3.19.41 \(\int \frac {x^2 (-4 b+a x^5)}{(b+a x^5)^{3/4} (b+c x^4+a x^5)} \, dx\) [1841]

3.19.41.1 Optimal result

Integrand size = 38, antiderivative size = 125 \[ \int \frac {x^2 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {b+a x^5}}\right )}{c^{3/4}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{c^{3/4}} \]

output

-2^(1/2)*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)))/c^(3/4)+2^(1/2)*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))/c^(3/4)
 

3.19.41.2 Mathematica [A] (verified)

Time = 7.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (-4 b+a x^5\right )}{\left (b+a x^5\right )^{3/4} \left (b+c x^4+a x^5\right )} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{\sqrt {c} x^2-\sqrt {b+a x^5}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}\right )\right )}{c^{3/4}} \]

input

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

output

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

3.19.41.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.19.41.3.1 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] 
]
 

3.19.41.4 Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}+b}}{\sqrt {a \,x^{5}+b}-\left (a \,x^{5}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right )\right )}{2 c^{\frac {3}{4}}}\)	\(142\)

input

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

output

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

3.19.41.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.19.41.6 Sympy [F]

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

input

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

output

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

3.19.41.7 Maxima [F]

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

input

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

output

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

3.19.41.8 Giac [F]

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

input

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

output

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

3.19.41.9 Mupad [F(-1)]

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

input

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

output

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