\(\int \frac {(a+b x^2)^{3/2} (e+f x^2)^{3/2}}{(c+d x^2)^{9/2}} \, dx\) [432]

Optimal result

Integrand size = 34, antiderivative size = 735 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=-\frac {(b c-a d) (d e-c f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 c d^2 \left (c+d x^2\right )^{7/2}}+\frac {2 (b c (d e-5 c f)+a d (3 d e+c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (b^2 c^2 \left (2 d^2 e^2+2 c d e f-5 c^2 f^2\right )-a^2 d^2 \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )+a b c d \left (5 d^2 e^2-5 c d e f+2 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^3 d^2 (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {2 \sqrt {e} (b c e-2 a d e+a c f) \left (b^2 c^2 e^2+2 a b c e (2 d e-3 c f)-a^2 \left (4 d^2 e^2-4 c d e f-c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{35 c^4 (b c-a d)^2 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) \left (b^2 c^2 e^2-a b c e (11 d e-9 c f)+a^2 \left (8 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{35 c^3 (b c-a d)^2 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/7*(-a*d+b*c)*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d^2/(d*x^2+ 
c)^(7/2)+2/35*(b*c*(-5*c*f+d*e)+a*d*(c*f+3*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+ 
e)^(1/2)/c^2/d^2/(d*x^2+c)^(5/2)+1/35*(b^2*c^2*(-5*c^2*f^2+2*c*d*e*f+2*d^2 
*e^2)-a^2*d^2*(-2*c^2*f^2-5*c*d*e*f+8*d^2*e^2)+a*b*c*d*(2*c^2*f^2-5*c*d*e* 
f+5*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^3/d^2/(-a*d+b*c)/(-c*f+d 
*e)/(d*x^2+c)^(3/2)+2/35*e^(1/2)*(a*c*f-2*a*d*e+b*c*e)*(b^2*c^2*e^2+2*a*b* 
c*e*(-3*c*f+2*d*e)-a^2*(-c^2*f^2-4*c*d*e*f+4*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))/c^4/(-a*d+b*c)^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)-1/35*e^(1/2)*(-a*f+b*e 
)*(b^2*c^2*e^2-a*b*c*e*(-9*c*f+11*d*e)+a^2*(-2*c^2*f^2-5*c*d*e*f+8*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))/c^3/(-a* 
d+b*c)^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)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]
 

Output:

Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2), 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}}dx\)

Input:

Int[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {9}{2}}}d x\]

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="fr 
icas")
 

Output:

integral((b*f*x^4 + (b*e + a*f)*x^2 + a*e)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c) 
*sqrt(f*x^2 + e)/(d^5*x^10 + 5*c*d^4*x^8 + 10*c^2*d^3*x^6 + 10*c^3*d^2*x^4 
 + 5*c^4*d*x^2 + c^5), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)*(f*x**2+e)**(3/2)/(d*x**2+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x \] Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)