\(\int \frac {4 b+a x^5}{(-b+a x^5) \sqrt [4]{-b+c x^4+a x^5}} \, dx\) [885]

Optimal result

Integrand size = 39, antiderivative size = 67 \[ \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}} \] Output:

-2*arctan(c^(1/4)*x/(a*x^5+c*x^4-b)^(1/4))/c^(1/4)-2*arctanh(c^(1/4)*x/(a* 
x^5+c*x^4-b)^(1/4))/c^(1/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87 \[ \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )\right )}{\sqrt [4]{c}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {5 b}{\left (a x^5-b\right ) \sqrt [4]{a x^5-b+c x^4}}+\frac {1}{\sqrt [4]{a x^5-b+c x^4}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )-\ln \left (\frac {-c^{\frac {1}{4}} x -\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x -\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\right )}{c^{\frac {1}{4}}}\)	\(87\)

Input:

int((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

1/c^(1/4)*(2*arctan(1/c^(1/4)/x*(a*x^5+c*x^4-b)^(1/4))-ln((-c^(1/4)*x-(a*x 
^5+c*x^4-b)^(1/4))/(c^(1/4)*x-(a*x^5+c*x^4-b)^(1/4))))
 

Fricas [F(-1)]

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

integrate((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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