\(\int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx\) [713]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{6\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{6\ 2^{2/3}} \] Output:

-1/18*arctan(1/3*3^(1/2)+1/3*2^(2/3)*(4+3*x)*3^(1/2)/(27*x^2+54*x+28)^(1/3 
))*2^(1/3)*3^(1/2)-1/12*ln(2+3*x)*2^(1/3)+1/12*ln(-108-81*x+27*2^(1/3)*(27 
*x^2+54*x+28)^(1/3))*2^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {4\ 2^{2/3}+3\ 2^{2/3} x+\sqrt [3]{28+54 x+27 x^2}}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )-2 \log \left (4\ 2^{2/3}+3\ 2^{2/3} x-2 \sqrt [3]{28+54 x+27 x^2}\right )+\log \left (16 \sqrt [3]{2}+24 \sqrt [3]{2} x+9 \sqrt [3]{2} x^2+2^{2/3} (4+3 x) \sqrt [3]{28+54 x+27 x^2}+2 \left (28+54 x+27 x^2\right )^{2/3}\right )}{18\ 2^{2/3}} \] Input:

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

Output:

-1/18*(2*Sqrt[3]*ArcTan[(4*2^(2/3) + 3*2^(2/3)*x + (28 + 54*x + 27*x^2)^(1 
/3))/(Sqrt[3]*(28 + 54*x + 27*x^2)^(1/3))] - 2*Log[4*2^(2/3) + 3*2^(2/3)*x 
 - 2*(28 + 54*x + 27*x^2)^(1/3)] + Log[16*2^(1/3) + 24*2^(1/3)*x + 9*2^(1/ 
3)*x^2 + 2^(2/3)*(4 + 3*x)*(28 + 54*x + 27*x^2)^(1/3) + 2*(28 + 54*x + 27* 
x^2)^(2/3)])/2^(2/3)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 103, 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 = {1176}

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.

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

\(\Big \downarrow \) 1176

\(\displaystyle -\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac {\log (3 x+2)}{6\ 2^{2/3}}\)

Input:

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

Output:

-1/3*ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 54*x + 27*x^2)^ 
(1/3))]/(2^(2/3)*Sqrt[3]) - Log[2 + 3*x]/(6*2^(2/3)) + Log[-108 - 81*x + 2 
7*2^(1/3)*(28 + 54*x + 27*x^2)^(1/3)]/(6*2^(2/3))
 

Defintions of rubi rules used

rule 1176

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*(Arc 
Tan[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] && NegQ[c*e^ 
2*(2*c*d - b*e)]
 

Maple [F]

\[\int \frac {1}{\left (3 x +2\right ) \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (80) = 160\).

Time = 1.86 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.02 \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (2 \cdot 4^{\frac {2}{3}} {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {2}{3}} {\left (3 \, x + 4\right )} + 4^{\frac {1}{3}} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} - 4 \, {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (9 \, x^{2} + 24 \, x + 16\right )}\right )}}{6 \, {\left (9 \, x^{3} + 54 \, x^{2} + 84 \, x + 40\right )}}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (9 \, x^{2} + 24 \, x + 16\right )} + 2 \, {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 4\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (3 \, x + 4\right )} - 2 \, {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}}{3 \, x + 2}\right ) \] Input:

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

Output:

-1/6*4^(1/6)*sqrt(1/3)*arctan(1/6*4^(1/6)*sqrt(1/3)*(2*4^(2/3)*(27*x^2 + 5 
4*x + 28)^(2/3)*(3*x + 4) + 4^(1/3)*(27*x^3 + 54*x^2 + 36*x + 8) - 4*(27*x 
^2 + 54*x + 28)^(1/3)*(9*x^2 + 24*x + 16))/(9*x^3 + 54*x^2 + 84*x + 40)) - 
 1/72*4^(2/3)*log((4^(2/3)*(27*x^2 + 54*x + 28)^(2/3) + 4^(1/3)*(9*x^2 + 2 
4*x + 16) + 2*(27*x^2 + 54*x + 28)^(1/3)*(3*x + 4))/(9*x^2 + 12*x + 4)) + 
1/36*4^(2/3)*log((4^(1/3)*(3*x + 4) - 2*(27*x^2 + 54*x + 28)^(1/3))/(3*x + 
 2))
 

Sympy [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{3 \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}} x +2 \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \] Input:

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

Output:

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