\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 (e+f x^2)^{5/2}} \, dx\) [336]

Optimal result

Integrand size = 37, antiderivative size = 592 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e x^3 \left (e+f x^2\right )^{3/2}}-\frac {\left (\frac {b}{a}+\frac {d}{c}-\frac {6 f}{e}\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e x \left (e+f x^2\right )^{3/2}}-\frac {f (b c e+a d e-8 a c f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e^3 \left (e+f x^2\right )^{3/2}}+\frac {\left (b^2 c e^2 (d e-c f)+a b e \left (d^2 e^2-16 c d e f+16 c^2 f^2\right )-a^2 f \left (d^2 e^2-16 c d e f+16 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{3/2} e^4 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b c-a d) (a f (7 d e-8 c f)-b e (d e-c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{3/2} e^3 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/x^3/(f*x^2+e)^(3/2)-1/3*(b/a+d/c-6* 
f/e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/x/(f*x^2+e)^(3/2)-1/3*f*(-8*a*c*f+a 
*d*e+b*c*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^3/(f*x^2+e)^(3/2)+1/3* 
(b^2*c*e^2*(-c*f+d*e)+a*b*e*(16*c^2*f^2-16*c*d*e*f+d^2*e^2)-a^2*f*(16*c^2* 
f^2-16*c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*E 
llipticE((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^(3/2)/e^4/(a*f-b*e)^(1/2)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x 
^2+a)/a/(f*x^2+e))^(1/2)+1/3*(-a*d+b*c)*(a*f*(-8*c*f+7*d*e)-b*e*(-c*f+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^(3/2)/e^3 
/(a*f-b*e)^(1/2)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2 
)
 

Mathematica [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^4*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^4*(e + f*x^2)^(5/2)), 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^4*(e + f*x^2)^(5/2)),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]

\[\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{x^{4} \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x, algorithm 
="fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(f^3*x^10 + 3*e*f 
^2*x^8 + 3*e^2*f*x^6 + e^3*x^4), x)
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{x^{4} \left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/x**4/(f*x**2+e)**(5/2),x)
 

Output:

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)/(x**4*(e + f*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x, algorithm 
="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/((f*x^2 + e)^(5/2)*x^4), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x, algorithm 
="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/((f*x^2 + e)^(5/2)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{x^4\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(x^4*(e + f*x^2)^(5/2)),x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(x^4*(e + f*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{x^{4} \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e)^(5/2),x)