Integrand size = 35, antiderivative size = 905 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {e (B d-A e) x}{d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {\left (2 a B d \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )-A \left (b^3 d e^2+b c d \left (c d^2-3 a e^2\right )+4 a c e \left (c d^2-a e^2\right )-b^2 \left (2 c d^2 e-a e^3\right )\right )\right ) \sqrt {d+e x^2}}{\left (b^2-4 a c\right ) d \left (c d^2-b d e+a e^2\right )^2 x \sqrt {a+b x^2+c x^4}}+\frac {c \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) x \sqrt {d+e x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\left (2 a B d \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )-A \left (b^3 d e^2+b c d \left (c d^2-3 a e^2\right )+4 a c e \left (c d^2-a e^2\right )-b^2 \left (2 c d^2 e-a e^3\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} d \left (c d^2-b d e+a e^2\right )^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {2} \left (b^2 e (B d-A e)-b c d (B d+A e)+2 c \left (A c d^2-a B d e+2 a A e^2\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{a \sqrt {b^2-4 a c} d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:
-e*(-A*e+B*d)*x/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2 )+(2*a*B*d*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))-A*(b^3*d*e^2+b*c*d*(-3*a*e^2+ c*d^2)+4*a*c*e*(-a*e^2+c*d^2)-b^2*(-a*e^3+2*c*d^2*e)))*(e*x^2+d)^(1/2)/(-4 *a*c+b^2)/d/(a*e^2-b*d*e+c*d^2)^2/x/(c*x^4+b*x^2+a)^(1/2)+c*(b^2*e*(-2*A*e +B*d)-2*c*(-3*A*a*e^2+A*c*d^2+4*B*a*d*e)+b*(2*A*c*d*e+B*a*e^2+B*c*d^2))*x* (e*x^2+d)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)^(1/2)-1 /2*(2*a*B*d*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))-A*(b^3*d*e^2+b*c*d*(-3*a*e^2 +c*d^2)+4*a*c*e*(-a*e^2+c*d^2)-b^2*(-a*e^3+2*c*d^2*e)))*(-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)/a/(-4*a*c+b^2)^(1/2)/d/(a*e^2-b*d*e+c*d^ 2)^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)+2^(1/2)*(b^2*e*(-A*e+B*d)-b*c*d*(A*e+B*d)+2*c*(2*A*a*e^2+A*c*d^2-B* a*d*e))*(-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))/a/(-4*a*c+b^2)^(1/2)/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/ 2)/(c*x^4+b*x^2+a)^(1/2)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx \] Input:
Integrate[(A + B*x^2)/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),x]
Output:
Integrate[(A + B*x^2)/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(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.
| \(\displaystyle \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2260 |
| \(\displaystyle \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}}dx\) |
Input:
Int[(A + B*x^2)/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),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]
\[\int \frac {B \,x^{2}+A}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x\]
Input:
int((B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x)
Output:
int((B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fr icas")
Output:
integral(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(c^2*e^2*x^12 + 2*(c^2*d*e + b*c*e^2)*x^10 + (c^2*d^2 + 4*b*c*d*e + (b^2 + 2*a*c)*e^2)* x^8 + 2*(b*c*d^2 + a*b*e^2 + (b^2 + 2*a*c)*d*e)*x^6 + (4*a*b*d*e + a^2*e^2 + (b^2 + 2*a*c)*d^2)*x^4 + a^2*d^2 + 2*(a*b*d^2 + a^2*d*e)*x^2), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)/(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral((A + B*x**2)/((d + e*x**2)**(3/2)*(a + b*x**2 + c*x**4)**(3/2)), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="ma xima")
Output:
integrate((B*x^2 + A)/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)^(3/2)), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="gi ac")
Output:
integrate((B*x^2 + A)/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)^(3/2)), x)
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x^2+d\right )}^{3/2}\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int((A + B*x^2)/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)),x)
Output:
int((A + B*x^2)/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(3/2)), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} e^{2} x^{12}+2 b c \,e^{2} x^{10}+2 c^{2} d e \,x^{10}+2 a c \,e^{2} x^{8}+b^{2} e^{2} x^{8}+4 b c d e \,x^{8}+c^{2} d^{2} x^{8}+2 a b \,e^{2} x^{6}+4 a c d e \,x^{6}+2 b^{2} d e \,x^{6}+2 b c \,d^{2} x^{6}+a^{2} e^{2} x^{4}+4 a b d e \,x^{4}+2 a c \,d^{2} x^{4}+b^{2} d^{2} x^{4}+2 a^{2} d e \,x^{2}+2 a b \,d^{2} x^{2}+a^{2} d^{2}}d x \right ) b +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} e^{2} x^{12}+2 b c \,e^{2} x^{10}+2 c^{2} d e \,x^{10}+2 a c \,e^{2} x^{8}+b^{2} e^{2} x^{8}+4 b c d e \,x^{8}+c^{2} d^{2} x^{8}+2 a b \,e^{2} x^{6}+4 a c d e \,x^{6}+2 b^{2} d e \,x^{6}+2 b c \,d^{2} x^{6}+a^{2} e^{2} x^{4}+4 a b d e \,x^{4}+2 a c \,d^{2} x^{4}+b^{2} d^{2} x^{4}+2 a^{2} d e \,x^{2}+2 a b \,d^{2} x^{2}+a^{2} d^{2}}d x \right ) a \] Input:
int((B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**2)/(a**2*d**2 + 2*a**2* d*e*x**2 + a**2*e**2*x**4 + 2*a*b*d**2*x**2 + 4*a*b*d*e*x**4 + 2*a*b*e**2* x**6 + 2*a*c*d**2*x**4 + 4*a*c*d*e*x**6 + 2*a*c*e**2*x**8 + b**2*d**2*x**4 + 2*b**2*d*e*x**6 + b**2*e**2*x**8 + 2*b*c*d**2*x**6 + 4*b*c*d*e*x**8 + 2 *b*c*e**2*x**10 + c**2*d**2*x**8 + 2*c**2*d*e*x**10 + c**2*e**2*x**12),x)* b + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4))/(a**2*d**2 + 2*a**2*d *e*x**2 + a**2*e**2*x**4 + 2*a*b*d**2*x**2 + 4*a*b*d*e*x**4 + 2*a*b*e**2*x **6 + 2*a*c*d**2*x**4 + 4*a*c*d*e*x**6 + 2*a*c*e**2*x**8 + b**2*d**2*x**4 + 2*b**2*d*e*x**6 + b**2*e**2*x**8 + 2*b*c*d**2*x**6 + 4*b*c*d*e*x**8 + 2* b*c*e**2*x**10 + c**2*d**2*x**8 + 2*c**2*d*e*x**10 + c**2*e**2*x**12),x)*a