\(\int \frac {A+B x^2}{\sqrt {d+e x^2} (a+b x^2+c x^4)^{5/2}} \, dx\) [252]

Optimal result

Integrand size = 35, antiderivative size = 1344 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[(A + B*x^2)/(Sqrt[d + e*x^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.

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

\(\Big \downarrow \) 2260

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

Input:

Int[(A + B*x^2)/(Sqrt[d + e*x^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]

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

Input:

int((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x)
 

Output:

int((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x)
 

Fricas [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(c^3*e*x^14 + 
 (c^3*d + 3*b*c^2*e)*x^12 + 3*(b*c^2*d + (b^2*c + a*c^2)*e)*x^10 + (3*(b^2 
*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*x^8 + ((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a 
^2*c)*e)*x^6 + 3*(a^2*b*e + (a*b^2 + a^2*c)*d)*x^4 + a^3*d + (3*a^2*b*d + 
a^3*e)*x^2), x)
 

Sympy [F(-1)]

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

integrate((B*x**2+A)/(e*x**2+d)**(1/2)/(c*x**4+b*x**2+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

int((A + B*x^2)/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(5/2)),x)
 

Output:

int((A + B*x^2)/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(5/2)), x)
 

Reduce [F]

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

int((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x)
 

Output:

int((B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(5/2),x)