\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx\) [308]

Optimal result

Integrand size = 37, antiderivative size = 821 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=-\frac {a \sqrt {c+d x^2} \sqrt {e+f x^2}}{7 x^7 \sqrt {a+b x^2}}-\frac {(6 b c e+a d e+a c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}{35 c e x^5 \sqrt {a+b x^2}}+\frac {\left (b^2 c e-5 a b (d e+c f)+2 a^2 \left (\frac {2 d^2 e}{c}-d f+\frac {2 c f^2}{e}\right )\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 a c e x^3 \sqrt {a+b x^2}}-\frac {\left (4 b^3 c^3 e^3-3 a b^2 c^2 e^2 (d e+c f)-a^2 b c e \left (9 d^2 e^2-8 c d e f+9 c^2 f^2\right )+a^3 \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 a^2 c^3 e^3 x \sqrt {a+b x^2}}-\frac {\sqrt {b c-a d} \left (8 b^3 c^3 e^3-5 a b^2 c^2 e^2 (d e+c f)-a^2 b c e \left (5 d^2 e^2-6 c d e f+5 c^2 f^2\right )+a^3 \left (8 d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+8 c^3 f^3\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{105 a^3 c^{7/2} e^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {b c-a d} (d e-c f) \left (4 b^2 c^2 e^2+a b c e (d e-2 c f)-a^2 \left (8 d^2 e^2-c d e f-4 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{105 a^2 c^{7/2} e^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/7*a*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^7/(b*x^2+a)^(1/2)-1/35*(a*c*f+a*d 
*e+6*b*c*e)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/c/e/x^5/(b*x^2+a)^(1/2)+1/105* 
(b^2*c*e-5*a*b*(c*f+d*e)+2*a^2*(2*d^2*e/c-d*f+2*c*f^2/e))*(d*x^2+c)^(1/2)* 
(f*x^2+e)^(1/2)/a/c/e/x^3/(b*x^2+a)^(1/2)-1/105*(4*b^3*c^3*e^3-3*a*b^2*c^2 
*e^2*(c*f+d*e)-a^2*b*c*e*(9*c^2*f^2-8*c*d*e*f+9*d^2*e^2)+a^3*(8*c^3*f^3-5* 
c^2*d*e*f^2-5*c*d^2*e^2*f+8*d^3*e^3))*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a^2/ 
c^3/e^3/x/(b*x^2+a)^(1/2)-1/105*(-a*d+b*c)^(1/2)*(8*b^3*c^3*e^3-5*a*b^2*c^ 
2*e^2*(c*f+d*e)-a^2*b*c*e*(5*c^2*f^2-6*c*d*e*f+5*d^2*e^2)+a^3*(8*c^3*f^3-5 
*c^2*d*e*f^2-5*c*d^2*e^2*f+8*d^3*e^3))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x 
^2+a))^(1/2)*EllipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f 
+b*e)/(-a*d+b*c)/e)^(1/2))/a^3/c^(7/2)/e^2/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2) 
/(f*x^2+e)^(1/2)-1/105*(-a*d+b*c)^(1/2)*(-c*f+d*e)*(4*b^2*c^2*e^2+a*b*c*e* 
(-2*c*f+d*e)-a^2*(-4*c^2*f^2-c*d*e*f+8*d^2*e^2))*(d*x^2+c)^(1/2)*(a*(f*x^2 
+e)/e/(b*x^2+a))^(1/2)*EllipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2 
),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/a^2/c^(7/2)/e^2/(a*(d*x^2+c)/c/(b*x^2 
+a))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx \] Input:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/x^8,x]
 

Output:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/x^8, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8}dx\)

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/x^8,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]

\[\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}{x^{8}}d x\]

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x)
 

Fricas [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{x^{8}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}{x^{8}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**(1/2)/x**8,x)
 

Output:

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*sqrt(e + f*x**2)/x**8, x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{x^{8}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x, algorithm 
="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{x^{8}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x, algorithm 
="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}}{x^8} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2))/x^8,x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^8} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{x^{8}}d x \] Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^8,x)