\(\int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} (c+b x+a x^2+x^3)} \, dx\) [1504]

Optimal result

Integrand size = 43, antiderivative size = 105 \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c+b x+a x^2}}{-2 x+\sqrt [3]{c+b x+a x^2}}\right )+\log \left (x+\sqrt [3]{c+b x+a x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{c+b x+a x^2}+\left (c+b x+a x^2\right )^{2/3}\right ) \] Output:

3^(1/2)*arctan(3^(1/2)*(a*x^2+b*x+c)^(1/3)/(-2*x+(a*x^2+b*x+c)^(1/3)))+ln( 
x+(a*x^2+b*x+c)^(1/3))-1/2*ln(x^2-x*(a*x^2+b*x+c)^(1/3)+(a*x^2+b*x+c)^(2/3 
))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c+x (b+a x)}}{-2 x+\sqrt [3]{c+x (b+a x)}}\right )+\log \left (x+\sqrt [3]{c+x (b+a x)}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{c+x (b+a x)}+(c+x (b+a x))^{2/3}\right ) \] Input:

Integrate[(3*c + 2*b*x + a*x^2)/((c + b*x + a*x^2)^(1/3)*(c + b*x + a*x^2 
+ x^3)),x]
 

Output:

Sqrt[3]*ArcTan[(Sqrt[3]*(c + x*(b + a*x))^(1/3))/(-2*x + (c + x*(b + a*x)) 
^(1/3))] + Log[x + (c + x*(b + a*x))^(1/3)] - Log[x^2 - x*(c + x*(b + a*x) 
)^(1/3) + (c + x*(b + a*x))^(2/3)]/2
 

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.

\(\displaystyle \int \frac {a x^2+2 b x+3 c}{\sqrt [3]{a x^2+b x+c} \left (a x^2+b x+c+x^3\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {a x^2}{\sqrt [3]{a x^2+b x+c} \left (a x^2+b x+c+x^3\right )}+\frac {2 b x}{\sqrt [3]{a x^2+b x+c} \left (a x^2+b x+c+x^3\right )}+\frac {3 c}{\sqrt [3]{a x^2+b x+c} \left (a x^2+b x+c+x^3\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 c \int \frac {1}{\sqrt [3]{a x^2+b x+c} \left (x^3+a x^2+b x+c\right )}dx+2 b \int \frac {x}{\sqrt [3]{a x^2+b x+c} \left (x^3+a x^2+b x+c\right )}dx+a \int \frac {x^2}{\sqrt [3]{a x^2+b x+c} \left (x^3+a x^2+b x+c\right )}dx\)

Input:

Int[(3*c + 2*b*x + a*x^2)/((c + b*x + a*x^2)^(1/3)*(c + b*x + a*x^2 + x^3) 
),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {a \,x^{2}+2 b x +3 c}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{3}} \left (a \,x^{2}+x^{3}+b x +c \right )}d x\]

Input:

int((a*x^2+2*b*x+3*c)/(a*x^2+b*x+c)^(1/3)/(a*x^2+x^3+b*x+c),x)
 

Output:

int((a*x^2+2*b*x+3*c)/(a*x^2+b*x+c)^(1/3)/(a*x^2+x^3+b*x+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x^2+2*b*x+3*c)/(a*x^2+b*x+c)^(1/3)/(a*x^2+x^3+b*x+c),x, algor 
ithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x**2+2*b*x+3*c)/(a*x**2+b*x+c)**(1/3)/(a*x**2+x**3+b*x+c),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*x^2+2*b*x+3*c)/(a*x^2+b*x+c)^(1/3)/(a*x^2+x^3+b*x+c),x, algor 
ithm="maxima")
 

Output:

integrate((a*x^2 + 2*b*x + 3*c)/((a*x^2 + x^3 + b*x + c)*(a*x^2 + b*x + c) 
^(1/3)), x)
 

Giac [F]

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

integrate((a*x^2+2*b*x+3*c)/(a*x^2+b*x+c)^(1/3)/(a*x^2+x^3+b*x+c),x, algor 
ithm="giac")
 

Output:

integrate((a*x^2 + 2*b*x + 3*c)/((a*x^2 + x^3 + b*x + c)*(a*x^2 + b*x + c) 
^(1/3)), x)
 

Mupad [F(-1)]

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

int((3*c + 2*b*x + a*x^2)/((c + b*x + a*x^2)^(1/3)*(c + b*x + a*x^2 + x^3) 
),x)
 

Output:

int((3*c + 2*b*x + a*x^2)/((c + b*x + a*x^2)^(1/3)*(c + b*x + a*x^2 + x^3) 
), x)
 

Reduce [F]

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

int((a*x^2+2*b*x+3*c)/(a*x^2+b*x+c)^(1/3)/(a*x^2+x^3+b*x+c),x)
 

Output:

int(x**2/((a*x**2 + b*x + c)**(1/3)*a*x**2 + (a*x**2 + b*x + c)**(1/3)*b*x 
 + (a*x**2 + b*x + c)**(1/3)*c + (a*x**2 + b*x + c)**(1/3)*x**3),x)*a + 2* 
int(x/((a*x**2 + b*x + c)**(1/3)*a*x**2 + (a*x**2 + b*x + c)**(1/3)*b*x + 
(a*x**2 + b*x + c)**(1/3)*c + (a*x**2 + b*x + c)**(1/3)*x**3),x)*b + 3*int 
(1/((a*x**2 + b*x + c)**(1/3)*a*x**2 + (a*x**2 + b*x + c)**(1/3)*b*x + (a* 
x**2 + b*x + c)**(1/3)*c + (a*x**2 + b*x + c)**(1/3)*x**3),x)*c