\(\int \frac {(4 b+a x^5) (-b+c x^4+a x^5)}{x^2 (-b+a x^5)^{3/4} (-b-c x^4+a x^5)} \, dx\) [1320]

Optimal result

Integrand size = 57, antiderivative size = 95 \[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\frac {4 \sqrt [4]{-b+a x^5}}{x}-4 \sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{c} x \left (-b+a x^5\right )^{3/4}}{b-a x^5}\right )+4 \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} x \left (-b+a x^5\right )^{3/4}}{b-a x^5}\right ) \] Output:

4*(a*x^5-b)^(1/4)/x-4*c^(1/4)*arctan(c^(1/4)*x*(a*x^5-b)^(3/4)/(-a*x^5+b)) 
+4*c^(1/4)*arctanh(c^(1/4)*x*(a*x^5-b)^(3/4)/(-a*x^5+b))
 

Mathematica [A] (verified)

Time = 6.90 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.79 \[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\frac {4 \sqrt [4]{-b+a x^5}}{x}+4 \sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^5}}\right )-4 \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^5}}\right ) \] Input:

Integrate[((4*b + a*x^5)*(-b + c*x^4 + a*x^5))/(x^2*(-b + a*x^5)^(3/4)*(-b 
 - c*x^4 + a*x^5)),x]
 

Output:

(4*(-b + a*x^5)^(1/4))/x + 4*c^(1/4)*ArcTan[(c^(1/4)*x)/(-b + a*x^5)^(1/4) 
] - 4*c^(1/4)*ArcTanh[(c^(1/4)*x)/(-b + a*x^5)^(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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c^4 \int \frac {x^4}{\left (-a x^5+c x^4+b\right ) \left (a x^5-b\right )^{3/4}}dx}{a^2}-\frac {2 b c^3 \int \frac {1}{\left (-a x^5+c x^4+b\right ) \left (a x^5-b\right )^{3/4}}dx}{a^2}+\frac {2 b c^2 \int \frac {x}{\left (a x^5-b\right )^{3/4} \left (a x^5-c x^4-b\right )}dx}{a}+10 b c \int \frac {x^2}{\left (a x^5-b\right )^{3/4} \left (a x^5-c x^4-b\right )}dx+\frac {2 c^3 x \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {3}{4},\frac {6}{5},\frac {a x^5}{b}\right )}{a^2 \left (a x^5-b\right )^{3/4}}+\frac {c^2 x^2 \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},\frac {3}{4},\frac {7}{5},\frac {a x^5}{b}\right )}{a \left (a x^5-b\right )^{3/4}}+\frac {2 c x^3 \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},\frac {3}{4},\frac {8}{5},\frac {a x^5}{b}\right )}{3 \left (a x^5-b\right )^{3/4}}-\frac {4 b \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {3}{4},\frac {4}{5},\frac {a x^5}{b}\right )}{x \left (a x^5-b\right )^{3/4}}+\frac {a x^4 \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {4}{5},\frac {9}{5},\frac {a x^5}{b}\right )}{4 \left (a x^5-b\right )^{3/4}}\)

Input:

Int[((4*b + a*x^5)*(-b + c*x^4 + a*x^5))/(x^2*(-b + a*x^5)^(3/4)*(-b - c*x 
^4 + a*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {-2 x \left (\ln \left (\frac {-c^{\frac {1}{4}} x -\left (a \,x^{5}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x -\left (a \,x^{5}-b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (a \,x^{5}-b \right )^{\frac {1}{4}}}{x \,c^{\frac {1}{4}}}\right )\right ) c^{\frac {1}{4}}+4 \left (a \,x^{5}-b \right )^{\frac {1}{4}}}{x}\)	\(90\)

Input:

int((a*x^5+4*b)*(a*x^5+c*x^4-b)/x^2/(a*x^5-b)^(3/4)/(a*x^5-c*x^4-b),x,meth 
od=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x^5+4*b)*(a*x^5+c*x^4-b)/x^2/(a*x^5-b)^(3/4)/(a*x^5-c*x^4-b), 
x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\int \frac {\left (a x^{5} + 4 b\right ) \left (a x^{5} - b + c x^{4}\right )}{x^{2} \left (a x^{5} - b\right )^{\frac {3}{4}} \left (a x^{5} - b - c x^{4}\right )}\, dx \] Input:

integrate((a*x**5+4*b)*(a*x**5+c*x**4-b)/x**2/(a*x**5-b)**(3/4)/(a*x**5-c* 
x**4-b),x)
 

Output:

Integral((a*x**5 + 4*b)*(a*x**5 - b + c*x**4)/(x**2*(a*x**5 - b)**(3/4)*(a 
*x**5 - b - c*x**4)), x)
 

