\(\int \frac {\sqrt {e+f x^2}}{(a+b x^2)^{5/2} \sqrt {c+d x^2}} \, dx\) [416]

Optimal result

Integrand size = 34, antiderivative size = 413 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {e \left (2 b^2 c e+3 a^2 d f-a b (4 d e+c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{3 a^2 \sqrt {c} (b c-a d)^{3/2} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {b e (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{3 a \sqrt {c} (b c-a d)^{3/2} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/3*b*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(3/2)+1/3*e 
*(2*b^2*c*e+3*a^2*d*f-a*b*(c*f+4*d*e))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x 
^2+a))^(1/2)*EllipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f 
+b*e)/(-a*d+b*c)/e)^(1/2))/a^2/c^(1/2)/(-a*d+b*c)^(3/2)/(-a*f+b*e)/(a*(d*x 
^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/3*b*e*(-c*f+d*e)*(d*x^2+c)^(1/2 
)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/(b* 
x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/a/c^(1/2)/(-a*d+b*c)^(3/2) 
/(-a*f+b*e)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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