\(\int \frac {(2 b+a x^6) (-b-x^4+a x^6)}{x^4 \sqrt [4]{-b+a x^6} (-b-2 x^4+a x^6)} \, dx\) [1323]

Optimal result

Integrand size = 56, antiderivative size = 95 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {\arctan \left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \] Output:

2/3*(a*x^6-b)^(3/4)/x^3+1/2*arctan(2^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2 
^(3/4)+1/2*arctanh(2^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(3/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 8.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}} \] Input:

Integrate[((2*b + a*x^6)*(-b - x^4 + a*x^6))/(x^4*(-b + a*x^6)^(1/4)*(-b - 
 2*x^4 + a*x^6)),x]
 

Output:

(2*(-b + a*x^6)^(3/4))/(3*x^3) - ArcTan[(2^(1/4)*x)/(-b + a*x^6)^(1/4)]/2^ 
(1/4) - ArcTanh[(2^(1/4)*x)/(-b + a*x^6)^(1/4)]/2^(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.

\(\displaystyle \int \frac {\left (a x^6+2 b\right ) \left (a x^6-b-x^4\right )}{x^4 \sqrt [4]{a x^6-b} \left (a x^6-b-2 x^4\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{\sqrt [4]{a x^6-b}}+\frac {-3 b-2 x^4}{\left (-a x^6+b+2 x^4\right ) \sqrt [4]{a x^6-b}}+\frac {2 b}{x^4 \sqrt [4]{a x^6-b}}+\frac {a x^2}{\sqrt [4]{a x^6-b}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 b \int \frac {1}{\left (-a x^6+2 x^4+b\right ) \sqrt [4]{a x^6-b}}dx+2 \int \frac {x^4}{\sqrt [4]{a x^6-b} \left (a x^6-2 x^4-b\right )}dx+\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{\sqrt [4]{a x^6-b}}+\frac {2 \left (a x^6-b\right )^{3/4}}{3 x^3}\)

Input:

Int[((2*b + a*x^6)*(-b - x^4 + a*x^6))/(x^4*(-b + a*x^6)^(1/4)*(-b - 2*x^4 
 + a*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {6 \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}} x^{3}-3 \ln \left (\frac {2^{\frac {1}{4}} x +\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (a \,x^{6}-b \right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} x^{3}+8 \left (a \,x^{6}-b \right )^{\frac {3}{4}}}{12 x^{3}}\)	\(95\)

Input:

int((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x,method 
=_RETURNVERBOSE)
 

Output:

1/12*(6*arctan(1/2*(a*x^6-b)^(1/4)/x*2^(3/4))*2^(3/4)*x^3-3*ln((2^(1/4)*x+ 
(a*x^6-b)^(1/4))/(-2^(1/4)*x+(a*x^6-b)^(1/4)))*2^(3/4)*x^3+8*(a*x^6-b)^(3/ 
4))/x^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x, 
 algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x**6+2*b)*(a*x**6-x**4-b)/x**4/(a*x**6-b)**(1/4)/(a*x**6-2*x* 
*4-b),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} - x^{4} - b\right )} {\left (a x^{6} + 2 \, b\right )}}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \] Input:

integrate((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x, 
 algorithm="maxima")
 

Output:

integrate((a*x^6 - x^4 - b)*(a*x^6 + 2*b)/((a*x^6 - 2*x^4 - b)*(a*x^6 - b) 
^(1/4)*x^4), x)
 

Giac [F]

\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} - x^{4} - b\right )} {\left (a x^{6} + 2 \, b\right )}}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \] Input:

integrate((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x, 
 algorithm="giac")
 

Output:

sage0*x
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\int \frac {\left (a\,x^6+2\,b\right )\,\left (-a\,x^6+x^4+b\right )}{x^4\,{\left (a\,x^6-b\right )}^{1/4}\,\left (-a\,x^6+2\,x^4+b\right )} \,d x \] Input:

int(((2*b + a*x^6)*(b - a*x^6 + x^4))/(x^4*(a*x^6 - b)^(1/4)*(b - a*x^6 + 
2*x^4)),x)
                                                                                    
                                                                                    
 

Output:

int(((2*b + a*x^6)*(b - a*x^6 + x^4))/(x^4*(a*x^6 - b)^(1/4)*(b - a*x^6 + 
2*x^4)), x)
 

Reduce [F]

\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\left (\int \frac {x^{8}}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} a \,x^{6}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} b -2 \left (a \,x^{6}-b \right )^{\frac {1}{4}} x^{4}}d x \right ) a^{2}-\left (\int \frac {x^{6}}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} a \,x^{6}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} b -2 \left (a \,x^{6}-b \right )^{\frac {1}{4}} x^{4}}d x \right ) a +\left (\int \frac {x^{2}}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} a \,x^{6}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} b -2 \left (a \,x^{6}-b \right )^{\frac {1}{4}} x^{4}}d x \right ) a b -2 \left (\int \frac {1}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} a \,x^{10}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} b \,x^{4}-2 \left (a \,x^{6}-b \right )^{\frac {1}{4}} x^{8}}d x \right ) b^{2}-2 \left (\int \frac {1}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} a \,x^{6}-\left (a \,x^{6}-b \right )^{\frac {1}{4}} b -2 \left (a \,x^{6}-b \right )^{\frac {1}{4}} x^{4}}d x \right ) b \] Input:

int((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x)
 

Output:

int(x**8/((a*x**6 - b)**(1/4)*a*x**6 - (a*x**6 - b)**(1/4)*b - 2*(a*x**6 - 
 b)**(1/4)*x**4),x)*a**2 - int(x**6/((a*x**6 - b)**(1/4)*a*x**6 - (a*x**6 
- b)**(1/4)*b - 2*(a*x**6 - b)**(1/4)*x**4),x)*a + int(x**2/((a*x**6 - b)* 
*(1/4)*a*x**6 - (a*x**6 - b)**(1/4)*b - 2*(a*x**6 - b)**(1/4)*x**4),x)*a*b 
 - 2*int(1/((a*x**6 - b)**(1/4)*a*x**10 - (a*x**6 - b)**(1/4)*b*x**4 - 2*( 
a*x**6 - b)**(1/4)*x**8),x)*b**2 - 2*int(1/((a*x**6 - b)**(1/4)*a*x**6 - ( 
a*x**6 - b)**(1/4)*b - 2*(a*x**6 - b)**(1/4)*x**4),x)*b