\(\int \frac {2+3 x}{\sqrt [3]{4+3 x^2} (-12+52 x+9 x^2)} \, dx\) [2432]

Optimal result

Integrand size = 29, antiderivative size = 197 \[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {10 \sqrt [3]{2}}{\sqrt {3} 7^{2/3}}-\frac {\sqrt [3]{2} \sqrt {3} x}{7^{2/3}}+\frac {\sqrt [3]{4+3 x^2}}{\sqrt {3}}}{\sqrt [3]{4+3 x^2}}\right )}{14 \sqrt [3]{14}}+\frac {\log \left (-10 \sqrt [3]{14}+3 \sqrt [3]{14} x+14 \sqrt [3]{4+3 x^2}\right )}{14 \sqrt [3]{14}}-\frac {\log \left (100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+\left (140 \sqrt [3]{14}-42 \sqrt [3]{14} x\right ) \sqrt [3]{4+3 x^2}+196 \left (4+3 x^2\right )^{2/3}\right )}{28 \sqrt [3]{14}} \] Output:

-1/196*3^(1/2)*arctan((10/21*2^(1/3)*3^(1/2)*7^(1/3)-1/7*2^(1/3)*3^(1/2)*x 
*7^(1/3)+1/3*(3*x^2+4)^(1/3)*3^(1/2))/(3*x^2+4)^(1/3))*14^(2/3)+1/196*ln(- 
10*14^(1/3)+3*14^(1/3)*x+14*(3*x^2+4)^(1/3))*14^(2/3)-1/392*ln(100*14^(2/3 
)-60*14^(2/3)*x+9*14^(2/3)*x^2+(140*14^(1/3)-42*14^(1/3)*x)*(3*x^2+4)^(1/3 
)+196*(3*x^2+4)^(2/3))*14^(2/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.80 \[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {10 \sqrt [3]{14}-3 \sqrt [3]{14} x+7 \sqrt [3]{4+3 x^2}}{7 \sqrt {3} \sqrt [3]{4+3 x^2}}\right )-2 \log \left (-10 \sqrt [3]{14}+3 \sqrt [3]{14} x+14 \sqrt [3]{4+3 x^2}\right )+\log \left (100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+196 \left (4+3 x^2\right )^{2/3}+14 (10-3 x) \sqrt [3]{56+42 x^2}\right )}{28 \sqrt [3]{14}} \] Input:

Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
 

Output:

-1/28*(2*Sqrt[3]*ArcTan[(10*14^(1/3) - 3*14^(1/3)*x + 7*(4 + 3*x^2)^(1/3)) 
/(7*Sqrt[3]*(4 + 3*x^2)^(1/3))] - 2*Log[-10*14^(1/3) + 3*14^(1/3)*x + 14*( 
4 + 3*x^2)^(1/3)] + Log[100*14^(2/3) - 60*14^(2/3)*x + 9*14^(2/3)*x^2 + 19 
6*(4 + 3*x^2)^(2/3) + 14*(10 - 3*x)*(56 + 42*x^2)^(1/3)])/14^(1/3)
 

Rubi [F]

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.

\(\displaystyle \int \frac {3 x+2}{\sqrt [3]{3 x^2+4} \left (9 x^2+52 x-12\right )} \, dx\)

\(\Big \downarrow \) 1376

\(\displaystyle \int \frac {3 x+2}{\sqrt [3]{3 x^2+4} \left (9 x^2+52 x-12\right )}dx\)

Input:

Int[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 14.46 (sec) , antiderivative size = 2106, normalized size of antiderivative = 10.69

method	result	size
trager	\(\text {Expression too large to display}\)	\(2106\)

Input:

int((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x,method=_RETURNVERBOSE)
 

Output:

-1/2*ln((8299631488661226*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196) 
+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x^3-52131884648319*RootOf(RootOf(_Z^3-196 
)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x^3+1475490042428 
66240*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)^2*RootOf 
(_Z^3-196)^2*x^2+3210600401526000*(3*x^2+4)^(2/3)*RootOf(RootOf(_Z^3-196)^ 
2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^2*x-926789060414560*R 
ootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196 
)^3*x^2+276654382955374200*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196 
)+9604*_Z^2)^2*RootOf(_Z^3-196)^2*x-10702001338420000*(3*x^2+4)^(2/3)*Root 
Of(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^2 
+30218282325894726*(3*x^2+4)^(1/3)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf( 
_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)*x^2-1737729488277300*RootOf(RootOf(_ 
Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196)^3*x[phone] 
3890987*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x^2-201455215505964840*(3*x^2+4 
)^(1/3)*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf 
(_Z^3-196)*x+37771792285131702*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3 
-196)+9604*_Z^2)*x^3-1400440892606580*(3*x^2+4)^(1/3)*RootOf(_Z^3-196)^2*x 
-237253270950513*RootOf(_Z^3-196)*x^3+335758692509941400*(3*x^2+4)^(1/3)*R 
ootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9604*_Z^2)*RootOf(_Z^3-196 
)-739212983246339308*RootOf(RootOf(_Z^3-196)^2+98*_Z*RootOf(_Z^3-196)+9...
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (residue poly has multiple non-linear fac 
tors)
 

Sympy [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int \frac {3 x + 2}{\left (x + 6\right ) \left (9 x - 2\right ) \sqrt [3]{3 x^{2} + 4}}\, dx \] Input:

integrate((2+3*x)/(3*x**2+4)**(1/3)/(9*x**2+52*x-12),x)
 

Output:

Integral((3*x + 2)/((x + 6)*(9*x - 2)*(3*x**2 + 4)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int { \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="maxima")
 

Output:

integrate((3*x + 2)/((9*x^2 + 52*x - 12)*(3*x^2 + 4)^(1/3)), x)
 

Giac [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int { \frac {3 \, x + 2}{{\left (9 \, x^{2} + 52 \, x - 12\right )} {\left (3 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x, algorithm="giac")
 

Output:

integrate((3*x + 2)/((9*x^2 + 52*x - 12)*(3*x^2 + 4)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=\int \frac {3\,x+2}{{\left (3\,x^2+4\right )}^{1/3}\,\left (9\,x^2+52\,x-12\right )} \,d x \] Input:

int((3*x + 2)/((3*x^2 + 4)^(1/3)*(52*x + 9*x^2 - 12)),x)
 

Output:

int((3*x + 2)/((3*x^2 + 4)^(1/3)*(52*x + 9*x^2 - 12)), x)
 

Reduce [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx=3 \left (\int \frac {x}{9 \left (3 x^{2}+4\right )^{\frac {1}{3}} x^{2}+52 \left (3 x^{2}+4\right )^{\frac {1}{3}} x -12 \left (3 x^{2}+4\right )^{\frac {1}{3}}}d x \right )+2 \left (\int \frac {1}{9 \left (3 x^{2}+4\right )^{\frac {1}{3}} x^{2}+52 \left (3 x^{2}+4\right )^{\frac {1}{3}} x -12 \left (3 x^{2}+4\right )^{\frac {1}{3}}}d x \right ) \] Input:

int((2+3*x)/(3*x^2+4)^(1/3)/(9*x^2+52*x-12),x)
 

Output:

3*int(x/(9*(3*x**2 + 4)**(1/3)*x**2 + 52*(3*x**2 + 4)**(1/3)*x - 12*(3*x** 
2 + 4)**(1/3)),x) + 2*int(1/(9*(3*x**2 + 4)**(1/3)*x**2 + 52*(3*x**2 + 4)* 
*(1/3)*x - 12*(3*x**2 + 4)**(1/3)),x)