\(\int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} (-b-a x^3+x^4)} \, dx\) [439]

Optimal result

Integrand size = 36, antiderivative size = 35 \[ \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{b+a x^3}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b+a x^3}}{x}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{b+a x^3}}\right ) \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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