\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^{5/2} (e+f x^2)^{5/2}} \, dx\) [554]

Optimal result

Integrand size = 34, antiderivative size = 1002 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{5/2}} \, dx=-\frac {d^2 x \sqrt {a+b x^2}}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}-\frac {2 d^2 (b c (2 d e-5 c f)-a d (d e-4 c f)) x \sqrt {a+b x^2}}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}-\frac {f \left (a^2 d^2 f \left (2 d^2 e^2-9 c d e f-c^2 f^2\right )-a b d \left (2 d^3 e^3-5 c d^2 e^2 f-11 c^2 d e f^2-2 c^3 f^3\right )+b^2 c \left (4 d^3 e^3-11 c d^2 e^2 f-c^3 f^3\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 e (b e-a f) (d e-c f)^3 \left (e+f x^2\right )^{3/2}}+\frac {2 \left (a^3 d^2 f^2 \left (d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+c^3 f^3\right )-2 a^2 b d f \left (d^4 e^4-4 c d^3 e^3 f-6 c^2 d^2 e^2 f^2-4 c^3 d e f^3+c^4 f^4\right )-2 b^3 c e \left (d^4 e^4-3 c d^3 e^3 f-3 c^3 d e f^3+c^4 f^4\right )+a b^2 \left (d^5 e^5-c d^4 e^4 f-12 c^2 d^3 e^3 f^2-12 c^3 d^2 e^2 f^3-c^4 d e f^4+c^5 f^5\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} c (b c-a d)^2 e^2 (-b e+a f)^{3/2} (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left (a^2 d^2 f^2 \left (d^2 e^2-18 c d e f+c^2 f^2\right )-2 a b d f \left (d^3 e^3-9 c d^2 e^2 f-9 c^2 d e f^2+c^3 f^3\right )+b^2 \left (d^4 e^4-18 c^2 d^2 e^2 f^2+c^4 f^4\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} c (b c-a d) e (-b e+a f)^{3/2} (d e-c f)^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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