\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx\) [112]

Optimal result

Integrand size = 38, antiderivative size = 1295 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx =\text {Too large to display} \] Output:

-1/11*A*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^11-1/99*(A*a*e+A*b*d+11*B* 
a*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/d/x^9-1/693*(11*a*B*d*(a*e+b* 
d)-A*(8*b^2*d^2-2*a*b*d*e-2*a*(-4*a*e^2+9*c*d^2)))*(e*x^2+d)^(1/2)*(c*x^4+ 
b*x^2+a)^(1/2)/a^2/d^2/x^7+1/3465*(22*a*B*d*(3*b^2*d^2-a*b*d*e-a*(-3*a*e^2 
+7*c*d^2))-A*(48*b^3*d^3-13*a*b^2*d^2*e+16*a^2*e*(3*a*e^2+2*c*d^2)-a*b*d*( 
13*a*e^2+157*c*d^2)))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^3/d^3/x^5-1/ 
3465*(11*a*B*d*(8*b^3*d^3-3*a*b^2*d^2*e+8*a^2*e*(a*e^2+c*d^2)-3*a*b*d*(a*e 
^2+9*c*d^2))-2*A*(32*b^4*d^4-10*a*b^3*d^3*e+5*a^2*b*d*e*(-2*a*e^2+7*c*d^2) 
-3*a*b^2*d^2*(3*a*e^2+46*c*d^2)+a^2*(32*a^2*e^4+23*a*c*d^2*e^2+75*c^2*d^4) 
))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^4/d^4/x^3-1/6930*(-4*a*c+b^2)^( 
1/2)*(A*(128*b^5*d^5-56*a*b^4*d^4*e+a^2*b^2*d^2*e*(-37*a*e^2+258*c*d^2)-a* 
b^3*d^3*(37*a*e^2+696*c*d^2)+a^2*b*d*(-56*a^2*e^4+135*a*c*d^2*e^2+771*c^2* 
d^4)-4*a^3*e*(-32*a^2*e^4-27*a*c*d^2*e^2+39*c^2*d^4))-22*a*B*d*(8*b^4*d^4- 
4*a*b^3*d^3*e+a^2*b*d*e*(-4*a*e^2+15*c*d^2)-3*a*b^2*d^2*(a*e^2+12*c*d^2)+a 
^2*(8*a^2*e^4+9*a*c*d^2*e^2+21*c^2*d^4)))*(-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^5/d^5/(-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)+1/3465*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d 
*e+c*d^2)*(11*a*B*d*(8*b^3*d^3-27*a*b*c*d^3+3*a*b^2*d^2*e-2*a^2*e*(8*a*...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^12,x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^12, 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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])/x^12,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]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{12}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{12}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x, algorith 
m="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {d + e x^{2}} \sqrt {a + b x^{2} + c x^{4}}}{x^{12}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(1/2)*(c*x**4+b*x**2+a)**(1/2)/x**12,x)
 

Output:

Integral((A + B*x**2)*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)/x**12, x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{12}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x, algorith 
m="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^12, x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{12}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x, algorith 
m="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^12, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {e\,x^2+d}\,\sqrt {c\,x^4+b\,x^2+a}}{x^{12}} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(1/2))/x^12,x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)^(1/2))/x^12, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^{12}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{12}}d x \] Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^12,x)