\(\int \frac {A+B x^2}{\sqrt {d+e x^2} (a+b x^2+c x^4)^{3/2}} \, dx\) [246]

Optimal result

Integrand size = 35, antiderivative size = 611 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) \sqrt {d+e x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) x \sqrt {a+b x^2+c x^4}}+\frac {c (b (B d+A e)-2 (A c d+a B e)) x \sqrt {d+e x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} (b B-2 A c) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{a \sqrt {b^2-4 a c} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

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

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

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

\(\Big \downarrow \) 2260

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

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]

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

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]

\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x^{2} + d}} \,d x } \] 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*e*x^10 + 
 (c^2*d + 2*b*c*e)*x^8 + (2*b*c*d + (b^2 + 2*a*c)*e)*x^6 + (2*a*b*e + (b^2 
 + 2*a*c)*d)*x^4 + a^2*d + (2*a*b*d + a^2*e)*x^2), x)
 

Sympy [F]

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

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

Giac [F]

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

Mupad [F(-1)]

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

\[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} e \,x^{10}+2 b c e \,x^{8}+c^{2} d \,x^{8}+2 a c e \,x^{6}+b^{2} e \,x^{6}+2 b c d \,x^{6}+2 a b e \,x^{4}+2 a c d \,x^{4}+b^{2} d \,x^{4}+a^{2} e \,x^{2}+2 a b d \,x^{2}+a^{2} d}d x \right ) b +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} e \,x^{10}+2 b c e \,x^{8}+c^{2} d \,x^{8}+2 a c e \,x^{6}+b^{2} e \,x^{6}+2 b c d \,x^{6}+2 a b e \,x^{4}+2 a c d \,x^{4}+b^{2} d \,x^{4}+a^{2} e \,x^{2}+2 a b d \,x^{2}+a^{2} d}d x \right ) a \] Input:

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