\(\int \frac {b+a x^6}{(-b+a x^6) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\) [1980]

Optimal result

Integrand size = 39, antiderivative size = 140 \[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b+a^3 x^3+a x^6}}\right )}{\sqrt {3} a}+\frac {\log \left (-a x+\sqrt [3]{-b+a^3 x^3+a x^6}\right )}{3 a}-\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b+a^3 x^3+a x^6}+\left (-b+a^3 x^3+a x^6\right )^{2/3}\right )}{6 a} \] Output:

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

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.96 \[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b+a^3 x^3+a x^6}}\right )-2 \log \left (a \left (a x-\sqrt [3]{-b+a^3 x^3+a x^6}\right )\right )+\log \left (a^2 x^2+a x \sqrt [3]{-b+a^3 x^3+a x^6}+\left (-b+a^3 x^3+a x^6\right )^{2/3}\right )}{6 a} \] Input:

Integrate[(b + a*x^6)/((-b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]
 

Output:

-1/6*(2*Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-b + a^3*x^3 + a*x^6)^(1/3) 
)] - 2*Log[a*(a*x - (-b + a^3*x^3 + a*x^6)^(1/3))] + Log[a^2*x^2 + a*x*(-b 
 + a^3*x^3 + a*x^6)^(1/3) + (-b + a^3*x^3 + a*x^6)^(2/3)])/a
 

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^3\right ) \sqrt [3]{a x^6+a^3 x^3-b}}dx-\sqrt {b} \int \frac {1}{\left (\sqrt {a} x^3+\sqrt {b}\right ) \sqrt [3]{a x^6+a^3 x^3-b}}dx+\frac {x \sqrt [3]{\frac {2 \sqrt {a} x^3}{a^{5/2}-\sqrt {a^5+4 b}}+1} \sqrt [3]{\frac {2 \sqrt {a} x^3}{a^{5/2}+\sqrt {a^5+4 b}}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 a x^3}{a^3-\sqrt {a^6+4 b a}},-\frac {2 a x^3}{a^3+\sqrt {a^6+4 b a}}\right )}{\sqrt [3]{a^3 x^3+a x^6-b}}\)

Input:

Int[(b + a*x^6)/((-b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (a x +2 \left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}}\right )}{3 a x}\right )+2 \ln \left (\frac {-a x +\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {a^{2} x^{2}+a x \left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}}+\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {2}{3}}}{x^{2}}\right )}{6 a}\)	\(132\)

Input:

int((a*x^6+b)/(a*x^6-b)/(a*x^6+a^3*x^3-b)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/6*(2*3^(1/2)*arctan(1/3*3^(1/2)/a/x*(a*x+2*(a*x^6+a^3*x^3-b)^(1/3)))+2*l 
n((-a*x+(a*x^6+a^3*x^3-b)^(1/3))/x)-ln((a^2*x^2+a*x*(a*x^6+a^3*x^3-b)^(1/3 
)+(a*x^6+a^3*x^3-b)^(2/3))/x^2))/a
 

Fricas [F(-1)]

Timed out. \[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\text {Timed out} \] Input:

integrate((a*x^6+b)/(a*x^6-b)/(a*x^6+a^3*x^3-b)^(1/3),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int \frac {a x^{6} + b}{\left (a x^{6} - b\right ) \sqrt [3]{a^{3} x^{3} + a x^{6} - b}}\, dx \] Input:

integrate((a*x**6+b)/(a*x**6-b)/(a*x**6+a**3*x**3-b)**(1/3),x)
 

Output:

Integral((a*x**6 + b)/((a*x**6 - b)*(a**3*x**3 + a*x**6 - b)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} - b\right )}} \,d x } \] Input:

integrate((a*x^6+b)/(a*x^6-b)/(a*x^6+a^3*x^3-b)^(1/3),x, algorithm="maxima 
")
 

Output:

integrate((a*x^6 + b)/((a*x^6 + a^3*x^3 - b)^(1/3)*(a*x^6 - b)), x)
 

Giac [F]

\[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int { \frac {a x^{6} + b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} - b\right )}} \,d x } \] Input:

integrate((a*x^6+b)/(a*x^6-b)/(a*x^6+a^3*x^3-b)^(1/3),x, algorithm="giac")
 

Output:

integrate((a*x^6 + b)/((a*x^6 + a^3*x^3 - b)^(1/3)*(a*x^6 - b)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int -\frac {a\,x^6+b}{\left (b-a\,x^6\right )\,{\left (a^3\,x^3+a\,x^6-b\right )}^{1/3}} \,d x \] Input:

int(-(b + a*x^6)/((b - a*x^6)*(a*x^6 - b + a^3*x^3)^(1/3)),x)
                                                                                    
                                                                                    
 

Output:

int(-(b + a*x^6)/((b - a*x^6)*(a*x^6 - b + a^3*x^3)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {b+a x^6}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\left (\int \frac {x^{6}}{\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}} a \,x^{6}-\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}} b}d x \right ) a +\left (\int \frac {1}{\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}} a \,x^{6}-\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}} b}d x \right ) b \] Input:

int((a*x^6+b)/(a*x^6-b)/(a*x^6+a^3*x^3-b)^(1/3),x)
 

Output:

int(x**6/((a**3*x**3 + a*x**6 - b)**(1/3)*a*x**6 - (a**3*x**3 + a*x**6 - b 
)**(1/3)*b),x)*a + int(1/((a**3*x**3 + a*x**6 - b)**(1/3)*a*x**6 - (a**3*x 
**3 + a*x**6 - b)**(1/3)*b),x)*b