Integrand size = 44, antiderivative size = 691 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{9 a d x^9}-\frac {(9 B d-8 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{63 a d^2 x^7}-\frac {\left (49 A c d^2+63 a C d^2-54 a B d e+48 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a^2 d^3 x^5}-\frac {\left (75 B c d^3+105 a d^3 D-62 A c d^2 e-84 a C d^2 e+72 a B d e^2-64 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a^2 d^4 x^3}+\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (A \left (147 c^2 d^4+108 a c d^2 e^2+128 a^2 e^4\right )+3 a d \left (c d^2 (63 C d-44 B e)-2 a e \left (35 d^2 D-28 C d e+24 B e^2\right )\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^3 d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {c} \left (111 A c^2 d^4 e-a c d^2 \left (105 d^3 D-147 C d^2 e-76 A e^3\right )-2 a^2 e^2 \left (105 d^3 D-84 C d^2 e-64 A e^3\right )-3 B \left (25 c^2 d^5+32 a c d^3 e^2+48 a^2 d e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^{5/2} d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/9*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d/x^9-1/63*(-8*A*e+9*B*d)*(e*x^2 +d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^7-1/315*(48*A*a*e^2+49*A*c*d^2-54*B*a*d *e+63*C*a*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^3/x^5-1/315*(-64*A*a *e^3-62*A*c*d^2*e+72*B*a*d*e^2+75*B*c*d^3-84*C*a*d^2*e+105*D*a*d^3)*(e*x^2 +d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^4/x^3+1/315*c*(d+a^(1/2)*e/c^(1/2))*(A*(1 28*a^2*e^4+108*a*c*d^2*e^2+147*c^2*d^4)+3*a*d*(c*d^2*(-44*B*e+63*C*d)-2*a* e*(24*B*e^2-28*C*d*e+35*D*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)*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^3/d^5/(e*x^2+d)^(1/2 )/(-c*x^4+a)^(1/2)-1/315*c^(1/2)*(111*A*c^2*d^4*e-a*c*d^2*(-76*A*e^3-147*C *d^2*e+105*D*d^3)-2*a^2*e^2*(-64*A*e^3-84*C*d^2*e+105*D*d^3)-3*B*(48*a^2*d *e^4+32*a*c*d^3*e^2+25*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)*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)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*Sqrt[d + e*x^2]*Sqrt[a - c*x^4 ]),x]
Output:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*Sqrt[d + e*x^2]*Sqrt[a - c*x^4 ]), 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.
\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {a-c x^4} \sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 2251 |
\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {a-c x^4} \sqrt {d+e x^2}}dx\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*Sqrt[d + e*x^2]*Sqrt[a - c*x^4]),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]
\[\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{x^{10} \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}d x\]
Input:
int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
Output:
int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, a lgorithm="fricas")
Output:
integral(-(D*x^6 + C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c* e*x^16 + c*d*x^14 - a*e*x^12 - a*d*x^10), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4} + D x^{6}}{x^{10} \sqrt {a - c x^{4}} \sqrt {d + e x^{2}}}\, dx \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/x**10/(e*x**2+d)**(1/2)/(-c*x**4+a)**(1 /2),x)
Output:
Integral((A + B*x**2 + C*x**4 + D*x**6)/(x**10*sqrt(a - c*x**4)*sqrt(d + e *x**2)), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, a lgorithm="maxima")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)*x^ 10), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, a lgorithm="giac")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)*x^ 10), x)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^{10}\,\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)) ,x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)) , x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\text {too large to display} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
Output:
( - 2880*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c*d*e**6 - 11520*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*d**2*e**6*x**2 + 13824*sqrt(d + e*x**2)*sqrt (a - c*x**4)*a**5*d*e**7*x**4 - 8640*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a** 4*b*c*d*e**6*x**2 + 10368*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*c*e**7* x**4 - 3360*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c**2*d**2*e**5*x**2 - 5 184*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c**2*d*e**6*x**4 - 19440*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c*d**4*e**4*x**2 + 23328*sqrt(d + e*x**2) *sqrt(a - c*x**4)*a**4*c*d**3*e**5*x**4 - 2304*sqrt(d + e*x**2)*sqrt(a - c *x**4)*a**4*c*d**2*e**6*x**6 - 10800*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a** 3*b*c**2*d**3*e**4*x**2 + 12960*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c **2*d**2*e**5*x**4 - 1728*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c**2*d* e**6*x**6 + 20*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**3*d**5*e**2 - 280 0*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**3*d**4*e**3*x**2 + 3360*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**3*d**3*e**4*x**4 - 14496*sqrt(d + e*x* *2)*sqrt(a - c*x**4)*a**3*c**3*d**2*e**5*x**6 - 16920*sqrt(d + e*x**2)*sqr t(a - c*x**4)*a**3*c**2*d**6*e**2*x**2 + 20304*sqrt(d + e*x**2)*sqrt(a - c *x**4)*a**3*c**2*d**5*e**3*x**4 - 3888*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a **3*c**2*d**4*e**4*x**6 - 9540*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c* *3*d**5*e**2*x**2 + 11448*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**3*d* *4*e**3*x**4 - 2160*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**3*d**3*...