Maxima [F]

\[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + c x^{4} - b\right )} {\left (a x^{5} + 4 \, b\right )}}{{\left (a x^{5} - c x^{4} - b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}} x^{2}} \,d x } \] Input:

integrate((a*x^5+4*b)*(a*x^5+c*x^4-b)/x^2/(a*x^5-b)^(3/4)/(a*x^5-c*x^4-b), 
x, algorithm="maxima")
 

Output:

integrate((a*x^5 + c*x^4 - b)*(a*x^5 + 4*b)/((a*x^5 - c*x^4 - b)*(a*x^5 - 
b)^(3/4)*x^2), x)
 

Giac [F]

\[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + c x^{4} - b\right )} {\left (a x^{5} + 4 \, b\right )}}{{\left (a x^{5} - c x^{4} - b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}} x^{2}} \,d x } \] Input:

integrate((a*x^5+4*b)*(a*x^5+c*x^4-b)/x^2/(a*x^5-b)^(3/4)/(a*x^5-c*x^4-b), 
x, algorithm="giac")
 

Output:

integrate((a*x^5 + c*x^4 - b)*(a*x^5 + 4*b)/((a*x^5 - c*x^4 - b)*(a*x^5 - 
b)^(3/4)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\int -\frac {\left (a\,x^5+4\,b\right )\,\left (a\,x^5+c\,x^4-b\right )}{x^2\,{\left (a\,x^5-b\right )}^{3/4}\,\left (-a\,x^5+c\,x^4+b\right )} \,d x \] Input:

int(-((4*b + a*x^5)*(a*x^5 - b + c*x^4))/(x^2*(a*x^5 - b)^(3/4)*(b - a*x^5 
 + c*x^4)),x)
                                                                                    
                                                                                    
 

Output:

int(-((4*b + a*x^5)*(a*x^5 - b + c*x^4))/(x^2*(a*x^5 - b)^(3/4)*(b - a*x^5 
 + c*x^4)), x)
 

Reduce [F]

\[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\left (\int \frac {x^{8}}{\left (a \,x^{5}-b \right )^{\frac {3}{4}} a \,x^{5}-\left (a \,x^{5}-b \right )^{\frac {3}{4}} b -\left (a \,x^{5}-b \right )^{\frac {3}{4}} c \,x^{4}}d x \right ) a^{2}+\left (\int \frac {x^{7}}{\left (a \,x^{5}-b \right )^{\frac {3}{4}} a \,x^{5}-\left (a \,x^{5}-b \right )^{\frac {3}{4}} b -\left (a \,x^{5}-b \right )^{\frac {3}{4}} c \,x^{4}}d x \right ) a c +3 \left (\int \frac {x^{3}}{\left (a \,x^{5}-b \right )^{\frac {3}{4}} a \,x^{5}-\left (a \,x^{5}-b \right )^{\frac {3}{4}} b -\left (a \,x^{5}-b \right )^{\frac {3}{4}} c \,x^{4}}d x \right ) a b +4 \left (\int \frac {x^{2}}{\left (a \,x^{5}-b \right )^{\frac {3}{4}} a \,x^{5}-\left (a \,x^{5}-b \right )^{\frac {3}{4}} b -\left (a \,x^{5}-b \right )^{\frac {3}{4}} c \,x^{4}}d x \right ) b c -4 \left (\int \frac {1}{\left (a \,x^{5}-b \right )^{\frac {3}{4}} a \,x^{7}-\left (a \,x^{5}-b \right )^{\frac {3}{4}} b \,x^{2}-\left (a \,x^{5}-b \right )^{\frac {3}{4}} c \,x^{6}}d x \right ) b^{2} \] Input:

int((a*x^5+4*b)*(a*x^5+c*x^4-b)/x^2/(a*x^5-b)^(3/4)/(a*x^5-c*x^4-b),x)
 

Output:

int(x**8/((a*x**5 - b)**(3/4)*a*x**5 - (a*x**5 - b)**(3/4)*b - (a*x**5 - b 
)**(3/4)*c*x**4),x)*a**2 + int(x**7/((a*x**5 - b)**(3/4)*a*x**5 - (a*x**5 
- b)**(3/4)*b - (a*x**5 - b)**(3/4)*c*x**4),x)*a*c + 3*int(x**3/((a*x**5 - 
 b)**(3/4)*a*x**5 - (a*x**5 - b)**(3/4)*b - (a*x**5 - b)**(3/4)*c*x**4),x) 
*a*b + 4*int(x**2/((a*x**5 - b)**(3/4)*a*x**5 - (a*x**5 - b)**(3/4)*b - (a 
*x**5 - b)**(3/4)*c*x**4),x)*b*c - 4*int(1/((a*x**5 - b)**(3/4)*a*x**7 - ( 
a*x**5 - b)**(3/4)*b*x**2 - (a*x**5 - b)**(3/4)*c*x**6),x)*b**2