\(\int \frac {x^2 (4 b+a x^5)}{(-b+a x^5)^{3/4} (-b+c x^4+a x^5)} \, dx\) [1918]

Optimal result

Integrand size = 42, antiderivative size = 133 \[ \int \frac {x^2 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^{3/4} \left (-b+c x^4+a x^5\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )}{c^{3/4}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^5}}\right )}{c^{3/4}} \] 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)
 

Mathematica [A] (verified)

Time = 7.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {x^2 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^{3/4} \left (-b+c x^4+a x^5\right )} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{\sqrt {c} x^2-\sqrt {-b+a x^5}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}\right )\right )}{c^{3/4}} \] 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 - Sqr 
t[-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)
 

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

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

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
 

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}-b}}{-\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}-b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right )\right )}{2 c^{\frac {3}{4}}}\)	\(144\)

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

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
 

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)
 

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)
 

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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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 + 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