\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx\) [57]

Optimal result

Integrand size = 34, antiderivative size = 413 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{5 d x^5}-\frac {(5 B d-4 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{15 d^2 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 A c d^2+5 a B d e-4 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 a d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} (5 B d-4 A e) \left (c d^2-a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 \sqrt {a} d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/5*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/d/x^5-1/15*(-4*A*e+5*B*d)*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/d^2/x^3-2/15*c*(d+a^(1/2)*e/c^(1/2))*(-4*A*a*e^2+ 
3*A*c*d^2+5*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)*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)-2/15*c^(1/2)*(-4*A*e+5*B*d)*(-a*e^2+c*d^2)*(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^(1/2) 
/d^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^6*Sqrt[d + e*x^2]),x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^6*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.

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{x^6 \sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{x^6 \sqrt {d+e x^2}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[a - c*x^4])/(x^6*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]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {-c \,x^{4}+a}}{x^{6} \sqrt {e \,x^{2}+d}}d x\]

Input:

int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^6/(e*x^2+d)^(1/2),x)
 

Output:

int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^6/(e*x^2+d)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d} x^{6}} \,d x } \] Input:

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^6/(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^8 + d*x^6), x)
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{x^{6} \sqrt {d + e x^{2}}}\, dx \] Input:

integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/x**6/(e*x**2+d)**(1/2),x)
 

Output:

Integral((A + B*x**2)*sqrt(a - c*x**4)/(x**6*sqrt(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d} x^{6}} \,d x } \] Input:

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^6/(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^6), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d} x^{6}} \,d x } \] Input:

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^6/(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^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{x^6\,\sqrt {e\,x^2+d}} \,d x \] Input:

int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^6*(d + e*x^2)^(1/2)),x)
 

Output:

int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^6*(d + e*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^6 \sqrt {d+e x^2}} \, dx =\text {Too large to display} \] Input:

int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^6/(e*x^2+d)^(1/2),x)
 

Output:

( - 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*d*e**2 + 10*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*a*b*d*e**2*x**2 - 20*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a* 
b*e**3*x**4 - 3*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e*x**2 + 10*sqr 
t(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e**2*x**4 + 15*sqrt(d + e*x**2)*sqrt( 
a - c*x**4)*b*c*d**3*x**2 - 30*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2* 
e*x**4 - 40*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - 
 c*d*x**4 - c*e*x**6),x)*a*b*c*e**4*x**5 + 20*int((sqrt(d + e*x**2)*sqrt(a 
 - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c**2*d*e**3*x 
**5 - 60*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c* 
d*x**4 - c*e*x**6),x)*b*c**2*d**2*e**2*x**5 - 20*int((sqrt(d + e*x**2)*sqr 
t(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d*e**3 
*x**5 + 10*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - 
c*d*x**4 - c*e*x**6),x)*a*c**2*d**2*e**2*x**5 - 30*int((sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b*c**2*d** 
3*e*x**5 - 8*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**4 + a*e*x**6 
- c*d*x**8 - c*e*x**10),x)*a**3*d*e**3*x**5 + 40*int((sqrt(d + e*x**2)*sqr 
t(a - c*x**4))/(a*d*x**4 + a*e*x**6 - c*d*x**8 - c*e*x**10),x)*a**2*b*d**2 
*e**2*x**5 - 9*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**4 + a*e*x** 
6 - c*d*x**8 - c*e*x**10),x)*a**2*c*d**3*e*x**5 + 45*int((sqrt(d + e*x**2) 
*sqrt(a - c*x**4))/(a*d*x**4 + a*e*x**6 - c*d*x**8 - c*e*x**10),x)*a*b*...