Integrand size = 35, antiderivative size = 814 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {(5 B c d-3 b B e+4 A c e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{8 c^2 x}+\frac {B e x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\sqrt {b^2-4 a c} (5 B c d-3 b B e+4 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{8 \sqrt {2} c^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {b^2-4 a c} \left (a B e (7 c d-3 b e)-4 A c \left (2 c d^2-a e^2\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{4 \sqrt {2} a c^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {b^2-4 a c} \left (4 A c e (3 c d-b e)+B \left (3 c^2 d^2+3 b^2 e^2-2 c e (3 b d+2 a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} c^2 \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:
1/8*(4*A*c*e-3*B*b*e+5*B*c*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/x+ 1/4*B*e*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c-1/16*(-4*a*c+b^2)^(1/2)* (4*A*c*e-3*B*b*e+5*B*c*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2 +d)^(1/2)*EllipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2 ^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1 /2)/c^2/(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2 +a)^(1/2)+1/8*(-4*a*c+b^2)^(1/2)*(a*B*e*(-3*b*e+7*c*d)-4*A*c*(-a*e^2+2*c*d ^2))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2 )^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticF(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/ 2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d- 2*a*e))^(1/2))*2^(1/2)/a/c^2/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+1/4*(-4 *a*c+b^2)^(1/2)*(4*A*c*e*(-b*e+3*c*d)+B*(3*c^2*d^2+3*b^2*e^2-2*c*e*(2*a*e+ 3*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c +b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticPi(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2 )^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2)),2^(1/2) *((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/c^ 2/(b+(-4*a*c+b^2)^(1/2))/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2))/Sqrt[a + b*x^2 + c*x^4],x]
Output:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2))/Sqrt[a + b*x^2 + 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx\) |
| \(\Big \downarrow \) 2260 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}}dx\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^(3/2))/Sqrt[a + b*x^2 + c*x^4],x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\sqrt {c \,x^{4}+b \,x^{2}+a}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(1/2),x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(1/2),x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a)**(1/2),x)
Output:
Integral((A + B*x**2)*(d + e*x**2)**(3/2)/sqrt(a + b*x**2 + c*x**4), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="ma xima")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^(3/2)/sqrt(c*x^4 + b*x^2 + a), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="gi ac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^(3/2)/sqrt(c*x^4 + b*x^2 + a), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{3/2}}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4)^(1/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4)^(1/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, b e x +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) a c \,e^{2}-3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) b^{2} e^{2}+5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) b c d e -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) a b \,e^{2}+8 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) a c d e -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) b^{2} d e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) b c \,d^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) a b d e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c e \,x^{6}+b e \,x^{4}+c d \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d}d x \right ) a c \,d^{2}}{4 c} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(1/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*e*x + 4*int((sqrt(d + e*x**2 )*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*a*c*e**2 - 3*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x* *6),x)*b**2*e**2 + 5*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4) /(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*b*c*d*e - 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**2)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*a*b*e**2 + 8*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**2)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x **4 + c*d*x**4 + c*e*x**6),x)*a*c*d*e - 2*int((sqrt(d + e*x**2)*sqrt(a + b *x**2 + c*x**4)*x**2)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c *e*x**6),x)*b**2*d*e + 4*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x **2)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*b*c*d **2 - int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4))/(a*d + a*e*x**2 + b *d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*a*b*d*e + 4*int((sqrt(d + e*x **2)*sqrt(a + b*x**2 + c*x**4))/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c* d*x**4 + c*e*x**6),x)*a*c*d**2)/(4*c)