\(\int \frac {(a+b x^2)^{3/2} (e+f x^2)^{3/2}}{(c+d x^2)^{11/2}} \, dx\) [433]

Optimal result

Integrand size = 34, antiderivative size = 1218 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx =\text {Too large to display} \] Output:

-1/9*(-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)^(9/2)+2/63*(b*c*(-6*c*f+d*e)+a*d*(c*f+4*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+ 
e)^(1/2)/c^2/d^2/(d*x^2+c)^(7/2)+1/315*(b^2*c^2*(-15*c^2*f^2+2*c*d*e*f+10* 
d^2*e^2)-a^2*d^2*(-10*c^2*f^2-35*c*d*e*f+48*d^2*e^2)+a*b*c*d*(2*c^2*f^2-31 
*c*d*e*f+35*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)^(5/2)+2/315*(a*b^2*c^2*d*(-4*c^3*f^3+14*c^2*d*e*f^2- 
31*c*d^2*e^2*f+15*d^3*e^3)-a^2*b*c*d^2*(4*c^3*f^3+31*c^2*d*e*f^2-95*c*d^2* 
e^2*f+54*d^3*e^3)+b^3*c^3*(5*c^3*f^3-4*c^2*d*e*f^2-4*c*d^2*e^2*f+5*d^3*e^3 
)+a^3*d^3*(5*c^3*f^3+15*c^2*d*e*f^2-54*c*d^2*e^2*f+32*d^3*e^3))*x*(b*x^2+a 
)^(1/2)*(f*x^2+e)^(1/2)/c^4/d^2/(-a*d+b*c)^2/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+ 
1/315*e^(1/2)*(2*b^4*c^4*e^3*(-9*c*f+5*d*e)+a*b^3*c^3*e^2*(90*c^2*f^2-83*c 
*d*e*f+25*d^2*e^2)-3*a^2*b^2*c^2*e*(-30*c^3*f^3+186*c^2*d*e*f^2-221*c*d^2* 
e^2*f+81*d^3*e^3)+a^3*b*c*(-18*c^4*f^4-83*c^3*d*e*f^3+663*c^2*d^2*e^2*f^2- 
858*c*d^3*e^3*f+328*d^4*e^4)-a^4*d*(-10*c^4*f^4-25*c^3*d*e*f^3+243*c^2*d^2 
*e^2*f^2-328*c*d^3*e^3*f+128*d^4*e^4))*(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^5/(-a*d+b*c)^3/(-c*f+d*e)^(5/2)/(c*(b*x^2+a) 
/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/315*e^(1/2)*(-a*f+b*e)*(b^3*c^3*e^2* 
(-9*c*f+5*d*e)-3*a*b^2*c^2*e*(27*c^2*f^2-66*c*d*e*f+35*d^2*e^2)-2*a^3*d*(5 
*c^3*f^3+15*c^2*d*e*f^2-54*c*d^2*e^2*f+32*d^3*e^3)+3*a^2*b*c*(6*c^3*f^3...
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2),x]
 

Output:

Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2), 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}}dx\)

Input:

Int[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2),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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {11}{2}}}d x\]

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x, algorithm="f 
ricas")
 

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^6*x^12 + 6*c*d^5*x^10 + 15*c^2*d^4*x^8 + 20*c^3*d^3*x^ 
6 + 15*c^4*d^2*x^4 + 6*c^5*d*x^2 + c^6), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)*(f*x**2+e)**(3/2)/(d*x**2+c)**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x, algorithm="m 
axima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(11/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x, algorithm="g 
iac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{11/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {11}{2}}}d x \] Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)