\(\int \frac {A+B x^2}{(d+e x^2)^{3/2} (a+b x^2+c x^4)^{5/2}} \, dx\) [253]

Optimal result

Integrand size = 35, antiderivative size = 2023 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Output:

-1/3*x*(a*B*(2*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)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(3/2)-1/3*e*(2*a*B*d*(c^2*d^2+2*b 
^2*e^2-c*e*(7*a*e+b*d))-A*(b^3*d*e^2+b*c*d*(-3*a*e^2+c*d^2)+4*a*c*e*(-3*a* 
e^2+c*d^2)-b^2*(-3*a*e^3+2*c*d^2*e)))*x/a/(-4*a*c+b^2)/d/(a*e^2-b*d*e+c*d^ 
2)^2/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+1/3*(a*B*d*(b^5*d*e^3+b^3*c*d*e 
*(-10*a*e^2+3*c*d^2)+4*a*b*c^2*d*e*(14*a*e^2+5*c*d^2)-b^4*(-7*a*e^4+3*c*d^ 
2*e^2)-4*a*c^2*(-20*a^2*e^4+15*a*c*d^2*e^2+3*c^2*d^4)-b^2*c*(52*a^2*e^4-3* 
a*c*d^2*e^2+c^2*d^4))+A*(2*b^6*d^2*e^3-b^3*c*d*(-50*a^2*e^4-39*a*c*d^2*e^2 
+2*c^2*d^4)-a*b^2*c*e*(-24*a^2*e^4+71*a*c*d^2*e^2+47*c^2*d^4)+4*a*b*c^2*d* 
(-20*a^2*e^4-3*a*c*d^2*e^2+4*c^2*d^4)+4*a^2*c^2*e*(-12*a^2*e^4+27*a*c*d^2* 
e^2+7*c^2*d^4)-b^5*(7*a*d*e^4+6*c*d^3*e^2)+b^4*(-3*a^2*e^5-a*c*d^2*e^3+6*c 
^2*d^4*e)))*(e*x^2+d)^(1/2)/a/(-4*a*c+b^2)^2/d/(a*e^2-b*d*e+c*d^2)^3/x/(c* 
x^4+b*x^2+a)^(1/2)+1/3*c*(A*(b^5*d*e^3+b^3*c*d*e*(4*a*e^2+3*c*d^2)-4*a*b*c 
^2*d*e*(16*a*e^2+11*c*d^2)-3*b^4*(3*a*e^4+c*d^2*e^2)-b^2*c*(-62*a^2*e^4-21 
*a*c*d^2*e^2+c^2*d^4)+4*a*c^2*(-22*a^2*e^4+15*a*c*d^2*e^2+5*c^2*d^4))+a*B* 
(5*b^4*d*e^3+b^2*c*d*e*(-35*a*e^2+17*c*d^2)-4*a*c^2*d*e*(-31*a*e^2+c*d^2)- 
b^3*(-3*a*e^4+6*c*d^2*e^2)-4*b*c*(5*a^2*e^4+6*a*c*d^2*e^2+2*c^2*d^4)))*x*( 
e*x^2+d)^(1/2)/a/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(c*x^4+b*x^2+a)^(1/2 
)-1/6*(a*B*d*(b^5*d*e^3+b^3*c*d*e*(-10*a*e^2+3*c*d^2)+4*a*b*c^2*d*e*(14...
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(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^2*x^16 
 + (2*c^3*d*e + 3*b*c^2*e^2)*x^14 + (c^3*d^2 + 6*b*c^2*d*e + 3*(b^2*c + a* 
c^2)*e^2)*x^12 + (3*b*c^2*d^2 + 6*(b^2*c + a*c^2)*d*e + (b^3 + 6*a*b*c)*e^ 
2)*x^10 + (3*(b^2*c + a*c^2)*d^2 + 2*(b^3 + 6*a*b*c)*d*e + 3*(a*b^2 + a^2* 
c)*e^2)*x^8 + (3*a^2*b*e^2 + (b^3 + 6*a*b*c)*d^2 + 6*(a*b^2 + a^2*c)*d*e)* 
x^6 + a^3*d^2 + (6*a^2*b*d*e + a^3*e^2 + 3*(a*b^2 + a^2*c)*d^2)*x^4 + (3*a 
^2*b*d^2 + 2*a^3*d*e)*x^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(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)*(e*x^2 + d)^(3/2)), x)
 

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(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)*(e*x^2 + d)^(3/2)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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