\(\int \frac {x^4 (-2 b+a x^2)}{(-b+a x^2)^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx\) [1559]

Optimal result

Integrand size = 42, antiderivative size = 107 \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=-\frac {x \left (-b+a x^2+c x^4\right )^{3/4}}{2 c \left (b-a x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\frac {-\frac {2 \sqrt [4]{c} x \left (-b+a x^2+c x^4\right )^{3/4}}{b-a x^2}-\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \] Input:

Integrate[(x^4*(-2*b + a*x^2))/((-b + a*x^2)^2*(-b + a*x^2 + c*x^4)^(1/4)) 
,x]
 

Output:

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

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.

\(\displaystyle \int \frac {x^4 \left (a x^2-2 b\right )}{\left (a x^2-b\right )^2 \sqrt [4]{a x^2-b+c x^4}} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int \frac {x^4 \left (a x^2-2 b\right )}{\left (a x^2-b\right )^2 \sqrt [4]{a x^2-b+c x^4}}dx\)

Input:

Int[(x^4*(-2*b + a*x^2))/((-b + a*x^2)^2*(-b + a*x^2 + c*x^4)^(1/4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(\frac {4 \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {3}{4}} x \,c^{\frac {1}{4}}+2 \left (a \,x^{2}-b \right ) \left (\arctan \left (\frac {\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )-\frac {\ln \left (\frac {c^{\frac {1}{4}} x +\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}{-c^{\frac {1}{4}} x +\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}\right )}{2}\right )}{c^{\frac {5}{4}} \left (8 a \,x^{2}-8 b \right )}\)	\(127\)

Input:

int(x^4*(a*x^2-2*b)/(a*x^2-b)^2/(c*x^4+a*x^2-b)^(1/4),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.23 \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=-\frac {{\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (-\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x - {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + {\left (-i \, a c x^{2} + i \, b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {i \, c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + {\left (i \, a c x^{2} - i \, b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {-i \, c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (c x^{4} + a x^{2} - b\right )}^{\frac {3}{4}} x}{8 \, {\left (a c x^{2} - b c\right )}} \] Input:

integrate(x^4*(a*x^2-2*b)/(a*x^2-b)^2/(c*x^4+a*x^2-b)^(1/4),x, algorithm=" 
fricas")
 

Output:

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

Sympy [F]

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

integrate(x**4*(a*x**2-2*b)/(a*x**2-b)**2/(c*x**4+a*x**2-b)**(1/4),x)
 

Output:

Integral(x**4*(a*x**2 - 2*b)/((a*x**2 - b)**2*(a*x**2 - b + c*x**4)**(1/4) 
), x)
 

Maxima [F]

\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\int { \frac {{\left (a x^{2} - 2 \, b\right )} x^{4}}{{\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{2}} \,d x } \] Input:

integrate(x^4*(a*x^2-2*b)/(a*x^2-b)^2/(c*x^4+a*x^2-b)^(1/4),x, algorithm=" 
maxima")
 

Output:

integrate((a*x^2 - 2*b)*x^4/((c*x^4 + a*x^2 - b)^(1/4)*(a*x^2 - b)^2), x)
 

Giac [F]

\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\int { \frac {{\left (a x^{2} - 2 \, b\right )} x^{4}}{{\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{2}} \,d x } \] Input:

integrate(x^4*(a*x^2-2*b)/(a*x^2-b)^2/(c*x^4+a*x^2-b)^(1/4),x, algorithm=" 
giac")
 

Output:

integrate((a*x^2 - 2*b)*x^4/((c*x^4 + a*x^2 - b)^(1/4)*(a*x^2 - b)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=-\int \frac {x^4\,\left (2\,b-a\,x^2\right )}{{\left (b-a\,x^2\right )}^2\,{\left (c\,x^4+a\,x^2-b\right )}^{1/4}} \,d x \] Input:

int(-(x^4*(2*b - a*x^2))/((b - a*x^2)^2*(a*x^2 - b + c*x^4)^(1/4)),x)
 

Output:

-int((x^4*(2*b - a*x^2))/((b - a*x^2)^2*(a*x^2 - b + c*x^4)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\left (\int \frac {x^{6}}{\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}} a^{2} x^{4}-2 \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}} a b \,x^{2}+\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}} b^{2}}d x \right ) a -2 \left (\int \frac {x^{4}}{\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}} a^{2} x^{4}-2 \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}} a b \,x^{2}+\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}} b^{2}}d x \right ) b \] Input:

int(x^4*(a*x^2-2*b)/(a*x^2-b)^2/(c*x^4+a*x^2-b)^(1/4),x)
 

Output:

int(x**6/((a*x**2 - b + c*x**4)**(1/4)*a**2*x**4 - 2*(a*x**2 - b + c*x**4) 
**(1/4)*a*b*x**2 + (a*x**2 - b + c*x**4)**(1/4)*b**2),x)*a - 2*int(x**4/(( 
a*x**2 - b + c*x**4)**(1/4)*a**2*x**4 - 2*(a*x**2 - b + c*x**4)**(1/4)*a*b 
*x**2 + (a*x**2 - b + c*x**4)**(1/4)*b**2),x)*b