\(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^{5/2} \sqrt {e+f x^2}} \, dx\) [484]

Optimal result

Integrand size = 34, antiderivative size = 951 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {e+f x^2}}{3 a (b c-a d) (b e-a f) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 \left (b^2 c e+5 a^2 d f-2 a b (2 d e+c f)\right ) x \sqrt {e+f x^2}}{3 a^2 (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (2 a^3 b d^3 e f-a^4 d^3 f^2+2 b^4 c^2 e (d e-c f)-a b^3 c \left (9 d^2 e^2-5 c d e f-4 c^2 f^2\right )-a^2 b^2 d \left (d^2 e^2-11 c d e f+11 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 a^2 c (b c-a d)^3 (b e-a f)^2 (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {2 \sqrt {e} \left (a^5 d^4 f^2 (d e-2 c f)+b^5 c^3 e (d e-c f)^2-a b^4 c^2 (d e-c f)^2 (5 d e+2 c f)+a^3 b^2 d^3 e \left (d^2 e^2+8 c d e f-12 c^2 f^2\right )-a^4 b d^3 f \left (2 d^2 e^2+c d e f-6 c^2 f^2\right )-a^2 b^3 c d \left (5 d^3 e^3-12 c d^2 e^2 f+12 c^2 d e f^2-6 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c^2 (b c-a d)^4 (b e-a f)^2 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (2 a^3 b d^4 e f-a^4 d^4 f^2+18 a b^3 c d^2 e (d e-c f)-b^4 c^2 (d e-c f)^2-a^2 b^2 d^2 \left (d^2 e^2+18 c d e f-18 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c (b c-a d)^4 (b e-a f) (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/3*b^2*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(3/2)/(d*x^2+c 
)^(3/2)+2/3*b^2*(b^2*c*e+5*a^2*d*f-2*a*b*(c*f+2*d*e))*x*(f*x^2+e)^(1/2)/a^ 
2/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*d*(2*a^3*b 
*d^3*e*f-a^4*d^3*f^2+2*b^4*c^2*e*(-c*f+d*e)-a*b^3*c*(-4*c^2*f^2-5*c*d*e*f+ 
9*d^2*e^2)-a^2*b^2*d*(11*c^2*f^2-11*c*d*e*f+d^2*e^2))*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)^2/(-c*f+d*e)/(d*x^2+c)^(3/2)+2 
/3*e^(1/2)*(a^5*d^4*f^2*(-2*c*f+d*e)+b^5*c^3*e*(-c*f+d*e)^2-a*b^4*c^2*(-c* 
f+d*e)^2*(2*c*f+5*d*e)+a^3*b^2*d^3*e*(-12*c^2*f^2+8*c*d*e*f+d^2*e^2)-a^4*b 
*d^3*f*(-6*c^2*f^2+c*d*e*f+2*d^2*e^2)-a^2*b^3*c*d*(-6*c^3*f^3+12*c^2*d*e*f 
^2-12*c*d^2*e^2*f+5*d^3*e^3))*(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)^2/(-c*f+d*e)^(3/2)/(c*( 
b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/3*e^(1/2)*(2*a^3*b*d^4*e*f-a 
^4*d^4*f^2+18*a*b^3*c*d^2*e*(-c*f+d*e)-b^4*c^2*(-c*f+d*e)^2-a^2*b^2*d^2*(- 
18*c^2*f^2+18*c*d*e*f+d^2*e^2))*(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/(-a*f+b*e)/(-c*f+d*e)^(3/2)/(c*(b* 
x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*Sqrt[e + f*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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}dx\)

Input:

Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*Sqrt[e + f*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]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (x^{2} d +c \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

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

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^3*d^3*f*x^14 + 
 (b^3*d^3*e + 3*(b^3*c*d^2 + a*b^2*d^3)*f)*x^12 + 3*((b^3*c*d^2 + a*b^2*d^ 
3)*e + (b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f)*x^10 + (3*(b^3*c^2*d + 3 
*a*b^2*c*d^2 + a^2*b*d^3)*e + (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a 
^3*d^3)*f)*x^8 + a^3*c^3*e + ((b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a 
^3*d^3)*e + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f)*x^6 + 3*((a*b^2*c 
^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e + (a^2*b*c^3 + a^3*c^2*d)*f)*x^4 + (a^3* 
c^3*f + 3*(a^2*b*c^3 + a^3*c^2*d)*e)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}d x \] Input:

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

Output:

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