3.21.10 \(\int \frac {(-4 b+a x^5) (b-c x^4+a x^5)}{x^2 (b+a x^5)^{3/4} (b+c x^4+a x^5)} \, dx\) [2010]

3.21.10.1 Optimal result

Integrand size = 51, antiderivative size = 142 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b-c x^4+a x^5\right )}{x^2 \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx=\frac {4 \sqrt [4]{b+a x^5}}{x}+2 \sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {b+a x^5}}\right )-2 \sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right ) \]

output

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

3.21.10.2 Mathematica [A] (verified)

Time = 7.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b-c x^4+a x^5\right )}{x^2 \left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx=\frac {4 \sqrt [4]{b+a x^5}}{x}+2 \sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {b+a x^5}}\right )-2 \sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}\right ) \]

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 + 2*Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(b + 
 a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^5])] - 2*Sqrt[2]*c^(1/4)*Arc 
Tanh[(Sqrt[c]*x^2 + Sqrt[b + a*x^5])/(Sqrt[2]*c^(1/4)*x*(b + a*x^5)^(1/4)) 
]
 

3.21.10.3 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

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

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
 

3.21.10.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.21.10.4 Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {-x \sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )+\ln \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}+b}}{\sqrt {a \,x^{5}+b}-\left (a \,x^{5}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )\right ) c^{\frac {1}{4}}+4 \left (a \,x^{5}+b \right )^{\frac {1}{4}}}{x}\)	\(159\)

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

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

3.21.10.5 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
 

3.21.10.6 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)
 

3.21.10.7 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)
 

3.21.10.8 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)
 

3.21.10.9 Mupad [F(-1)]

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

input

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

output

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