Integrand size = 34, antiderivative size = 518 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {(B d-A e) x \sqrt {a-c x^4}}{d^2 \sqrt {d+e x^2}}-\frac {(B d-A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{d^2 e x}-\frac {c (B d-2 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{d^2 e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} (B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{d^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {B c \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
(-A*e+B*d)*x*(-c*x^4+a)^(1/2)/d^2/(e*x^2+d)^(1/2)-(-A*e+B*d)*(e*x^2+d)^(1/ 2)*(-c*x^4+a)^(1/2)/d^2/e/x-c*(-2*A*e+B*d)*(d+a^(1/2)*e/c^(1/2))*(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))/d^2/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+a^(1/2)*c^(1/2)*(-2*A*e+B *d)*(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))/d^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-B*c*(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)*EllipticPi (1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/ 2)))^(1/2))/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^2*(d + e*x^2)^(3/2)),x]
Output:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(x^2*(d + e*x^2)^(3/2)), 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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{x^2 \left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2251 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{x^2 \left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(x^2*(d + e*x^2)^(3/2)),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]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {-c \,x^{4}+a}}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(e*x^2+d)^(3/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(e*x^2+d)^(3/2),x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(e*x^2+d)^(3/2),x, algorithm="fri cas")
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e^2*x^6 + 2*d*e*x^4 + d^2*x^2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/x**2/(e*x**2+d)**(3/2),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)/(x**2*(d + e*x**2)**(3/2)), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(e*x^2+d)^(3/2),x, algorithm="max ima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/((e*x^2 + d)^(3/2)*x^2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(e*x^2+d)^(3/2),x, algorithm="gia c")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/((e*x^2 + d)^(3/2)*x^2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^2*(d + e*x^2)^(3/2)),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(x^2*(d + e*x^2)^(3/2)), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b c d e x -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b c \,e^{2} x^{3}-2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c d e x -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c \,e^{2} x^{3}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b c \,d^{2} x -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b c d e \,x^{3}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{10}-2 c d e \,x^{8}+a \,e^{2} x^{6}-c \,d^{2} x^{6}+2 a d e \,x^{4}+a \,d^{2} x^{2}}d x \right ) a^{2} d e x +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{10}-2 c d e \,x^{8}+a \,e^{2} x^{6}-c \,d^{2} x^{6}+2 a d e \,x^{4}+a \,d^{2} x^{2}}d x \right ) a^{2} e^{2} x^{3}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{10}-2 c d e \,x^{8}+a \,e^{2} x^{6}-c \,d^{2} x^{6}+2 a d e \,x^{4}+a \,d^{2} x^{2}}d x \right ) a b \,d^{2} x -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{10}-2 c d e \,x^{8}+a \,e^{2} x^{6}-c \,d^{2} x^{6}+2 a d e \,x^{4}+a \,d^{2} x^{2}}d x \right ) a b d e \,x^{3}}{2 e x \left (e \,x^{2}+d \right )} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/x^2/(e*x^2+d)^(3/2),x)
Output:
( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*b - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(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)*b*c*d*e*x - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x* *4)*x**4)/(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)*b*c*e**2*x**3 - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x** 4)*x**2)/(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*d*e*x - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x* *2)/(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*e**2*x**3 - int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2) /(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)*b*c*d**2*x - int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(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)*b*c*d*e*x**3 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2*x**2 + 2*a*d*e*x**4 + a*e**2*x**6 - c*d**2*x**6 - 2*c*d*e*x**8 - c*e**2*x**10) ,x)*a**2*d*e*x + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2*x**2 + 2*a*d*e*x**4 + a*e**2*x**6 - c*d**2*x**6 - 2*c*d*e*x**8 - c*e**2*x**10),x) *a**2*e**2*x**3 - int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2*x**2 + 2 *a*d*e*x**4 + a*e**2*x**6 - c*d**2*x**6 - 2*c*d*e*x**8 - c*e**2*x**10),x)* a*b*d**2*x - int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2*x**2 + 2*a*d* e*x**4 + a*e**2*x**6 - c*d**2*x**6 - 2*c*d*e*x**8 - c*e**2*x**10),x)*a*...