\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{x^{10} \sqrt {d+e x^2}} \, dx\) [59]

Optimal result

Integrand size = 34, antiderivative size = 588 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^{10} \sqrt {d+e x^2}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{9 d x^9}-\frac {(9 B d-8 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{63 d^2 x^7}+\frac {2 \left (7 A c d^2+27 a B d e-24 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a d^3 x^5}+\frac {2 \left (15 B c d^3-11 A c d^2 e-36 a B d e^2+32 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a d^4 x^3}+\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 a B d e \left (13 c d^2-24 a e^2\right )-A \left (21 c^2 d^4+30 a c d^2 e^2-64 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^2 d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (15 B c d^3-18 A c d^2 e+72 a B d e^2-64 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^{3/2} d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/9*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/d/x^9-1/63*(-8*A*e+9*B*d)*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/d^2/x^7+2/315*(-24*A*a*e^2+7*A*c*d^2+27*B*a*d*e)* 
(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^3/x^5+2/315*(32*A*a*e^3-11*A*c*d^2*e- 
36*B*a*d*e^2+15*B*c*d^3)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^4/x^3+2/315* 
c*(d+a^(1/2)*e/c^(1/2))*(3*a*B*d*e*(-24*a*e^2+13*c*d^2)-A*(-64*a^2*e^4+30* 
a*c*d^2*e^2+21*c^2*d^4))*(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^2/d^5/(e*x^2+d)^(1/2)/(-c*x^ 
4+a)^(1/2)-2/315*c^(1/2)*(-a*e^2+c*d^2)*(-64*A*a*e^3-18*A*c*d^2*e+72*B*a*d 
*e^2+15*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)*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^5/(e*x^2+d)^(1/2)/(-c*x^4+a) 
^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^10*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.

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

\(\Big \downarrow \) 2251

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

Input:

Int[((A + B*x^2)*Sqrt[a - c*x^4])/(x^10*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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^10/(e*x^2+d)^(1/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e*x^12 + d*x^10), x 
)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^10/(e*x^2+d)^(1/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^10/(e*x^2+d)^(1/2),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^{10} \sqrt {d+e x^2}} \, dx=\text {too large to display} \] Input:

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

Output:

( - 1440*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*d*e**6 + 1440*sqrt(d + e*x 
**2)*sqrt(a - c*x**4)*a**4*b*d*e**6*x**2 - 1728*sqrt(d + e*x**2)*sqrt(a - 
c*x**4)*a**4*b*e**7*x**4 - 1680*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c*d 
**2*e**5*x**2 + 2592*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c*d*e**6*x**4 
+ 4320*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c*d**3*e**4*x**2 - 5184*sq 
rt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c*d**2*e**5*x**4 + 288*sqrt(d + e*x 
**2)*sqrt(a - c*x**4)*a**3*b*c*d*e**6*x**6 + 10*sqrt(d + e*x**2)*sqrt(a - 
c*x**4)*a**3*c**2*d**5*e**2 - 1400*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3* 
c**2*d**4*e**3*x**2 + 1680*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d** 
3*e**4*x**4 + 528*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d**2*e**5*x* 
*6 + 3690*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**2*d**5*e**2*x**2 - 4 
428*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**2*d**4*e**3*x**4 + 864*sqr 
t(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**2*d**3*e**4*x**6 - 105*sqrt(d + e 
*x**2)*sqrt(a - c*x**4)*a**2*c**3*d**6*e*x**2 + 122*sqrt(d + e*x**2)*sqrt( 
a - c*x**4)*a**2*c**3*d**5*e**2*x**4 - 280*sqrt(d + e*x**2)*sqrt(a - c*x** 
4)*a**2*c**3*d**4*e**3*x**6 + 945*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c* 
*3*d**7*x**2 - 1134*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**3*d**6*e*x**4 
 + 738*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**3*d**5*e**2*x**6 - 27*sqrt 
(d + e*x**2)*sqrt(a - c*x**4)*a*c**4*d**6*e*x**6 + 189*sqrt(d + e*x**2)*sq 
rt(a - c*x**4)*b*c**4*d**7*x**6 - 1658880*int((sqrt(d + e*x**2)*sqrt(a ...