\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^8} \, dx\) [40]

Optimal result

Integrand size = 34, antiderivative size = 488 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^8} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{7 x^7}-\frac {(7 B d+A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{35 d x^5}+\frac {\left (10 A c d^2-7 a B d e+4 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{105 a d^2 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (21 B c d^3+8 A c d^2 e+7 a B d e^2-4 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (5 A c d^2+7 a B d e-4 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a^{3/2} d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/7*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^7-1/35*(A*e+7*B*d)*(e*x^2+d)^(1/ 
2)*(-c*x^4+a)^(1/2)/d/x^5+1/105*(4*A*a*e^2+10*A*c*d^2-7*B*a*d*e)*(e*x^2+d) 
^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^3-2/105*c*(d+a^(1/2)*e/c^(1/2))*(-4*A*a*e^ 
3+8*A*c*d^2*e+7*B*a*d*e^2+21*B*c*d^3)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^ 
2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticE(1/2*(1-a^(1/2)/c^(1/2)/x^2 
)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/a/d^3/(e*x^2+d)^( 
1/2)/(-c*x^4+a)^(1/2)-2/105*c^(1/2)*(-a*e^2+c*d^2)*(-4*A*a*e^2+5*A*c*d^2+7 
*B*a*d*e)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x 
^2)^(1/2)*EllipticF(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/( 
d+a^(1/2)*e/c^(1/2)))^(1/2))/a^(3/2)/d^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2251

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*d*e**4 - 84*sqrt(d + e*x**2) 
*sqrt(a - c*x**4)*a**3*b*d*e**4*x**2 + 2*sqrt(d + e*x**2)*sqrt(a - c*x**4) 
*a**3*c*d**3*e**2 - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c*d**2*e**3* 
x**2 + 48*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c*d*e**4*x**4 - 24*sqrt(d 
 + e*x**2)*sqrt(a - c*x**4)*a**3*c*e**5*x**6 + 91*sqrt(d + e*x**2)*sqrt(a 
- c*x**4)*a**2*b*c*d**3*e**2*x**2 - 168*sqrt(d + e*x**2)*sqrt(a - c*x**4)* 
a**2*b*c*d**2*e**3*x**4 + 252*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c*d 
*e**4*x**6 + 30*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**2*d**4*e*x**2 - 
60*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**2*d**3*e**2*x**4 + 42*sqrt(d 
+ e*x**2)*sqrt(a - c*x**4)*a**2*c**2*d**2*e**3*x**6 + 35*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*a*b*c**2*d**5*x**2 - 70*sqrt(d + e*x**2)*sqrt(a - c*x**4) 
*a*b*c**2*d**4*e*x**4 + 399*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**2*d** 
3*e**2*x**6 + 90*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c**3*d**4*e*x**6 + 10 
5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**3*d**5*x**6 - 576*int((sqrt(d + e 
*x**2)*sqrt(a - c*x**4)*x**4)/(12*a**2*d*e**2 + 12*a**2*e**3*x**2 - a*c*d* 
*3 - a*c*d**2*e*x**2 - 12*a*c*d*e**2*x**4 - 12*a*c*e**3*x**6 + c**2*d**3*x 
**4 + c**2*d**2*e*x**6),x)*a**4*c**2*e**8*x**7 + 6048*int((sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*x**4)/(12*a**2*d*e**2 + 12*a**2*e**3*x**2 - a*c*d**3 - 
a*c*d**2*e*x**2 - 12*a*c*d*e**2*x**4 - 12*a*c*e**3*x**6 + c**2*d**3*x**4 + 
 c**2*d**2*e*x**6),x)*a**3*b*c**2*d*e**7*x**7 + 1056*int((sqrt(d + e*x*...