\(\int \frac {(A+B x^2) \sqrt {d+e x^2}}{(a+b x^2+c x^4)^{3/2}} \, dx\) [245]

Optimal result

Integrand size = 35, antiderivative size = 533 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (A b^2-a b B-2 a A c+(A b-2 a B) c x^2\right ) \sqrt {d+e x^2}}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {(A b-2 a B) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) x}+\frac {(A b-2 a B) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} (b B d-2 A c d+A b e-2 a B e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{a \sqrt {b^2-4 a c} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

x*(A*b^2-B*a*b-2*A*a*c+(A*b-2*B*a)*c*x^2)*(e*x^2+d)^(1/2)/a/(-4*a*c+b^2)/( 
c*x^4+b*x^2+a)^(1/2)-(A*b-2*B*a)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/( 
-4*a*c+b^2)/x+1/2*(A*b-2*B*a)*(-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)/(-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)*(A*b*e-2*A*c*d-2*B*a*e+B*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*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)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4)^(3/2), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2260

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]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(3/2),x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(3/2),x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)/(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*x^8 + 2* 
b*c*x^6 + (b^2 + 2*a*c)*x^4 + 2*a*b*x^2 + a^2), x)
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {d + e x^{2}}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(1/2)/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral((A + B*x**2)*sqrt(d + e*x**2)/(a + b*x**2 + c*x**4)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((B*x^2 + A)*sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((B*x^2 + A)*sqrt(e*x^2 + d)/(c*x^4 + b*x^2 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {e\,x^2+d}}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^(1/2))/(a + b*x^2 + c*x^4)^(3/2),x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(1/2))/(a + b*x^2 + c*x^4)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)/(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*x + sqrt(d + e*x**2)*sqrt( 
a + b*x**2 + c*x**4)*b*d*x + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x** 
4)*x**4)/(a**2*d + a**2*e*x**2 + 2*a*b*d*x**2 + 2*a*b*e*x**4 + 2*a*c*d*x** 
4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2*e*x**6 + 2*b*c*d*x**6 + 2*b*c*e*x**8 
 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*b*e**2 + int((sqrt(d + e*x**2)*sqrt 
(a + b*x**2 + c*x**4)*x**4)/(a**2*d + a**2*e*x**2 + 2*a*b*d*x**2 + 2*a*b*e 
*x**4 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2*e*x**6 + 2*b*c*d* 
x**6 + 2*b*c*e*x**8 + c**2*d*x**8 + c**2*e*x**10),x)*a**2*c*d*e - int((sqr 
t(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2*d + a**2*e*x**2 + 2*a* 
b*d*x**2 + 2*a*b*e*x**4 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2 
*e*x**6 + 2*b*c*d*x**6 + 2*b*c*e*x**8 + c**2*d*x**8 + c**2*e*x**10),x)*a*b 
**2*d*e + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2*d + 
a**2*e*x**2 + 2*a*b*d*x**2 + 2*a*b*e*x**4 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + 
b**2*d*x**4 + b**2*e*x**6 + 2*b*c*d*x**6 + 2*b*c*e*x**8 + c**2*d*x**8 + c* 
*2*e*x**10),x)*a*b**2*e**2*x**2 + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + 
c*x**4)*x**4)/(a**2*d + a**2*e*x**2 + 2*a*b*d*x**2 + 2*a*b*e*x**4 + 2*a*c* 
d*x**4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2*e*x**6 + 2*b*c*d*x**6 + 2*b*c*e 
*x**8 + c**2*d*x**8 + c**2*e*x**10),x)*a*b*c*d**2 + int((sqrt(d + e*x**2)* 
sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2*d + a**2*e*x**2 + 2*a*b*d*x**2 + 2*a 
*b*e*x**4 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2*e*x**6 + 2...