Integrand size = 35, antiderivative size = 895 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (A b^2-a b B-2 a A c+(A b-2 a B) c x^2\right ) \left (d+e x^2\right )^{3/2}}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {(A c (b d-2 a e)-a B (2 c d-b e)) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{a c \left (b^2-4 a c\right ) x}-\frac {(A b-2 a B) e x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right )}+\frac {(A c (b d-2 a e)-a B (2 c d-b 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 )}{\sqrt {2} a c \sqrt {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}} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {2} \left (2 c \left (A c d^2+2 a B d e+a A e^2\right )-b \left (B c d^2+2 A c d e+a B 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 c \sqrt {b^2-4 a c} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} e^2 \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 )}{c \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:
x*(A*b^2-B*a*b-2*A*a*c+(A*b-2*B*a)*c*x^2)*(e*x^2+d)^(3/2)/a/(-4*a*c+b^2)/( c*x^4+b*x^2+a)^(1/2)-(A*c*(-2*a*e+b*d)-a*B*(-b*e+2*c*d))*(e*x^2+d)^(1/2)*( c*x^4+b*x^2+a)^(1/2)/a/c/(-4*a*c+b^2)/x-(A*b-2*B*a)*e*x*(e*x^2+d)^(1/2)*(c *x^4+b*x^2+a)^(1/2)/a/(-4*a*c+b^2)+1/2*(A*c*(-2*a*e+b*d)-a*B*(-b*e+2*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)/a/c/(-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)/(c*x^4+b*x^2+a)^(1 /2)+2^(1/2)*(2*c*(A*a*e^2+A*c*d^2+2*B*a*d*e)-b*(2*A*c*d*e+B*a*e^2+B*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))/a/c/(-4*a*c+b^2)^(1/2)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+2 *2^(1/2)*B*(-4*a*c+b^2)^(1/2)*e^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*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))/c/(b+(-4*a*c+b^2)^(1/2))/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1 /2)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx \] 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2260 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}}dx\) |
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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x\]
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)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] 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:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] 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)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] 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)*(e*x^2 + d)^(3/2)/(c*x^4 + b*x^2 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] 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)*(e*x^2 + d)^(3/2)/(c*x^4 + b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{3/2}}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] 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)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a)^(3/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*e**2*x + 2*sqrt(d + e*x**2)* sqrt(a + b*x**2 + c*x**4)*b*d*e*x + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**6)/(a**2*b*d*e + a**2*b*e**2*x**2 - a**2*c*d**2 - a**2*c*d*e* x**2 + 2*a*b**2*d*e*x**2 + 2*a*b**2*e**2*x**4 - 2*a*b*c*d**2*x**2 + 2*a*b* c*e**2*x**6 - 2*a*c**2*d**2*x**4 - 2*a*c**2*d*e*x**6 + b**3*d*e*x**4 + b** 3*e**2*x**6 - b**2*c*d**2*x**4 + b**2*c*d*e*x**6 + 2*b**2*c*e**2*x**8 - 2* b*c**2*d**2*x**6 - b*c**2*d*e*x**8 + b*c**2*e**2*x**10 - c**3*d**2*x**8 - c**3*d*e*x**10),x)*a*b**3*e**4 - 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**6)/(a**2*b*d*e + a**2*b*e**2*x**2 - a**2*c*d**2 - a**2*c*d*e*x **2 + 2*a*b**2*d*e*x**2 + 2*a*b**2*e**2*x**4 - 2*a*b*c*d**2*x**2 + 2*a*b*c *e**2*x**6 - 2*a*c**2*d**2*x**4 - 2*a*c**2*d*e*x**6 + b**3*d*e*x**4 + b**3 *e**2*x**6 - b**2*c*d**2*x**4 + b**2*c*d*e*x**6 + 2*b**2*c*e**2*x**8 - 2*b *c**2*d**2*x**6 - b*c**2*d*e*x**8 + b*c**2*e**2*x**10 - c**3*d**2*x**8 - c **3*d*e*x**10),x)*a*b**2*c*d*e**3 + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**6)/(a**2*b*d*e + a**2*b*e**2*x**2 - a**2*c*d**2 - a**2*c*d*e* x**2 + 2*a*b**2*d*e*x**2 + 2*a*b**2*e**2*x**4 - 2*a*b*c*d**2*x**2 + 2*a*b* c*e**2*x**6 - 2*a*c**2*d**2*x**4 - 2*a*c**2*d*e*x**6 + b**3*d*e*x**4 + b** 3*e**2*x**6 - b**2*c*d**2*x**4 + b**2*c*d*e*x**6 + 2*b**2*c*e**2*x**8 - 2* b*c**2*d**2*x**6 - b*c**2*d*e*x**8 + b*c**2*e**2*x**10 - c**3*d**2*x**8 - c**3*d*e*x**10),x)*a*b*c**2*d**2*e**2 + int((sqrt(d + e*x**2)*sqrt(a + ...