\(\int \frac {x (8 b+5 a x^3)}{\sqrt [4]{b+a x^3} (-b-a x^3+x^8)} \, dx\) [469]

Optimal result

Integrand size = 38, antiderivative size = 37 \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {x^2}{\sqrt [4]{b+a x^3}}\right ) \] Output:

2*arctan((a*x^3+b)^(1/4)/x^2)-2*arctanh(x^2/(a*x^3+b)^(1/4))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x^2}\right )-2 \text {arctanh}\left (\frac {x^2}{\sqrt [4]{b+a x^3}}\right ) \] Input:

Integrate[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]
 

Output:

2*ArcTan[(b + a*x^3)^(1/4)/x^2] - 2*ArcTanh[x^2/(b + a*x^3)^(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.

\(\displaystyle \int \frac {x \left (5 a x^3+8 b\right )}{\sqrt [4]{a x^3+b} \left (-a x^3-b+x^8\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {8 b x}{\sqrt [4]{a x^3+b} \left (a x^3+b-x^8\right )}-\frac {5 a x^4}{\sqrt [4]{a x^3+b} \left (a x^3+b-x^8\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -8 b \int \frac {x}{\sqrt [4]{a x^3+b} \left (-x^8+a x^3+b\right )}dx-5 a \int \frac {x^4}{\sqrt [4]{a x^3+b} \left (-x^8+a x^3+b\right )}dx\)

Input:

Int[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {x \left (5 a \,x^{3}+8 b \right )}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{3}-b \right )}d x\]

Input:

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)
 

Output:

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\text {Timed out} \] Input:

integrate(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="fric 
as")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int \frac {x \left (5 a x^{3} + 8 b\right )}{\sqrt [4]{a x^{3} + b} \left (- a x^{3} - b + x^{8}\right )}\, dx \] Input:

integrate(x*(5*a*x**3+8*b)/(a*x**3+b)**(1/4)/(x**8-a*x**3-b),x)
 

Output:

Integral(x*(5*a*x**3 + 8*b)/((a*x**3 + b)**(1/4)*(-a*x**3 - b + x**8)), x)
 

Maxima [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int { \frac {{\left (5 \, a x^{3} + 8 \, b\right )} x}{{\left (x^{8} - a x^{3} - b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="maxi 
ma")
 

Output:

integrate((5*a*x^3 + 8*b)*x/((x^8 - a*x^3 - b)*(a*x^3 + b)^(1/4)), x)
 

Giac [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int { \frac {{\left (5 \, a x^{3} + 8 \, b\right )} x}{{\left (x^{8} - a x^{3} - b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x, algorithm="giac 
")
 

Output:

integrate((5*a*x^3 + 8*b)*x/((x^8 - a*x^3 - b)*(a*x^3 + b)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=\int -\frac {x\,\left (5\,a\,x^3+8\,b\right )}{{\left (a\,x^3+b\right )}^{1/4}\,\left (-x^8+a\,x^3+b\right )} \,d x \] Input:

int(-(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)),x)
 

Output:

int(-(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(b + a*x^3 - x^8)), x)
 

Reduce [F]

\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx=-5 \left (\int \frac {x^{4}}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} a \,x^{3}+\left (a \,x^{3}+b \right )^{\frac {1}{4}} b -\left (a \,x^{3}+b \right )^{\frac {1}{4}} x^{8}}d x \right ) a -8 \left (\int \frac {x}{\left (a \,x^{3}+b \right )^{\frac {1}{4}} a \,x^{3}+\left (a \,x^{3}+b \right )^{\frac {1}{4}} b -\left (a \,x^{3}+b \right )^{\frac {1}{4}} x^{8}}d x \right ) b \] Input:

int(x*(5*a*x^3+8*b)/(a*x^3+b)^(1/4)/(x^8-a*x^3-b),x)
 

Output:

 - 5*int(x**4/((a*x**3 + b)**(1/4)*a*x**3 + (a*x**3 + b)**(1/4)*b - (a*x** 
3 + b)**(1/4)*x**8),x)*a - 8*int(x/((a*x**3 + b)**(1/4)*a*x**3 + (a*x**3 + 
 b)**(1/4)*b - (a*x**3 + b)**(1/4)*x**8),x)*b