Integrand size = 44, antiderivative size = 529 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) \sqrt {a-c x^4}}{d e \left (c d^2-a e^2\right ) x^3 \sqrt {d+e x^2}}-\frac {\left (\frac {A c d}{a}-3 C d+\frac {3 d^2 D}{e}+3 B e-\frac {4 A e^2}{d}\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3 d \left (c d^2-a e^2\right ) x^3}+\frac {\sqrt {c} \left (3 B c d^3-3 a d^3 D-5 A c d^2 e+3 a C d^2 e-6 a B d e^2+8 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 a d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (3 a d (C d-2 B e)+A \left (c d^2+8 a e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 a^{3/2} d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
(-A*e^3+B*d*e^2-C*d^2*e+D*d^3)*(-c*x^4+a)^(1/2)/d/e/(-a*e^2+c*d^2)/x^3/(e* x^2+d)^(1/2)-1/3*(A*c*d/a-3*C*d+3*d^2*D/e+3*B*e-4*A*e^2/d)*(e*x^2+d)^(1/2) *(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/x^3+1/3*c^(1/2)*(8*A*a*e^3-5*A*c*d^2*e- 6*B*a*d*e^2+3*B*c*d^3+3*C*a*d^2*e-3*D*a*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)*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/(c^( 1/2)*d-a^(1/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/3*c^(1/2)*(3*a*d*(-2* B*e+C*d)+A*(8*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^(3/2)/d^3/(e*x^2+d)^(1/2) /(-c*x^4+a)^(1/2)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(d + e*x^2)^(3/2)*Sqrt[a - c*x^ 4]),x]
Output:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(d + e*x^2)^(3/2)*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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^4 \sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2251 |
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^4 \sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^4*(d + e*x^2)^(3/2)*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]
\[\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}d x\]
Input:
int((D*x^6+C*x^4+B*x^2+A)/x^4/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
Output:
int((D*x^6+C*x^4+B*x^2+A)/x^4/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, al gorithm="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^2*x^12 + 2*c*d*e*x^10 - 2*a*d*e*x^6 + (c*d^2 - a*e^2)*x^8 - a*d^2*x^4), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4} + D x^{6}}{x^{4} \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/x**4/(e*x**2+d)**(3/2)/(-c*x**4+a)**(1/ 2),x)
Output:
Integral((A + B*x**2 + C*x**4 + D*x**6)/(x**4*sqrt(a - c*x**4)*(d + e*x**2 )**(3/2)), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, al gorithm="maxima")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3/2)* x^4), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/x^4/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, al gorithm="giac")
Output:
integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3/2)* x^4), x)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^4\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^4*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2)), x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/(x^4*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2)), x)
\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\text {too large to display} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/x^4/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
Output:
( - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c*d*e**2 - 8*sqrt(d + e*x**2) *sqrt(a - c*x**4)*a**2*d**2*e**2*x**2 - 16*sqrt(d + e*x**2)*sqrt(a - c*x** 4)*a**2*d*e**3*x**4 - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d*e**2*x** 2 - 8*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*e**3*x**4 - 2*sqrt(d + e*x** 2)*sqrt(a - c*x**4)*a*c**2*d**2*e*x**2 + 12*sqrt(d + e*x**2)*sqrt(a - c*x* *4)*a*c**2*d*e**2*x**4 - sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**4*x**2 - 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**3*e*x**4 + sqrt(d + e*x**2)*sq rt(a - c*x**4)*b*c**2*d**3*x**2 + 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c* *2*d**2*e*x**4 - 32*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x) *a**2*c*d**2*e**4*x**3 - 32*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/( a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2* x**8),x)*a**2*c*d*e**5*x**5 - 16*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x* *6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c* e**2*x**8),x)*a*b*c**2*d*e**4*x**3 - 16*int((sqrt(d + e*x**2)*sqrt(a - c*x **4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x* *6 - c*e**2*x**8),x)*a*b*c**2*e**5*x**5 + 24*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d *e*x**6 - c*e**2*x**8),x)*a*c**3*d**2*e**3*x**3 + 24*int((sqrt(d + e*x**2) *sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x...