\(\int \frac {(A+B x^2) (d+e x^2)^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx\) [118]

Optimal result

Integrand size = 38, antiderivative size = 1013 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx =\text {Too large to display} \] Output:

-1/7*A*d*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^7-1/35*(8*A*a*e+A*b*d+7*B 
*a*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/x^5-1/105*(7*a*B*d*(6*a*e+b* 
d)-A*(4*b^2*d^2-9*a*b*d*e-a*(3*a*e^2+10*c*d^2)))*(e*x^2+d)^(1/2)*(c*x^4+b* 
x^2+a)^(1/2)/a^2/d/x^3-1/210*(-4*a*c+b^2)^(1/2)*(A*(-2*a*e+b*d)*(8*b^2*d^2 
-3*a*b*d*e-a*(-3*a*e^2+29*c*d^2))-7*a*B*d*(2*b^2*d^2-7*a*b*d*e-3*a*(a*e^2+ 
2*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)*Ellip 
ticE(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^3/d^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 
/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(7*a*B*d*(3*a^2*e^3-2*a*b*d*e^2-12*a*c*d^2 
*e-b^2*d^2*e+b*c*d^3)+2*A*(2*b^3*d^3*e-2*a*b*d*e*(-3*a*e^2+c*d^2)-b^2*(5*a 
*d^2*e^2+2*c*d^4)+a*(-3*a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)))*(-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))/a^3/d 
^2/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+2*2^(1/2)*B*c*(-4*a*c+b^2)^(1/2)* 
e^2*(-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*e))^(1/2))/(b+(-4*a*c+b...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx \] Input:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4])/x^8,x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4])/x^8, 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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8}dx\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4])/x^8,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ 
q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol 
yQ[Px, x]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{8}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}} \sqrt {a + b x^{2} + c x^{4}}}{x^{8}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(c*x**4+b*x**2+a)**(1/2)/x**8,x)
 

Output:

Integral((A + B*x**2)*(d + e*x**2)**(3/2)*sqrt(a + b*x**2 + c*x**4)/x**8, 
x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x, algorithm 
="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x, algorithm 
="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{3/2}\,\sqrt {c\,x^4+b\,x^2+a}}{x^8} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(1/2))/x^8,x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(1/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{8}}d x \] Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^8,x)