\(\int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{(c+d x^2)^{5/2}} \, dx\) [386]

Optimal result

Integrand size = 34, antiderivative size = 388 \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} (b c e-2 a d e+a c f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 c^2 (b c-a d) \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 c (b c-a d) \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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