\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{(e+f x^2)^{5/2}} \, dx\) [537]

Optimal result

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

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

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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