\(\int \frac {\sqrt {e+f x^2}}{(a+b x^2)^{5/2} (c+d x^2)^{5/2}} \, dx\) [418]

Optimal result

Integrand size = 34, antiderivative size = 786 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {b x \sqrt {e+f x^2}}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {b \left (2 b^2 c e+7 a^2 d f-a b (8 d e+c f)\right ) x \sqrt {e+f x^2}}{3 a^2 (b c-a d)^2 (b e-a f) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (2 b^3 c^2 e+a^3 d^2 f-a^2 b d (d e-8 c f)-a b^2 c (9 d e+c f)\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 a^2 c (b c-a d)^3 (b e-a f) \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (2 b^4 c^3 e (d e-c f)-a^4 d^3 f (2 d e-c f)+a^3 b d^2 \left (2 d^2 e^2+9 c d e f-9 c^2 f^2\right )-a b^3 c^2 \left (10 d^2 e^2-9 c d e f-c^2 f^2\right )-a^2 b^2 c d \left (10 d^2 e^2-18 c d e f+9 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c^2 (b c-a d)^4 (b e-a f) \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (a^3 d^3 f-b^3 c^2 (d e-c f)+9 a b^2 c d (2 d e-c f)-a^2 b d^2 (d e+9 c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c (b c-a d)^4 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/3*b*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)+1/3*b 
*(2*b^2*c*e+7*a^2*d*f-a*b*(c*f+8*d*e))*x*(f*x^2+e)^(1/2)/a^2/(-a*d+b*c)^2/ 
(-a*f+b*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*d*(2*b^3*c^2*e+a^3*d^2*f-a^ 
2*b*d*(-8*c*f+d*e)-a*b^2*c*(c*f+9*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/ 
a^2/c/(-a*d+b*c)^3/(-a*f+b*e)/(d*x^2+c)^(3/2)+1/3*e^(1/2)*(2*b^4*c^3*e*(-c 
*f+d*e)-a^4*d^3*f*(-c*f+2*d*e)+a^3*b*d^2*(-9*c^2*f^2+9*c*d*e*f+2*d^2*e^2)- 
a*b^3*c^2*(-c^2*f^2-9*c*d*e*f+10*d^2*e^2)-a^2*b^2*c*d*(9*c^2*f^2-18*c*d*e* 
f+10*d^2*e^2))*(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))/a^2/c^2/(-a*d+b*c)^4/(-a*f+b*e)/(-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^3*d^3*f-b^3*c^2*(-c*f+d*e)+9*a*b 
^2*c*d*(-c*f+2*d*e)-a^2*b*d^2*(9*c*f+d*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))/a^2/c/(-a*d+b*c)^4/(-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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(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)/(b^3*d^3*x^12 + 3 
*(b^3*c*d^2 + a*b^2*d^3)*x^10 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)* 
x^8 + (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*x^6 + a^3*c^3 + 
3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*x^4 + 3*(a^2*b*c^3 + a^3*c^2*d)* 
x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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