\(\int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [343]

Optimal result

Integrand size = 37, antiderivative size = 608 \[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\left (\frac {8 b^2 c e^2}{a}+b e (7 d e-3 c f)+a \left (\frac {8 d^2 e^2}{c}-3 d e f-2 c f^2\right )\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a^2 c^2 e x \sqrt {e+f x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{5 a c x^5}+\frac {(4 b c e+4 a d e-a c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 a^2 c^2 e x^3}-\frac {\sqrt {-b e+a f} \left (8 b^2 c^2 e^2+a b c e (7 d e-3 c f)+a^2 \left (8 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 a^{7/2} c^2 e^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\sqrt {-b e+a f} \left (8 b^2 c^2 e+a^2 d (4 d e-c f)+a b c (3 d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 a^{7/2} c^2 e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/15*(8*b^2*c*e^2/a+b*e*(-3*c*f+7*d*e)+a*(8*d^2*e^2/c-3*d*e*f-2*c*f^2))*( 
b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e/x/(f*x^2+e)^(1/2)-1/5*(b*x^2+a)^( 
1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/c/x^5+1/15*(-a*c*f+4*a*d*e+4*b*c*e) 
*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a^2/c^2/e/x^3-1/15*(a*f-b 
*e)^(1/2)*(8*b^2*c^2*e^2+a*b*c*e*(-3*c*f+7*d*e)+a^2*(-2*c^2*f^2-3*c*d*e*f+ 
8*d^2*e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticE((a*f 
-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a 
^(7/2)/c^2/e^2/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/15*(a*f-b 
*e)^(1/2)*(8*b^2*c^2*e+a^2*d*(-c*f+4*d*e)+a*b*c*(c*f+3*d*e))*(b*x^2+a)^(1/ 
2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((a*f-b*e)^(1/2)*x/a^(1/2)/(f* 
x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(7/2)/c^2/e/(d*x^2+c)^(1 
/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx \] Input:

Integrate[Sqrt[e + f*x^2]/(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

Integrate[Sqrt[e + f*x^2]/(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x]
 

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.

\(\displaystyle \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}dx\)

Input:

Int[Sqrt[e + f*x^2]/(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [F]

\[\int \frac {\sqrt {f \,x^{2}+e}}{x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}d x\]

Input:

int((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Output:

int((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}} \,d x } \] Input:

integrate((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm 
="fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b*d*x^10 + (b*c 
+ a*d)*x^8 + a*c*x^6), x)
 

Sympy [F]

\[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {e + f x^{2}}}{x^{6} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate((f*x**2+e)**(1/2)/x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(sqrt(e + f*x**2)/(x**6*sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}} \,d x } \] Input:

integrate((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm 
="maxima")
 

Output:

integrate(sqrt(f*x^2 + e)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^6), x)
 

Giac [F]

\[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}} \,d x } \] Input:

integrate((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm 
="giac")
 

Output:

integrate(sqrt(f*x^2 + e)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {f\,x^2+e}}{x^6\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((e + f*x^2)^(1/2)/(x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((e + f*x^2)^(1/2)/(x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {e+f x^2}}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}}{x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}d x \] Input:

int((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Output:

int((f*x^2+e)^(1/2)/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)