Integrand size = 34, antiderivative size = 700 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=-\frac {A d \sqrt {d+e x^2} \sqrt {a-c x^4}}{11 x^{11}}-\frac {(11 B d+12 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{99 x^9}+\frac {\left (18 A c d^2-110 a B d e-3 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{693 a d x^7}+\frac {\left (154 B c d^3+186 A c d^2 e-33 a B d e^2+18 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3465 a d^2 x^5}+\frac {2 \left (75 A c^2 d^4+209 a B c d^3 e+21 a A c d^2 e^2+22 a^2 B d e^3-12 a^2 A e^4\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3465 a^2 d^3 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 A e \left (103 c^2 d^4-15 a c d^2 e^2+8 a^2 e^4\right )+11 B \left (21 c^2 d^5+15 a c d^3 e^2-4 a^2 d e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3465 a^2 d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (44 a B d e \left (3 c d^2-a e^2\right )+3 A \left (25 c^2 d^4-9 a c d^2 e^2+8 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3465 a^{5/2} d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/11*A*d*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^11-1/99*(12*A*e+11*B*d)*(e*x^ 2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^9+1/693*(-3*A*a*e^2+18*A*c*d^2-110*B*a*d*e)* (e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d/x^7+1/3465*(18*A*a*e^3+186*A*c*d^2*e- 33*B*a*d*e^2+154*B*c*d^3)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^5+2/346 5*(-12*A*a^2*e^4+21*A*a*c*d^2*e^2+75*A*c^2*d^4+22*B*a^2*d*e^3+209*B*a*c*d^ 3*e)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^3/x^3-2/3465*c*(d+a^(1/2)*e/c^ (1/2))*(3*A*e*(8*a^2*e^4-15*a*c*d^2*e^2+103*c^2*d^4)+11*B*(-4*a^2*d*e^4+15 *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^4/(e*x^2+d)^(1/2)/(-c*x ^4+a)^(1/2)-2/3465*c^(1/2)*(-a*e^2+c*d^2)*(44*a*B*d*e*(-a*e^2+3*c*d^2)+3*A *(8*a^2*e^4-9*a*c*d^2*e^2+25*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^4/(e*x ^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^12,x]
Output:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^12, x]
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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 2251 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^{12}}dx\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^12,x]
Output:
$Aborted
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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}{x^{12}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^12,x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^12,x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{12}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^12,x, algorithm="fr icas")
Output:
integral((B*e*x^4 + (B*d + A*e)*x^2 + A*d)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d )/x^12, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{12}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(1/2)/x**12,x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)*(d + e*x**2)**(3/2)/x**12, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{12}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^12,x, algorithm="ma xima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^12, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{12}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^12,x, algorithm="gi ac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^12, x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}}{x^{12}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^12,x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^12, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{12}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^12,x)
Output:
( - 120960*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**7*d**3*e**8 - 221760*sqrt( d + e*x**2)*sqrt(a - c*x**4)*a**7*d**2*e**9*x**2 + 63360*sqrt(d + e*x**2)* sqrt(a - c*x**4)*a**7*d*e**10*x**4 - 221760*sqrt(d + e*x**2)*sqrt(a - c*x* *4)*a**6*b*d**3*e**8*x**2 - 126720*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**6* b*d**2*e**9*x**4 + 2016*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**6*c*d**5*e**6 + 37296*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**6*c*d**4*e**7*x**2 - 4896*sq rt(d + e*x**2)*sqrt(a - c*x**4)*a**6*c*d**3*e**8*x**4 - 158400*sqrt(d + e* x**2)*sqrt(a - c*x**4)*a**6*c*d**2*e**9*x**6 + 316800*sqrt(d + e*x**2)*sqr t(a - c*x**4)*a**6*c*d*e**10*x**8 - 380160*sqrt(d + e*x**2)*sqrt(a - c*x** 4)*a**6*c*e**11*x**10 + 3696*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d* *5*e**6*x**2 + 2112*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d**4*e**7*x **4 - 348480*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d**3*e**8*x**6 + 6 96960*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d**2*e**9*x**8 - 570240*s qrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*b*c*d*e**10*x**10 + 3528*sqrt(d + e* x**2)*sqrt(a - c*x**4)*a**5*c**2*d**7*e**4 + 101164*sqrt(d + e*x**2)*sqrt( a - c*x**4)*a**5*c**2*d**6*e**5*x**2 - 110648*sqrt(d + e*x**2)*sqrt(a - c* x**4)*a**5*c**2*d**5*e**6*x**4 - 88560*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a **5*c**2*d**4*e**7*x**6 + 177120*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c* *2*d**3*e**8*x**8 - 849024*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c**2*d** 2*e**9*x**10 + 215292*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*c**2*d**...