\(\int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} (b-a x^3+x^4)} \, dx\) [625]

Optimal result

Integrand size = 36, antiderivative size = 49 \[ \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )+2 \text {arctanh}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )+2 \text {arctanh}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right ) \] 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)^(3/4))/(b - a*x 
^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.

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

\(\Big \downarrow \) 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

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

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

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

Output:

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