\(\int \frac {(-4 b+a x^5) (b+a x^5)^{3/4}}{x^4 (2 b+c x^4+2 a x^5)} \, dx\) [2123]

Optimal result

Integrand size = 41, antiderivative size = 154 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {2 \left (b+a x^5\right )^{3/4}}{3 x^3}+\frac {c^{3/4} \arctan \left (\frac {2^{3/4} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {2} \sqrt {b+a x^5}}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{2^{3/4}}+\frac {\sqrt {b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{2 \sqrt [4]{2}} \] Output:

2/3*(a*x^5+b)^(3/4)/x^3+1/4*c^(3/4)*arctan(2^(3/4)*c^(1/4)*x*(a*x^5+b)^(1/ 
4)/(-c^(1/2)*x^2+2^(1/2)*(a*x^5+b)^(1/2)))*2^(3/4)+1/4*c^(3/4)*arctanh((1/ 
2*c^(1/4)*x^2*2^(1/4)+1/2*(a*x^5+b)^(1/2)*2^(3/4)/c^(1/4))/x/(a*x^5+b)^(1/ 
4))*2^(3/4)
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {1}{12} \left (\frac {8 \left (b+a x^5\right )^{3/4}}{x^3}+3\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}-\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )+3\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}+\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )\right ) \] Input:

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

Output:

((8*(b + a*x^5)^(3/4))/x^3 + 3*2^(3/4)*c^(3/4)*ArcTan[(c^(1/4)*x)/(2^(3/4) 
*(b + a*x^5)^(1/4)) - (b + a*x^5)^(1/4)/(2^(1/4)*c^(1/4)*x)] + 3*2^(3/4)*c 
^(3/4)*ArcTanh[(c^(1/4)*x)/(2^(3/4)*(b + a*x^5)^(1/4)) + (b + a*x^5)^(1/4) 
/(2^(1/4)*c^(1/4)*x)])/12
 

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {2 \left (a \,x^{5}+b \right )^{\frac {3}{4}}}{3 x^{3}}-\frac {\ln \left (\frac {\sqrt {2}\, \sqrt {c}\, x^{2}-2 \,2^{\frac {1}{4}} c^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} x +2 \sqrt {a \,x^{5}+b}}{\sqrt {2}\, \sqrt {c}\, x^{2}+2 \,2^{\frac {1}{4}} c^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} x +2 \sqrt {a \,x^{5}+b}}\right ) c^{\frac {3}{4}} 2^{\frac {3}{4}}}{8}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right ) c^{\frac {3}{4}} 2^{\frac {3}{4}}}{4}+\frac {\arctan \left (-\frac {2^{\frac {3}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right ) c^{\frac {3}{4}} 2^{\frac {3}{4}}}{4}\)	\(169\)

Input:

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

Output:

2/3*(a*x^5+b)^(3/4)/x^3-1/8*ln((2^(1/2)*c^(1/2)*x^2-2*2^(1/4)*c^(1/4)*(a*x 
^5+b)^(1/4)*x+2*(a*x^5+b)^(1/2))/(2^(1/2)*c^(1/2)*x^2+2*2^(1/4)*c^(1/4)*(a 
*x^5+b)^(1/4)*x+2*(a*x^5+b)^(1/2)))*c^(3/4)*2^(3/4)-1/4*arctan(2^(3/4)/c^( 
1/4)*(a*x^5+b)^(1/4)/x+1)*c^(3/4)*2^(3/4)+1/4*arctan(-2^(3/4)/c^(1/4)*(a*x 
^5+b)^(1/4)/x+1)*c^(3/4)*2^(3/4)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {4 \left (a \,x^{5}+b \right )^{\frac {3}{4}}+12 \left (\int \frac {\left (a \,x^{5}+b \right )^{\frac {3}{4}}}{2 a^{2} x^{10}+a c \,x^{9}+4 a b \,x^{5}+b c \,x^{4}+2 b^{2}}d x \right ) b c \,x^{3}-3 \left (\int \frac {\left (a \,x^{5}+b \right )^{\frac {3}{4}} x^{5}}{2 a^{2} x^{10}+a c \,x^{9}+4 a b \,x^{5}+b c \,x^{4}+2 b^{2}}d x \right ) a c \,x^{3}}{6 x^{3}} \] Input:

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

Output:

(4*(a*x**5 + b)**(3/4) + 12*int((a*x**5 + b)**(3/4)/(2*a**2*x**10 + 4*a*b* 
x**5 + a*c*x**9 + 2*b**2 + b*c*x**4),x)*b*c*x**3 - 3*int(((a*x**5 + b)**(3 
/4)*x**5)/(2*a**2*x**10 + 4*a*b*x**5 + a*c*x**9 + 2*b**2 + b*c*x**4),x)*a* 
c*x**3)/(6*x**3)