 \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}} x^{3}+54 \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}} x^{2}+36 \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}} x +8 \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \] Input:

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

Output:

int(1/(27*(27*x**2 - 54*x + 52)**(1/3)*x**3 + 54*(27*x**2 - 54*x + 52)**(1 
/3)*x**2 + 36*(27*x**2 - 54*x + 52)**(1/3)*x + 8*(27*x**2 - 54*x + 52)**(1 
/3)),x)