\(\int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} (e+f x^2)^{5/2}} \, dx\) [538]

Optimal result

Integrand size = 34, antiderivative size = 420 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=-\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e (d e-c f) \left (e+f x^2\right )^{3/2}}-\frac {c (2 a f (2 d e-c f)-b e (3 d e-c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e^2 \sqrt {-b e+a f} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {(b c-a d) (3 d e-c f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e \sqrt {-b e+a f} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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