\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx\) [42]

Optimal result

Integrand size = 34, antiderivative size = 700 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{11 x^{11}}-\frac {(11 B d+A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{99 d x^9}+\frac {\left (18 A c d^2-11 a B d e+8 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{693 a d^2 x^7}+\frac {2 \left (77 B c d^3+16 A c d^2 e+33 a B d e^2-24 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3465 a d^3 x^5}+\frac {2 \left (44 a B d e \left (c d^2-a e^2\right )+A \left (75 c^2 d^4-23 a c d^2 e^2+32 a^2 e^4\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3465 a^2 d^4 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (2 A e \left (39 c^2 d^4+27 a c d^2 e^2-32 a^2 e^4\right )+11 B \left (21 c^2 d^5-9 a c d^3 e^2+8 a^2 d e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3465 a^2 d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (11 a B d e \left (3 c d^2-8 a e^2\right )-A \left (75 c^2 d^4+6 a c d^2 e^2-64 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3465 a^{5/2} d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/11*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^11-1/99*(A*e+11*B*d)*(e*x^2+d)^ 
(1/2)*(-c*x^4+a)^(1/2)/d/x^9+1/693*(8*A*a*e^2+18*A*c*d^2-11*B*a*d*e)*(e*x^ 
2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^7+2/3465*(-24*A*a*e^3+16*A*c*d^2*e+33* 
B*a*d*e^2+77*B*c*d^3)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^3/x^5+2/3465*(4 
4*a*B*d*e*(-a*e^2+c*d^2)+A*(32*a^2*e^4-23*a*c*d^2*e^2+75*c^2*d^4))*(e*x^2+ 
d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^4/x^3-2/3465*c*(d+a^(1/2)*e/c^(1/2))*(2*A* 
e*(-32*a^2*e^4+27*a*c*d^2*e^2+39*c^2*d^4)+11*B*(8*a^2*d*e^4-9*a*c*d^3*e^2+ 
21*c^2*d^5))*(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 
/3465*c^(1/2)*(-a*e^2+c*d^2)*(11*a*B*d*e*(-8*a*e^2+3*c*d^2)-A*(-64*a^2*e^4 
+6*a*c*d^2*e^2+75*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)*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^(5/2)/d^5/(e*x^2+d)^(1/2) 
/(-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4])/x^12,x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4])/x^12, 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.

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \sqrt {d+e x^2}}{x^{12}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \sqrt {d+e x^2}}{x^{12}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4])/x^12,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]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{x^{12}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^12,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^12,x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{12}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^12,x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \sqrt {d + e x^{2}}}{x^{12}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(1/2)*(-c*x**4+a)**(1/2)/x**12,x)
 

Output:

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

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{12}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^12,x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{x^{12}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^12,x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}}{x^{12}} \,d x \] Input:

int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2))/x^12,x)
 

Output:

int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2))/x^12, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{12}} \, dx=\text {too large to display} \] Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^12,x)
 

Output:

( - 120960*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**7*d**2*e**8 - 221760*sqrt( 
d + e*x**2)*sqrt(a - c*x**4)*a**6*b*d**2*e**8*x**2 + 63360*sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*a**6*b*d*e**9*x**4 + 2016*sqrt(d + e*x**2)*sqrt(a - c*x 
**4)*a**6*c*d**4*e**6 - 40320*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**6*c*d** 
3*e**7*x**2 + 80640*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**6*c*d**2*e**8*x** 
4 + 77616*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d**4*e**6*x**2 - 8553 
6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d**3*e**7*x**4 - 158400*sqrt( 
d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d**2*e**8*x**6 + 316800*sqrt(d + e*x 
**2)*sqrt(a - c*x**4)*a**5*b*c*d*e**9*x**8 - 380160*sqrt(d + e*x**2)*sqrt( 
a - c*x**4)*a**5*b*c*e**10*x**10 + 3528*sqrt(d + e*x**2)*sqrt(a - c*x**4)* 
a**5*c**2*d**6*e**4 - 20496*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c**2*d* 
*5*e**5*x**2 + 22848*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c**2*d**4*e**6 
*x**4 - 80640*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c**2*d**3*e**7*x**6 + 
 161280*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c**2*d**2*e**8*x**8 + 12166 
0*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*c**2*d**6*e**4*x**2 - 133496*sq 
rt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*c**2*d**5*e**5*x**4 - 7920*sqrt(d + 
 e*x**2)*sqrt(a - c*x**4)*a**4*b*c**2*d**4*e**6*x**6 + 15840*sqrt(d + e*x* 
*2)*sqrt(a - c*x**4)*a**4*b*c**2*d**3*e**7*x**8 - 849024*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*a**4*b*c**2*d**2*e**8*x**10 + 210*sqrt(d + e*x**2)*sqrt(a 
 - c*x**4)*a**4*c**3*d**8*e**2 + 75180*sqrt(d + e*x**2)*sqrt(a - c*x**4...