\(\int \frac {A+B x^2}{(d+e x^2)^{5/2} \sqrt {a+b x^2+c x^4}} \, dx\) [242]

Optimal result

Integrand size = 35, antiderivative size = 660 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{3 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^{3/2}}-\frac {\left (2 A e \left (3 c d^2-e (2 b d-a e)\right )-B d \left (3 c d^2-e (b d+a e)\right )\right ) \sqrt {a+b x^2+c x^4}}{3 d \left (c d^2-b d e+a e^2\right )^2 x \sqrt {d+e x^2}}+\frac {\sqrt {b^2-4 a c} \left (2 A e \left (3 c d^2-e (2 b d-a e)\right )-B d \left (3 c d^2-e (b d+a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{3 \sqrt {2} d^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (a B d e+A \left (3 c d^2-e (3 b d-2 a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{3 a d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/3*e*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^ 
(3/2)-1/3*(2*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(3*c*d^2-e*(a*e+b*d)))*(c*x^ 
4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c*d^2)^2/x/(e*x^2+d)^(1/2)+1/6*(-4*a*c+b^2 
)^(1/2)*(2*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(3*c*d^2-e*(a*e+b*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)/d^2/(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)-1 
/3*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*B*d*e+A*(3*c*d^2-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*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/d^2/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx \] Input:

Integrate[(A + B*x^2)/((d + e*x^2)^(5/2)*Sqrt[a + b*x^2 + c*x^4]),x]
 

Output:

Integrate[(A + B*x^2)/((d + e*x^2)^(5/2)*Sqrt[a + b*x^2 + c*x^4]), 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.

\(\displaystyle \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}}dx\)

Input:

Int[(A + B*x^2)/((d + e*x^2)^(5/2)*Sqrt[a + b*x^2 + c*x^4]),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]

\[\int \frac {B \,x^{2}+A}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}}d x\]

Input:

int((B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int((B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+b*x^2+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(c*e^3*x^10 + 
 (3*c*d*e^2 + b*e^3)*x^8 + (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*x^6 + (c*d^3 + 
3*b*d^2*e + 3*a*d*e^2)*x^4 + a*d^3 + (b*d^3 + 3*a*d^2*e)*x^2), x)
 

Sympy [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {A + B x^{2}}{\left (d + e x^{2}\right )^{\frac {5}{2}} \sqrt {a + b x^{2} + c x^{4}}}\, dx \] Input:

integrate((B*x**2+A)/(e*x**2+d)**(5/2)/(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x**2)/((d + e*x**2)**(5/2)*sqrt(a + b*x**2 + c*x**4)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)^(5/2)), x)
 

Giac [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x^2+d\right )}^{5/2}\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \] Input:

int((A + B*x^2)/((d + e*x^2)^(5/2)*(a + b*x^2 + c*x^4)^(1/2)),x)
 

Output:

int((A + B*x^2)/((d + e*x^2)^(5/2)*(a + b*x^2 + c*x^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a+b x^2+c x^4}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,e^{3} x^{10}+b \,e^{3} x^{8}+3 c d \,e^{2} x^{8}+a \,e^{3} x^{6}+3 b d \,e^{2} x^{6}+3 c \,d^{2} e \,x^{6}+3 a d \,e^{2} x^{4}+3 b \,d^{2} e \,x^{4}+c \,d^{3} x^{4}+3 a \,d^{2} e \,x^{2}+b \,d^{3} x^{2}+a \,d^{3}}d x \right ) b +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,e^{3} x^{10}+b \,e^{3} x^{8}+3 c d \,e^{2} x^{8}+a \,e^{3} x^{6}+3 b d \,e^{2} x^{6}+3 c \,d^{2} e \,x^{6}+3 a d \,e^{2} x^{4}+3 b \,d^{2} e \,x^{4}+c \,d^{3} x^{4}+3 a \,d^{2} e \,x^{2}+b \,d^{3} x^{2}+a \,d^{3}}d x \right ) a \] Input:

int((B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**2)/(a*d**3 + 3*a*d**2*e 
*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 + b*d**3*x**2 + 3*b*d**2*e*x**4 + 3* 
b*d*e**2*x**6 + b*e**3*x**8 + c*d**3*x**4 + 3*c*d**2*e*x**6 + 3*c*d*e**2*x 
**8 + c*e**3*x**10),x)*b + int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4) 
)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 + b*d**3*x**2 
+ 3*b*d**2*e*x**4 + 3*b*d*e**2*x**6 + b*e**3*x**8 + c*d**3*x**4 + 3*c*d**2 
*e*x**6 + 3*c*d*e**2*x**8 + c*e**3*x**10),x)*a