\(\int \frac {(A+B x^2) (a+b x^2+c x^4)^{3/2}}{(d+e x^2)^{5/2}} \, dx\) [234]

Optimal result

Integrand size = 35, antiderivative size = 1035 \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Output:

-1/3*(-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 
)^(3/2)+1/3*(B*d*(9*c*d^2-e*(-a*e+5*b*d))-2*A*e*(3*c*d^2-e*(a*e+b*d)))*x*( 
c*x^4+b*x^2+a)^(1/2)/d^2/e^3/(e*x^2+d)^(1/2)-1/24*(B*d*(105*c*d^2-e*(-8*a* 
e+55*b*d))-4*A*e*(15*c*d^2-4*e*(a*e+b*d)))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a) 
^(1/2)/d^2/e^4/x+1/4*B*c*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/e^3+1/48* 
(-4*a*c+b^2)^(1/2)*(B*d*(105*c*d^2-e*(-8*a*e+55*b*d))-4*A*e*(15*c*d^2-4*e* 
(a*e+b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*Elli 
pticE(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^2/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)- 
1/24*(-4*a*c+b^2)^(1/2)*(B*d*(35*c*d^2-e*(-8*a*e+23*b*d))-4*A*e*(5*c*d^2-2 
*e*(2*a*e+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*EllipticF(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^2/e^3/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a) 
^(1/2)-1/4*(-4*a*c+b^2)^(1/2)*(4*A*c*e*(-3*b*e+5*c*d)-B*(35*c^2*d^2+3*b^2* 
e^2-6*c*e*(-2*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+(-4*a*c+b 
^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*...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx \] Input:

Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(5/2),x]
 

Output:

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

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}}dx\)

Input:

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

\[\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x, algorithm="ma 
xima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x, algorithm="gi 
ac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(5/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x \] Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(5/2),x)