Integrand size = 31, antiderivative size = 546 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\left (\frac {A}{d}-\frac {B}{e}\right ) x \sqrt {a-c x^4}}{\sqrt {d+e x^2}}+\frac {(3 B d-2 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 d e^2 x}+\frac {\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) (3 B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 d e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \sqrt {c} (B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 d e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {c (3 B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
(A/d-B/e)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^(1/2)+1/2*(-2*A*e+3*B*d)*(e*x^2+d)^ (1/2)*(-c*x^4+a)^(1/2)/d/e^2/x+1/2*c^(1/2)*(c^(1/2)*d+a^(1/2)*e)*(-2*A*e+3 *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)*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/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-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/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^( 1/2)+1/2*c*(-2*A*e+3*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)*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^2/(e*x^2+d)^(1/2)/(-c*x^ 4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx \] Input:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(3/2),x]
Output:
Integrate[((A + B*x^2)*Sqrt[a - c*x^4])/(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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right )}{\left (d+e x^2\right )^{3/2}}dx\) |
Input:
Int[((A + B*x^2)*Sqrt[a - c*x^4])/(d + e*x^2)^(3/2),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {-c \,x^{4}+a}}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="fricas" )
Output:
integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(1/2)/(e*x**2+d)**(3/2),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)/(d + e*x**2)**(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="maxima" )
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)/(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(3/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2))/(d + e*x^2)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a x +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c d e +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a c \,e^{2} x^{2}-3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b c \,d^{2}-3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b c d e \,x^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a b \,d^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a b d e \,x^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a^{2} d^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a^{2} d e \,x^{2}}{3 d \left (e \,x^{2}+d \right )} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(3/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*x + 2*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*d*e + 2*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*e**2*x**2 - 3*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)*b*c*d**2 - 3*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)*b*c*d*e*x**2 + 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)*a*b*d**2 + 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 )*a*b*d*e*x**2 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*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)*a**2* d**2 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*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)*a**2*d*e*x**2)/ (3*d*(d + e*x**2))