\(\int \frac {(A+B x^2) (d+e x^2)^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx\) [119]

Optimal result

Integrand size = 38, antiderivative size = 954 \[ \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^{10}} \, dx=-\frac {A d \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{9 x^9}-\frac {(A b d+9 a B d+10 a A e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{63 a x^7}-\frac {\left (9 a B d (b d+8 a e)-A \left (6 b^2 d^2-11 a b d e-a \left (14 c d^2+3 a e^2\right )\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{315 a^2 d x^5}-\frac {\left (A \left (8 b^3 d^3-15 a b^2 d^2 e+38 a^2 c d^2 e-4 a^3 e^3-3 a b d \left (9 c d^2-a e^2\right )\right )-3 a B d \left (4 b^2 d^2-9 a b d e-a \left (10 c d^2+3 a e^2\right )\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{315 a^3 d^2 x^3}-\frac {\sqrt {b^2-4 a c} \left (3 a B d (b d-2 a e) \left (8 b^2 d^2-3 a b d e-a \left (29 c d^2-3 a e^2\right )\right )-A \left (16 b^4 d^4-32 a b^3 d^3 e-9 a b^2 d^2 \left (8 c d^2-a e^2\right )+a^2 b d e \left (117 c d^2+7 a e^2\right )+2 a^2 \left (21 c^2 d^4-15 a c d^2 e^2-4 a^2 e^4\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 )}{315 \sqrt {2} a^4 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 (6 a B d \left (2 b^2 d^2-3 a b d e-a \left (5 c d^2-3 a e^2\right )\right )-A \left (8 b^3 d^3-9 a b^2 d^2 e+8 a^2 e \left (3 c d^2+a e^2\right )-3 a b d \left (9 c d^2+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 )}{315 a^4 d^3 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/9*A*d*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^9-1/63*(10*A*a*e+A*b*d+9* 
B*a*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/x^7-1/315*(9*a*B*d*(8*a*e+b 
*d)-A*(6*b^2*d^2-11*a*b*d*e-a*(3*a*e^2+14*c*d^2)))*(e*x^2+d)^(1/2)*(c*x^4+ 
b*x^2+a)^(1/2)/a^2/d/x^5-1/315*(A*(8*b^3*d^3-15*a*b^2*d^2*e+38*a^2*c*d^2*e 
-4*a^3*e^3-3*a*b*d*(-a*e^2+9*c*d^2))-3*a*B*d*(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^3/d^2/x^3-1/630*(-4 
*a*c+b^2)^(1/2)*(3*a*B*d*(-2*a*e+b*d)*(8*b^2*d^2-3*a*b*d*e-a*(-3*a*e^2+29* 
c*d^2))-A*(16*b^4*d^4-32*a*b^3*d^3*e-9*a*b^2*d^2*(-a*e^2+8*c*d^2)+a^2*b*d* 
e*(7*a*e^2+117*c*d^2)+2*a^2*(-4*a^2*e^4-15*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^4/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)-1/315*2^(1/2)*(-4 
*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(6*a*B*d*(2*b^2*d^2-3*a*b*d*e-a*(-3*a* 
e^2+5*c*d^2))-A*(8*b^3*d^3-9*a*b^2*d^2*e+8*a^2*e*(a*e^2+3*c*d^2)-3*a*b*d*( 
a*e^2+9*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^4/d^3/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx \] Input:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4])/x^10,x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4])/x^10, 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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}}dx\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4])/x^10,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]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{10}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x, algorith 
m="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^10, x)
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}} \sqrt {a + b x^{2} + c x^{4}}}{x^{10}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(c*x**4+b*x**2+a)**(1/2)/x**10,x)
 

Output:

Integral((A + B*x**2)*(d + e*x**2)**(3/2)*sqrt(a + b*x**2 + c*x**4)/x**10, 
 x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x, algorith 
m="maxima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^10, x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x, algorith 
m="giac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^10, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{3/2}\,\sqrt {c\,x^4+b\,x^2+a}}{x^{10}} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(1/2))/x^10,x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(1/2))/x^10, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^{10}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{10}}d x \] Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^10,x)