\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)^{3/2}}{(c+d x^2)^{9/2}} \, dx\) [438]

Optimal result

Integrand size = 34, antiderivative size = 1161 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Output:

1/7*(-a*d+b*c)^2*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d^3/(d*x^2 
+c)^(7/2)-1/35*(-a*d+b*c)*(b*c*(-17*c*f+9*d*e)+2*a*d*(c*f+3*d*e))*x*(b*x^2 
+a)^(1/2)*(f*x^2+e)^(1/2)/c^2/d^3/(d*x^2+c)^(5/2)+1/105*(a*b*c*d*(-20*c^2* 
f^2+c*d*e*f+13*d^2*e^2)+3*a^2*d^2*(-2*c^2*f^2-5*c*d*e*f+8*d^2*e^2)+b^2*c^2 
*(71*c^2*f^2-76*c*d*e*f+8*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^3/ 
d^3/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/105*e^(1/2)*(a*b^2*c^2*d*(-70*c^3*f^3+119 
*c^2*d*e*f^2-40*c*d^2*e^2*f+9*d^3*e^3)+a^2*b*c*d^2*(-14*c^3*f^3+23*c^2*d*e 
*f^2-43*c*d^2*e^2*f+16*d^3*e^3)-6*a^3*d^3*(c^3*f^3+2*c^2*d*e*f^2-12*c*d^2* 
e^2*f+8*d^3*e^3)+b^3*c^3*(105*c^3*f^3-175*c^2*d*e*f^2+56*c*d^2*e^2*f+8*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))/c^4/ 
d^4/(-a*d+b*c)/(-c*f+d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^ 
(1/2)-1/105*e^(1/2)*(105*b^4*c^5*f^2*(-c*f+d*e)+3*a^4*d^4*f*(-2*c^2*f^2-5* 
c*d*e*f+8*d^2*e^2)+a^2*b^2*c*d^2*(-70*c^3*f^3+91*c^2*d*e*f^2-17*c*d^2*e^2* 
f+5*d^3*e^3)-a^3*b*d^3*(14*c^3*f^3-19*c^2*d*e*f^2-10*c*d^2*e^2*f+24*d^3*e^ 
3)+a*b^3*c^2*d*(210*c^3*f^3-245*c^2*d*e*f^2+28*c*d^2*e^2*f+4*d^3*e^3))*(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/c^3/d^5/(-a* 
d+b*c)/(-c*f+d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+b^ 
3*c*e^(1/2)*f^2*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*Ellipti...
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(5/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]
 

Output:

Integrate[((a + b*x^2)^(5/2)*(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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}}dx\)

Input:

Int[((a + b*x^2)^(5/2)*(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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {9}{2}}}d x\]

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**(3/2)/(d*x**2+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)