\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{3/2}}{x^8} \, dx\) [316]

Optimal result

Integrand size = 37, antiderivative size = 803 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{x^8} \, dx=-\frac {(b c e+a d e-2 a c f) \left (8 b^2 c^2 e^2-a b c e (13 d e+3 c f)+a^2 \left (8 d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 a^3 c^3 e x \sqrt {e+f x^2}}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{7 x^7}-\frac {f \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 x^5}-\frac {(2 b c e+2 a d e-19 a c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{70 a c x^5}+\frac {\left (4 b^2 c^2 e^2-a b c e (2 d e+9 c f)+a^2 \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{105 a^2 c^2 e x^3}-\frac {\sqrt {-b e+a f} (b c e+a d e-2 a c f) \left (8 b^2 c^2 e^2-a b c e (13 d e+3 c f)+a^2 \left (8 d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{105 a^{7/2} c^2 e^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b c-a d) \sqrt {-b e+a f} \left (8 b^2 c^2 e^2-a b c e (d e+15 c f)-a^2 \left (4 d^2 e^2-9 c d e f-3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{105 a^{7/2} c^2 e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/105*(-2*a*c*f+a*d*e+b*c*e)*(8*b^2*c^2*e^2-a*b*c*e*(3*c*f+13*d*e)+a^2*(3 
*c^2*f^2-3*c*d*e*f+8*d^2*e^2))*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^3/c^3/e/x 
/(f*x^2+e)^(1/2)-1/7*e*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^7 
-1/2*f*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/x^5-1/70*(-19*a*c*f 
+2*a*d*e+2*b*c*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/c/x^5+ 
1/105*(4*b^2*c^2*e^2-a*b*c*e*(9*c*f+2*d*e)+a^2*(-3*c^2*f^2-9*c*d*e*f+4*d^2 
*e^2))*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a^2/c^2/e/x^3-1/105 
*(a*f-b*e)^(1/2)*(-2*a*c*f+a*d*e+b*c*e)*(8*b^2*c^2*e^2-a*b*c*e*(3*c*f+13*d 
*e)+a^2*(3*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)+1/105*(-a*d+b*c)*(a*f-b*e)^(1/2)*(8*b^2*c^2*e^2-a*b*c*e* 
(15*c*f+d*e)-a^2*(-3*c^2*f^2-9*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/(d*x^2+c)^(1/2)/(e*(b* 
x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/x^8,x]
 

Output:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{x^8} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{x^8}dx\)

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/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]

\[\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{x^{8}}d x\]

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x)
 

Fricas [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x, algorithm 
="fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)/x^8, x)
 

Sympy [F]

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

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**(3/2)/x**8,x)
 

Output:

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**(3/2)/x**8, x)
 

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x, algorithm 
="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)/x^8, x)
 

Giac [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x, algorithm 
="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)/x^8, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{x^8} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{3/2}}{x^8} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(3/2))/x^8,x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(3/2))/x^8, x)
 

Reduce [F]

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

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(3/2)/x^8,x)