\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{x^8 \sqrt {d+e x^2}} \, dx\) [58]

Optimal result

Integrand size = 34, antiderivative size = 492 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^8 \sqrt {d+e x^2}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{7 d x^7}-\frac {(7 B d-6 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{35 d^2 x^5}+\frac {2 \left (5 A c d^2+14 a B d e-12 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{105 a d^3 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (21 B c d^3-13 A c d^2 e-28 a B d e^2+24 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (5 A c d^2-28 a B d e+24 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a^{3/2} d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/7*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/d/x^7-1/35*(-6*A*e+7*B*d)*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/d^2/x^5+2/105*(-12*A*a*e^2+5*A*c*d^2+14*B*a*d*e)* 
(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^3/x^3-2/105*c*(d+a^(1/2)*e/c^(1/2))*( 
24*A*a*e^3-13*A*c*d^2*e-28*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^4/ 
(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-2/105*c^(1/2)*(-a*e^2+c*d^2)*(24*A*a*e^2+ 
5*A*c*d^2-28*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^4/(e*x^2+d)^(1/2)/(-c*x 
^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e*x^10 + d*x^8), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^8/(e*x^2+d)^(1/2),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. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^8 \sqrt {d+e x^2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{x^8\,\sqrt {e\,x^2+d}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

( - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*d*e**4 + 2*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*a**3*c*d**3*e**2 + 60*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a 
**3*c*d**2*e**3*x**2 - 120*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c*d*e**4 
*x**4 + 144*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c*e**5*x**6 - 84*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 - 168*sqrt(d + e*x**2)*sqrt(a - 
c*x**4)*a**2*b*c*d*e**4*x**6 - 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c* 
*2*d**4*e*x**2 + 10*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**2*d**3*e**2* 
x**4 + 168*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 - 126*sqrt(d + e*x**2)*sqrt(a - c* 
x**4)*a*b*c**2*d**3*e**2*x**6 - 15*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c** 
3*d**4*e*x**6 + 105*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**3*d**5*x**6 + 3 
456*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 - 4032*i 
nt((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 + 3744*in 
t((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(12*a**2*d*e**2 + 12*a**2*e*...