\(\int \frac {(A+B x^2) (a+b x^2+c x^4)^{3/2}}{(d+e x^2)^{9/2}} \, dx\) [236]

Optimal result

Integrand size = 35, antiderivative size = 1429 \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Output:

-1/7*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/d/e^3/(e*x^2+d 
)^(7/2)+1/35*(B*d*(17*c*d^2-e*(-a*e+9*b*d))-2*A*e*(5*c*d^2-e*(3*a*e+b*d))) 
*x*(c*x^4+b*x^2+a)^(1/2)/d^2/e^3/(e*x^2+d)^(5/2)+1/105*(3*A*e*(5*c^2*d^4-c 
*d^2*e*(-9*a*e+2*b*d)-e^2*(-8*a^2*e^2+5*a*b*d*e+2*b^2*d^2))-B*d*(71*c^2*d^ 
4-c*d^2*e*(-55*a*e+76*b*d)+e^2*(-4*a^2*e^2-a*b*d*e+8*b^2*d^2)))*x*(c*x^4+b 
*x^2+a)^(1/2)/d^3/e^3/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(3/2)-1/105*(6*A*e^4*( 
-2*a*e+b*d)*(b^2*d^2+4*a*b*d*e-4*a*(a*e^2+2*c*d^2))+B*d*(105*c^3*d^6-7*c^2 
*d^4*e*(-26*a*e+25*b*d)+c*d^2*e^2*(37*a^2*e^2-120*a*b*d*e+56*b^2*d^2)+e^3* 
(8*a^3*e^3-5*a^2*b*d*e^2-5*a*b^2*d^2*e+8*b^3*d^3)))*(c*x^4+b*x^2+a)^(1/2)/ 
d^3/e^4/(a*e^2-b*d*e+c*d^2)^2/x/(e*x^2+d)^(1/2)+1/210*(-4*a*c+b^2)^(1/2)*( 
6*A*e^4*(-2*a*e+b*d)*(b^2*d^2+4*a*b*d*e-4*a*(a*e^2+2*c*d^2))+B*d*(105*c^3* 
d^6-7*c^2*d^4*e*(-26*a*e+25*b*d)+c*d^2*e^2*(37*a^2*e^2-120*a*b*d*e+56*b^2* 
d^2)+e^3*(8*a^3*e^3-5*a^2*b*d*e^2-5*a*b^2*d^2*e+8*b^3*d^3)))*(-a*(c+a/x^4+ 
b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*EllipticE(1/2*(1+(b+2*a/x^2)/ 
(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a 
*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/d^4/e^4/(a*e^2-b*d*e+c*d^2)^2/(-a*( 
e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)+1/1 
05*2^(1/2)*(-4*a*c+b^2)^(1/2)*(3*A*e^3*(b^2*d^2+16*a*b*d*e-4*a*(4*a*e^2+5* 
c*d^2))-B*d*(35*c^2*d^4-c*d^2*e*(-31*a*e+28*b*d)-e^2*(-8*a^2*e^2+a*b*d*e+4 
*b^2*d^2)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx \] Input:

Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(9/2),x]
 

Output:

Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}}dx\)

Input:

Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(9/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2260

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_.), x_Symbol] :> Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] 
 /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x\]

Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{9/2}} \,d x \] Input:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(9/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x \] Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(9/2),x)