\(\int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} (-c-b x-a x^2+x^4)} \, dx\) [531]

Optimal result

Integrand size = 48, antiderivative size = 41 \[ \int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{c+b x+a x^2}}{x}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{c+b x+a x^2}}\right ) \] Output:

2*arctan((a*x^2+b*x+c)^(1/4)/x)-2*arctanh(x/(a*x^2+b*x+c)^(1/4))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx=2 \left (\arctan \left (\frac {\sqrt [4]{c+x (b+a x)}}{x}\right )-\text {arctanh}\left (\frac {x}{\sqrt [4]{c+x (b+a x)}}\right )\right ) \] Input:

Integrate[(4*c + 3*b*x + 2*a*x^2)/((c + b*x + a*x^2)^(1/4)*(-c - b*x - a*x 
^2 + x^4)),x]
 

Output:

2*(ArcTan[(c + x*(b + a*x))^(1/4)/x] - ArcTanh[x/(c + x*(b + a*x))^(1/4)])
 

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.

\(\displaystyle \int \frac {2 a x^2+3 b x+4 c}{\sqrt [4]{a x^2+b x+c} \left (-a x^2-b x-c+x^4\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 a x^2}{\sqrt [4]{a x^2+b x+c} \left (a x^2+b x+c-x^4\right )}-\frac {3 b x}{\sqrt [4]{a x^2+b x+c} \left (a x^2+b x+c-x^4\right )}-\frac {4 c}{\sqrt [4]{a x^2+b x+c} \left (a x^2+b x+c-x^4\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 c \int \frac {1}{\sqrt [4]{a x^2+b x+c} \left (-x^4+a x^2+b x+c\right )}dx-3 b \int \frac {x}{\sqrt [4]{a x^2+b x+c} \left (-x^4+a x^2+b x+c\right )}dx-2 a \int \frac {x^2}{\sqrt [4]{a x^2+b x+c} \left (-x^4+a x^2+b x+c\right )}dx\)

Input:

Int[(4*c + 3*b*x + 2*a*x^2)/((c + b*x + a*x^2)^(1/4)*(-c - b*x - a*x^2 + x 
^4)),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {2 a \,x^{2}+3 b x +4 c}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}} \left (x^{4}-a \,x^{2}-b x -c \right )}d x\]

Input:

int((2*a*x^2+3*b*x+4*c)/(a*x^2+b*x+c)^(1/4)/(x^4-a*x^2-b*x-c),x)
 

Output:

int((2*a*x^2+3*b*x+4*c)/(a*x^2+b*x+c)^(1/4)/(x^4-a*x^2-b*x-c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((2*a*x^2+3*b*x+4*c)/(a*x^2+b*x+c)^(1/4)/(x^4-a*x^2-b*x-c),x, alg 
orithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((2*a*x**2+3*b*x+4*c)/(a*x**2+b*x+c)**(1/4)/(x**4-a*x**2-b*x-c),x 
)
 

Output:

Timed out
 

Maxima [F]

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

integrate((2*a*x^2+3*b*x+4*c)/(a*x^2+b*x+c)^(1/4)/(x^4-a*x^2-b*x-c),x, alg 
orithm="maxima")
 

Output:

integrate((2*a*x^2 + 3*b*x + 4*c)/((x^4 - a*x^2 - b*x - c)*(a*x^2 + b*x + 
c)^(1/4)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((2*a*x^2+3*b*x+4*c)/(a*x^2+b*x+c)^(1/4)/(x^4-a*x^2-b*x-c),x, alg 
orithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(-(4*c + 3*b*x + 2*a*x^2)/((c + b*x + a*x^2)^(1/4)*(c + b*x + a*x^2 - x 
^4)),x)
 

Output:

int(-(4*c + 3*b*x + 2*a*x^2)/((c + b*x + a*x^2)^(1/4)*(c + b*x + a*x^2 - x 
^4)), x)
 

Reduce [F]

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

int((2*a*x^2+3*b*x+4*c)/(a*x^2+b*x+c)^(1/4)/(x^4-a*x^2-b*x-c),x)
 

Output:

 - 2*int(x**2/((a*x**2 + b*x + c)**(1/4)*a*x**2 + (a*x**2 + b*x + c)**(1/4 
)*b*x + (a*x**2 + b*x + c)**(1/4)*c - (a*x**2 + b*x + c)**(1/4)*x**4),x)*a 
 - 3*int(x/((a*x**2 + b*x + c)**(1/4)*a*x**2 + (a*x**2 + b*x + c)**(1/4)*b 
*x + (a*x**2 + b*x + c)**(1/4)*c - (a*x**2 + b*x + c)**(1/4)*x**4),x)*b - 
4*int(1/((a*x**2 + b*x + c)**(1/4)*a*x**2 + (a*x**2 + b*x + c)**(1/4)*b*x 
+ (a*x**2 + b*x + c)**(1/4)*c - (a*x**2 + b*x + c)**(1/4)*x**4),x)*c