\(\int \frac {(A+B x^2) (d+e x^2)^{7/2}}{(a+b x^2+c x^4)^{5/2}} \, dx\) [248]

Optimal result

Integrand size = 35, antiderivative size = 1562 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Output:

1/3*x*(A*b^2-B*a*b-2*A*a*c+(A*b-2*B*a)*c*x^2)*(e*x^2+d)^(7/2)/a/(-4*a*c+b^ 
2)/(c*x^4+b*x^2+a)^(3/2)+1/3*x*(e*x^2+d)^(5/2)*(a*B*(16*a^2*c*e-8*a*b^2*e+ 
4*a*b*c*d+b^3*d)+A*(-12*a^2*b*c*e+20*a^2*c^2*d+5*a*b^3*e-17*a*b^2*c*d+2*b^ 
4*d)+c*(a*B*(-8*a*b*e+12*a*c*d+b^2*d)+2*A*(-4*a^2*c*e+3*a*b^2*e-8*a*b*c*d+ 
b^3*d))*x^2)/a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)-1/3*(2*A*c^2*(-2*a*e 
+b*d)*(b^2*d^2+4*a*b*d*e-4*a*(a*e^2+2*c*d^2))+a*B*(3*a*b^3*e^3+b^2*c*d*(a* 
e^2+c*d^2)-4*a*b*c*e*(5*a*e^2+6*c*d^2)+4*a*c^2*d*(11*a*e^2+3*c*d^2)))*(e*x 
^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^2/c^2/(-4*a*c+b^2)^2/x+1/3*e*(2*a*B*(1 
2*a*b*c*d*e-b^2*(-a*e^2+c*d^2)-12*a*c*(a*e^2+c*d^2))-A*c*(4*b^3*d^2+11*a*b 
^2*d*e+4*a^2*c*d*e-8*a*b*(a*e^2+4*c*d^2)))*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+ 
a)^(1/2)/a^2/c/(-4*a*c+b^2)^2-1/3*e^2*(a*B*(-8*a*b*e+12*a*c*d+b^2*d)+2*A*( 
-4*a^2*c*e+3*a*b^2*e-8*a*b*c*d+b^3*d))*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a) 
^(1/2)/a^2/(-4*a*c+b^2)^2+1/6*(2*A*c^2*(-2*a*e+b*d)*(b^2*d^2+4*a*b*d*e-4*a 
*(a*e^2+2*c*d^2))+a*B*(3*a*b^3*e^3+b^2*c*d*(a*e^2+c*d^2)-4*a*b*c*e*(5*a*e^ 
2+6*c*d^2)+4*a*c^2*d*(11*a*e^2+3*c*d^2)))*(-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)/a^2/c^2/(-4*a*c+b^2)^(3/2)/(-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)-1/3*2^(1/2)*(A*c^2*(b^3*d 
^3*e-4*a*b*d*e*(8*a*e^2+9*c*d^2)-b^2*(-15*a*d^2*e^2+c*d^4)+4*a*(4*a^2*e...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx \] Input:

Integrate[((A + B*x^2)*(d + e*x^2)^(7/2))/(a + b*x^2 + c*x^4)^(5/2),x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^(7/2))/(a + b*x^2 + c*x^4)^(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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}}dx\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^(7/2))/(a + b*x^2 + c*x^4)^(5/2),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]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="fr 
icas")
 

Output:

integral((B*e^3*x^8 + (3*B*d*e^2 + A*e^3)*x^6 + 3*(B*d^2*e + A*d*e^2)*x^4 
+ A*d^3 + (B*d^3 + 3*A*d^2*e)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d) 
/(c^3*x^12 + 3*b*c^2*x^10 + 3*(b^2*c + a*c^2)*x^8 + (b^3 + 6*a*b*c)*x^6 + 
3*a^2*b*x^2 + 3*(a*b^2 + a^2*c)*x^4 + a^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(7/2)/(c*x**4+b*x**2+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="ma 
xima")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^(7/2)/(c*x^4 + b*x^2 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="gi 
ac")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^(7/2)/(c*x^4 + b*x^2 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{7/2}}{{\left (c\,x^4+b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(7/2))/(a + b*x^2 + c*x^4)^(5/2),x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(7/2))/(a + b*x^2 + c*x^4)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{7/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(7/2)/(c*x^4+b*x^2+a)^(5/2),x)