\(\int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\) [706]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{3 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{6\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{6\ 10^{2/3}} \] Output:

1/90*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)-1/60*ln(2+3*x)*10^(1/3)+1/60*ln(216-81*x-27*1 
0^(1/3)*(27*x^2-54*x+52)^(1/3))*10^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {8\ 10^{2/3}-3\ 10^{2/3} x+5 \sqrt [3]{52-54 x+27 x^2}}{5 \sqrt {3} \sqrt [3]{52-54 x+27 x^2}}\right )-2 \log \left (-8 10^{2/3}+3\ 10^{2/3} x+10 \sqrt [3]{52-54 x+27 x^2}\right )+\log \left (-64 \sqrt [3]{10}+48 \sqrt [3]{10} x-9 \sqrt [3]{10} x^2+10^{2/3} (-8+3 x) \sqrt [3]{52-54 x+27 x^2}-10 \left (52-54 x+27 x^2\right )^{2/3}\right )}{18\ 10^{2/3}} \] Input:

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

Output:

-1/18*(2*Sqrt[3]*ArcTan[(8*10^(2/3) - 3*10^(2/3)*x + 5*(52 - 54*x + 27*x^2 
)^(1/3))/(5*Sqrt[3]*(52 - 54*x + 27*x^2)^(1/3))] - 2*Log[-8*10^(2/3) + 3*1 
0^(2/3)*x + 10*(52 - 54*x + 27*x^2)^(1/3)] + Log[-64*10^(1/3) + 48*10^(1/3 
)*x - 9*10^(1/3)*x^2 + 10^(2/3)*(-8 + 3*x)*(52 - 54*x + 27*x^2)^(1/3) - 10 
*(52 - 54*x + 27*x^2)^(2/3)])/10^(2/3)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1175}

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.

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

\(\Big \downarrow \) 1175

\(\displaystyle -\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}}\)

Input:

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

Output:

-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))
 

Defintions of rubi rules used

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)]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 17.14 (sec) , antiderivative size = 2201, normalized size of antiderivative = 20.38

method	result	size
trager	\(\text {Expression too large to display}\)	\(2201\)

Input:

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

Output:

1/3*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*ln(-(12013105 
49293725*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_ 
Z^3-10)^3*x^3+321317013531015450*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^ 
3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3+2217804091003800*RootOf(RootOf(_Z^ 
3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^2+593200640364 
951600*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_ 
Z^3-10)^2*x^2+35356252380606720*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3- 
10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^2*x+1971381414225600 
*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^ 
3*x+527289458102179200*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900* 
_Z^2)^2*RootOf(_Z^3-10)^2*x-94283339681617920*(27*x^2-54*x+52)^(2/3)*RootO 
f(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^2-3800 
74665718431*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2-10606875714182016 
0*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+90 
0*_Z^2)*RootOf(_Z^3-10)*x^2+2027064883831632*(27*x^2-54*x+52)^(1/3)*RootOf 
(_Z^3-10)^2*x+565700038089707520*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3 
-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)*x[phone]*R 
ootOf(_Z^3-10)*x^3+212563562797440990*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootO 
f(_Z^3-10)+900*_Z^2)*x^3-2702753178442176*(27*x^2-54*x+52)^(1/3)*RootOf(_Z 
^3-10)^2-754266717452943360*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (83) = 166\).

Time = 1.78 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {1}{30} \cdot 100^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (\frac {100^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (2 \cdot 100^{\frac {2}{3}} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} {\left (3 \, x - 8\right )} + 100^{\frac {1}{3}} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 20 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )}\right )}}{30 \, {\left (9 \, x^{3} - 162 \, x^{2} + 372 \, x - 344\right )}}\right ) - \frac {1}{1800} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {2}{3}} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} + 100^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )} - 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x - 8\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac {1}{900} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {1}{3}} {\left (3 \, x - 8\right )} + 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}{3 \, x + 2}\right ) \] Input:

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

Output:

-1/30*100^(1/6)*sqrt(1/3)*arctan(1/30*100^(1/6)*sqrt(1/3)*(2*100^(2/3)*(27 
*x^2 - 54*x + 52)^(2/3)*(3*x - 8) + 100^(1/3)*(27*x^3 + 54*x^2 + 36*x + 8) 
 + 20*(27*x^2 - 54*x + 52)^(1/3)*(9*x^2 - 48*x + 64))/(9*x^3 - 162*x^2 + 3 
72*x - 344)) - 1/1800*100^(2/3)*log((100^(2/3)*(27*x^2 - 54*x + 52)^(2/3) 
+ 100^(1/3)*(9*x^2 - 48*x + 64) - 10*(27*x^2 - 54*x + 52)^(1/3)*(3*x - 8)) 
/(9*x^2 + 12*x + 4)) + 1/900*100^(2/3)*log((100^(1/3)*(3*x - 8) + 10*(27*x 
^2 - 54*x + 52)^(1/3))/(3*x + 2))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(2+3*x)/(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)), x)
 

Giac [F]

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

integrate(1/(2+3*x)/(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)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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