\(\int \frac {x^2 (-2 b+a x^6)}{(b+a x^6)^{3/4} (b+c x^4+a x^6)} \, dx\) [1843]

Optimal result

Integrand size = 38, antiderivative size = 125 \[ \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{-\sqrt {c} x^2+\sqrt {b+a x^6}}\right )}{\sqrt {2} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \] Output:

-1/2*arctan(2^(1/2)*c^(1/4)*x*(a*x^6+b)^(1/4)/(-c^(1/2)*x^2+(a*x^6+b)^(1/2 
)))*2^(1/2)/c^(3/4)+1/2*arctanh((1/2*c^(1/4)*x^2*2^(1/2)+1/2*(a*x^6+b)^(1/ 
2)*2^(1/2)/c^(1/4))/x/(a*x^6+b)^(1/4))*2^(1/2)/c^(3/4)
 

Mathematica [A] (verified)

Time = 8.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{\sqrt {c} x^2-\sqrt {b+a x^6}}\right )+\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \] Input:

Integrate[(x^2*(-2*b + a*x^6))/((b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)),x]
 

Output:

(ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^6)^(1/4))/(Sqrt[c]*x^2 - Sqrt[b + a*x^ 
6])] + ArcTanh[(Sqrt[c]*x^2 + Sqrt[b + a*x^6])/(Sqrt[2]*c^(1/4)*x*(b + a*x 
^6)^(1/4))])/(Sqrt[2]*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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 a b x^2+b c+c^2 x^4}{a \left (a x^6+b\right )^{3/4} \left (a x^6+b+c x^4\right )}-\frac {c}{a \left (a x^6+b\right )^{3/4}}+\frac {x^2}{\left (a x^6+b\right )^{3/4}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(x^2*(-2*b + a*x^6))/((b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(-\frac {\left (-\frac {\ln \left (\frac {\left (a \,x^{6}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{6}+b}}{-\left (a \,x^{6}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{6}+b}}\right )}{2}-\arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right )+\arctan \left (-\frac {\sqrt {2}\, \left (a \,x^{6}+b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}+1\right )\right ) \sqrt {2}}{2 c^{\frac {3}{4}}}\)	\(132\)

Input:

int(x^2*(a*x^6-2*b)/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b),x,method=_RETURNVERBOS 
E)
 

Output:

-1/2/c^(3/4)*(-1/2*ln(((a*x^6+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^ 
6+b)^(1/2))/(-(a*x^6+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^6+b)^(1/2 
)))-arctan(2^(1/2)/c^(1/4)*(a*x^6+b)^(1/4)/x+1)+arctan(-2^(1/2)/c^(1/4)*(a 
*x^6+b)^(1/4)/x+1))*2^(1/2)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(x^2*(a*x^6-2*b)/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(x**2*(a*x**6-2*b)/(a*x**6+b)**(3/4)/(a*x**6+c*x**4+b),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*(a*x^6-2*b)/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b),x, algorithm="ma 
xima")
 

Output:

integrate((a*x^6 - 2*b)*x^2/((a*x^6 + c*x^4 + b)*(a*x^6 + b)^(3/4)), x)
 

Giac [F]

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

integrate(x^2*(a*x^6-2*b)/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b),x, algorithm="gi 
ac")
 

Output:

sage0*x
 

Mupad [F(-1)]

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

int(-(x^2*(2*b - a*x^6))/((b + a*x^6)^(3/4)*(b + a*x^6 + c*x^4)),x)
                                                                                    
                                                                                    
 

Output:

-int((x^2*(2*b - a*x^6))/((b + a*x^6)^(3/4)*(b + a*x^6 + c*x^4)), x)
 

Reduce [F]

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

int(x^2*(a*x^6-2*b)/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b),x)
 

Output:

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