\(\int \frac {(A+B x^2) (a+b x^2+c x^4)^{3/2}}{(d+e x^2)^{3/2}} \, dx\) [233]

Optimal result

Integrand size = 35, antiderivative size = 1097 \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

-(-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)^(1 
/2)-1/48*(6*A*c*e*(15*c*d^2-e*(-8*a*e+13*b*d))-B*d*(105*c^2*d^2+3*b^2*e^2- 
20*c*e*(-4*a*e+5*b*d)))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/d/e^4/x-1/ 
24*(-6*A*c*e-7*B*b*e+11*B*c*d)*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/e^3 
+1/6*B*c*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/e^2+1/96*(-4*a*c+b^2)^( 
1/2)*(6*A*c*e*(15*c*d^2-e*(-8*a*e+13*b*d))-B*d*(105*c^2*d^2+3*b^2*e^2-20*c 
*e*(-4*a*e+5*b*d)))*(-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)/c/ 
d/e^4/(-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)+1/48*(-4*a*c+b^2)^(1/2)*(B*d*(32*a*c*e^2+3*b^2*e^2-38*b*c*d*e+35*c 
^2*d^2)-6*A*c*e*(5*c*d^2-e*(-8*a*e+5*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^ 
2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*Ellipt 
icF(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)/c/d/e^3/(e*x 
^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+1/8*(-4*a*c+b^2)^(1/2)*(6*A*c*e*(4*a*c*e 
^2+b^2*e^2-6*b*c*d*e+5*c^2*d^2)-B*(35*c^3*d^3+b^3*e^3+3*b*c*e^2*(-4*a*e+3* 
b*d)-9*c^2*d*e*(-4*a*e+5*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(- 
a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticPi(1/2*(1+ 
(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(...
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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