\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^{3/2} (e+f x^2)^{5/2}} \, dx\) [556]

Optimal result

Integrand size = 34, antiderivative size = 960 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\frac {b^2 x}{a (b c-a d) (b e-a f) \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}+\frac {d \left (a b d^2 e-a^2 d^2 f+b^2 c (d e-c f)\right ) x \sqrt {a+b x^2}}{a c (b c-a d)^2 (b e-a f) (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}+\frac {f \left (3 b^3 c e (d e-c f)^2+a^3 d^2 f^2 (3 d e+c f)-2 a^2 b d f \left (3 d^2 e^2+c^2 f^2\right )+a b^2 \left (3 d^3 e^3+c^3 f^3\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c (b c-a d)^2 e (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )^{3/2}}-\frac {\left (3 b^4 c e^2 (d e-c f)^3-a^4 d^2 f^3 \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )+a^3 b d f^2 \left (9 d^3 e^3+12 c d^2 e^2 f+7 c^2 d e f^2-4 c^3 f^3\right )+a b^3 e \left (3 d^4 e^4+12 c^3 d e f^3-7 c^4 f^4\right )-a^2 b^2 f \left (9 d^4 e^4+24 c^2 d^2 e^2 f^2-7 c^3 d e f^3-2 c^4 f^4\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{3/2} (b c-a d)^2 e^2 (-b e+a f)^{5/2} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left (3 b^3 e (d e-c f)^3+a^3 d^2 f^3 (9 d e-c f)-a^2 b d f^2 \left (15 d^2 e^2+3 c d e f-2 c^2 f^2\right )+a b^2 f \left (3 d^3 e^3+9 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{3/2} (b c-a d) e (-b e+a f)^{5/2} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

b^2*x/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3 
/2)+d*(a*b*d^2*e-a^2*d^2*f+b^2*c*(-c*f+d*e))*x*(b*x^2+a)^(1/2)/a/c/(-a*d+b 
*c)^2/(-a*f+b*e)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2)+1/3*f*(3*b^3*c 
*e*(-c*f+d*e)^2+a^3*d^2*f^2*(c*f+3*d*e)-2*a^2*b*d*f*(c^2*f^2+3*d^2*e^2)+a* 
b^2*(c^3*f^3+3*d^3*e^3))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/(-a*d+b*c)^ 
2/e/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x^2+e)^(3/2)-1/3*(3*b^4*c*e^2*(-c*f+d*e)^ 
3-a^4*d^2*f^3*(-2*c^2*f^2+7*c*d*e*f+3*d^2*e^2)+a^3*b*d*f^2*(-4*c^3*f^3+7*c 
^2*d*e*f^2+12*c*d^2*e^2*f+9*d^3*e^3)+a*b^3*e*(-7*c^4*f^4+12*c^3*d*e*f^3+3* 
d^4*e^4)-a^2*b^2*f*(-2*c^4*f^4-7*c^3*d*e*f^3+24*c^2*d^2*e^2*f^2+9*d^4*e^4) 
)*(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^(3/2)/(-a 
*d+b*c)^2/e^2/(a*f-b*e)^(5/2)/(-c*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/ 
(f*x^2+e))^(1/2)+1/3*(3*b^3*e*(-c*f+d*e)^3+a^3*d^2*f^3*(-c*f+9*d*e)-a^2*b* 
d*f^2*(-2*c^2*f^2+3*c*d*e*f+15*d^2*e^2)+a*b^2*f*(-c^3*f^3-3*c^2*d*e*f^2+9* 
c*d^2*e^2*f+3*d^3*e^3))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*El 
lipticF((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^(3/2)/(-a*d+b*c)/e/(a*f-b*e)^(5/2)/(-c*f+d*e)^3/(d*x^2+c)^(1/ 
2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(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]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (x^{2} d +c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(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^2*d^2*f^3*x^14 
 + (3*b^2*d^2*e*f^2 + 2*(b^2*c*d + a*b*d^2)*f^3)*x^12 + (3*b^2*d^2*e^2*f + 
 6*(b^2*c*d + a*b*d^2)*e*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^3)*x^10 + 
 (b^2*d^2*e^3 + 6*(b^2*c*d + a*b*d^2)*e^2*f + 3*(b^2*c^2 + 4*a*b*c*d + a^2 
*d^2)*e*f^2 + 2*(a*b*c^2 + a^2*c*d)*f^3)*x^8 + a^2*c^2*e^3 + (a^2*c^2*f^3 
+ 2*(b^2*c*d + a*b*d^2)*e^3 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f + 6* 
(a*b*c^2 + a^2*c*d)*e*f^2)*x^6 + (3*a^2*c^2*e*f^2 + (b^2*c^2 + 4*a*b*c*d + 
 a^2*d^2)*e^3 + 6*(a*b*c^2 + a^2*c*d)*e^2*f)*x^4 + (3*a^2*c^2*e^2*f + 2*(a 
*b*c^2 + a^2*c*d)*e^3)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x, algorithm=" 
maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x, algorithm=" 
giac")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)),x)
 

Output:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)