\(\int \frac {(A+B x^2) (d+e x^2)^{5/2}}{(a+b x^2+c x^4)^{5/2}} \, dx\) [249]

Optimal result

Integrand size = 35, antiderivative size = 956 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{5/2}}{\left (a+b x^2+c x^4\right )^{5/2}} \, 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 )^{5/2}}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {x \left (d+e x^2\right )^{3/2} \left (a B \left (b^3 d+4 a b c d-6 a b^2 e+8 a^2 c e\right )+A \left (2 b^4 d-17 a b^2 c d+20 a^2 c^2 d+3 a b^3 e-4 a^2 b c e\right )+c \left (a B \left (b^2 d+12 a c d-8 a b e\right )+2 A b \left (b^2 d-8 a c d+2 a b e\right )\right ) x^2\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\left (A \left (2 b^3 d^2+3 a b^2 d e+20 a^2 c d e-8 a b \left (2 c d^2+a e^2\right )\right )+a B \left (b^2 d^2-16 a b d e+4 a \left (3 c d^2+4 a e^2\right )\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{3 a^2 \left (b^2-4 a c\right )^2 x}-\frac {e \left (a B \left (b^2 d+12 a c d-8 a b e\right )+2 A b \left (b^2 d-8 a c d+2 a b e\right )\right ) x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{3 a^2 \left (b^2-4 a c\right )^2}+\frac {\left (A \left (2 b^3 d^2+3 a b^2 d e+20 a^2 c d e-8 a b \left (2 c d^2+a e^2\right )\right )+a B \left (b^2 d^2-16 a b d e+4 a \left (3 c d^2+4 a e^2\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 )}{3 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \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 (c d^2-b d e+a e^2\right ) \left (8 a B (b d-2 a e)+A \left (b^2 d-20 a c d+8 a b 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^2 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

1/3*x*(A*b^2-B*a*b-2*A*a*c+(A*b-2*B*a)*c*x^2)*(e*x^2+d)^(5/2)/a/(-4*a*c+b^ 
2)/(c*x^4+b*x^2+a)^(3/2)+1/3*x*(e*x^2+d)^(3/2)*(a*B*(8*a^2*c*e-6*a*b^2*e+4 
*a*b*c*d+b^3*d)+A*(-4*a^2*b*c*e+20*a^2*c^2*d+3*a*b^3*e-17*a*b^2*c*d+2*b^4* 
d)+c*(a*B*(-8*a*b*e+12*a*c*d+b^2*d)+2*A*b*(2*a*b*e-8*a*c*d+b^2*d))*x^2)/a^ 
2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)-1/3*(A*(2*b^3*d^2+3*a*b^2*d*e+20*a^ 
2*c*d*e-8*a*b*(a*e^2+2*c*d^2))+a*B*(b^2*d^2-16*a*b*d*e+4*a*(4*a*e^2+3*c*d^ 
2)))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^2/(-4*a*c+b^2)^2/x-1/3*e*(a*B 
*(-8*a*b*e+12*a*c*d+b^2*d)+2*A*b*(2*a*b*e-8*a*c*d+b^2*d))*x*(e*x^2+d)^(1/2 
)*(c*x^4+b*x^2+a)^(1/2)/a^2/(-4*a*c+b^2)^2+1/6*(A*(2*b^3*d^2+3*a*b^2*d*e+2 
0*a^2*c*d*e-8*a*b*(a*e^2+2*c*d^2))+a*B*(b^2*d^2-16*a*b*d*e+4*a*(4*a*e^2+3* 
c*d^2)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*Ellipti 
cE(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^2/(-4*a*c+b 
^2)^(3/2)/(-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)*(a*e^2-b*d*e+c*d^2)*(8*a*B*(-2*a*e+b*d)+A*(8*a*b*e 
-20*a*c*d+b^2*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^2/(-4*a*c+b^2)^(3/2)/(e*x^2+d)^(1/2)/(c*x^4+ 
b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((B*e^2*x^6 + (2*B*d*e + A*e^2)*x^4 + A*d^2 + (B*d^2 + 2*A*d*e)*x^ 
2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)/(c^3*x^12 + 3*b*c^2*x^10 + 3*(b 
^2*c + a*c^2)*x^8 + (b^3 + 6*a*b*c)*x^6 + 3*a^2*b*x^2 + 3*(a*b^2 + a^2*c)* 
x^4 + a^3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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