\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^8 \sqrt {e+f x^2}} \, dx\) [323]

Optimal result

Integrand size = 37, antiderivative size = 828 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^8 \sqrt {e+f x^2}} \, dx=-\frac {\left (8 b^3 c^3 e^3-a b^2 c^2 e^2 (5 d e-9 c f)-a^2 b c e \left (5 d^2 e^2+8 c d e f-16 c^2 f^2\right )+a^3 \left (8 d^3 e^3+9 c d^2 e^2 f+16 c^2 d e f^2-48 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 a^3 c^3 e^3 x \sqrt {e+f x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{7 e x^7}-\frac {\left (\frac {b}{a}+\frac {d}{c}-\frac {6 f}{e}\right ) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{35 e x^5}+\frac {\left (\frac {4 b^2 c e}{a}-2 b d e+\frac {4 a d^2 e}{c}+5 b c f+5 a d f-\frac {24 a c f^2}{e}\right ) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 a c e^2 x^3}-\frac {\sqrt {-b e+a f} \left (8 b^3 c^3 e^3-a b^2 c^2 e^2 (5 d e-9 c f)-a^2 b c e \left (5 d^2 e^2+8 c d e f-16 c^2 f^2\right )+a^3 \left (8 d^3 e^3+9 c d^2 e^2 f+16 c^2 d e f^2-48 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{105 a^{7/2} c^2 e^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b c-a d) \sqrt {-b e+a f} \left (8 b^2 c^2 e^2-a b c e (d e-13 c f)-a^2 \left (4 d^2 e^2+5 c d e f-24 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{105 a^{7/2} c^2 e^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/105*(8*b^3*c^3*e^3-a*b^2*c^2*e^2*(-9*c*f+5*d*e)-a^2*b*c*e*(-16*c^2*f^2+ 
8*c*d*e*f+5*d^2*e^2)+a^3*(-48*c^3*f^3+16*c^2*d*e*f^2+9*c*d^2*e^2*f+8*d^3*e 
^3))*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^3/c^3/e^3/x/(f*x^2+e)^(1/2)-1/7*(b* 
x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/e/x^7-1/35*(b/a+d/c-6*f/e)*(b 
*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/e/x^5+1/105*(4*b^2*c*e/a-2*b 
*d*e+4*a*d^2*e/c+5*b*c*f+5*a*d*f-24*a*c*f^2/e)*(b*x^2+a)^(1/2)*(d*x^2+c)^( 
1/2)*(f*x^2+e)^(1/2)/a/c/e^2/x^3-1/105*(a*f-b*e)^(1/2)*(8*b^3*c^3*e^3-a*b^ 
2*c^2*e^2*(-9*c*f+5*d*e)-a^2*b*c*e*(-16*c^2*f^2+8*c*d*e*f+5*d^2*e^2)+a^3*( 
-48*c^3*f^3+16*c^2*d*e*f^2+9*c*d^2*e^2*f+8*d^3*e^3))*(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^4/(d*x^2+c)^(1/2)/(e 
*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/105*(-a*d+b*c)*(a*f-b*e)^(1/2)*(8*b^2*c^2* 
e^2-a*b*c*e*(-13*c*f+d*e)-a^2*(-24*c^2*f^2+5*c*d*e*f+4*d^2*e^2))*(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^3/(d*x^2 
+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 450

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^8/(f*x^2+e)^(1/2),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. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^8 \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{x^8\,\sqrt {f\,x^2+e}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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