\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx\) [110]

Optimal result

Integrand size = 38, antiderivative size = 726 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{x^8} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{7 x^7}-\frac {(A b d+7 a B d+a A e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{35 a d x^5}-\frac {\left (7 a B d (b d+a e)-A \left (4 b^2 d^2-2 a b d e-2 a \left (5 c d^2-2 a e^2\right )\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{105 a^2 d^2 x^3}+\frac {\sqrt {b^2-4 a c} \left (14 a B d \left (b^2 d^2-a b d e-a \left (3 c d^2-a e^2\right )\right )-A \left (8 b^3 d^3-5 a b^2 d^2 e+8 a^2 e \left (2 c d^2+a e^2\right )-a b d \left (29 c d^2+5 a e^2\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{105 \sqrt {2} a^3 d^3 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (7 a B d (b d-2 a e)-A \left (4 b^2 d^2+a b d e-2 a \left (5 c d^2+4 a e^2\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{105 a^3 d^3 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/7*A*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^7-1/35*(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/d/x^5-1/105*(7*a*B*d*(a*e+b*d)-A 
*(4*b^2*d^2-2*a*b*d*e-2*a*(-2*a*e^2+5*c*d^2)))*(e*x^2+d)^(1/2)*(c*x^4+b*x^ 
2+a)^(1/2)/a^2/d^2/x^3+1/210*(-4*a*c+b^2)^(1/2)*(14*a*B*d*(b^2*d^2-a*b*d*e 
-a*(-a*e^2+3*c*d^2))-A*(8*b^3*d^3-5*a*b^2*d^2*e+8*a^2*e*(a*e^2+2*c*d^2)-a* 
b*d*(5*a*e^2+29*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)*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^3/d^3/(-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)*(a*e^2-b*d*e+c*d^2)*(7*a*B*d 
*(-2*a*e+b*d)-A*(4*b^2*d^2+a*b*d*e-2*a*(4*a*e^2+5*c*d^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*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 
^3/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2250

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(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)*sqrt(e*x^2 + d)/x^8, x)
 

Giac [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(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)*sqrt(e*x^2 + d)/x^8, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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