\(\int \frac {(A+B x^2) \sqrt {d+e x^2}}{(a+b x^2+c x^4)^{5/2}} \, dx\) [251]

Optimal result

Integrand size = 35, antiderivative size = 965 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^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 ) \sqrt {d+e x^2}}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {\left (a B \left (b^3 d e+12 a b c d e-b^2 \left (c d^2+2 a e^2\right )-4 a c \left (3 c d^2+2 a e^2\right )\right )+A \left (2 b^4 d e-17 a b^2 c d e+4 a^2 c^2 d e-b^3 \left (2 c d^2+a e^2\right )+4 a b c \left (4 c d^2+3 a e^2\right )\right )\right ) \sqrt {d+e x^2}}{3 a \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {a+b x^2+c x^4}}+\frac {c \left (a B \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )-A \left (b^2 c d^2-b^3 d e+20 a b c d e-4 a c \left (5 c d^2+4 a e^2\right )\right )\right ) x \sqrt {d+e x^2}}{3 a \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (a B \left (b^3 d e+12 a b c d e-b^2 \left (c d^2+2 a e^2\right )-4 a c \left (3 c d^2+2 a e^2\right )\right )+A \left (2 b^4 d e-17 a b^2 c d e+4 a^2 c^2 d e-b^3 \left (2 c d^2+a e^2\right )+4 a b c \left (4 c d^2+3 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} \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} \left (2 a B \left (4 b c d-b^2 e-4 a c e\right )+A \left (b^2 c d-20 a c^2 d-b^3 e+12 a b c 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)^(1/2)/a/(-4*a*c+b^ 
2)/(c*x^4+b*x^2+a)^(3/2)+1/3*(a*B*(b^3*d*e+12*a*b*c*d*e-b^2*(2*a*e^2+c*d^2 
)-4*a*c*(2*a*e^2+3*c*d^2))+A*(2*b^4*d*e-17*a*b^2*c*d*e+4*a^2*c^2*d*e-b^3*( 
a*e^2+2*c*d^2)+4*a*b*c*(3*a*e^2+4*c*d^2)))*(e*x^2+d)^(1/2)/a/(-4*a*c+b^2)^ 
2/(a*e^2-b*d*e+c*d^2)/x/(c*x^4+b*x^2+a)^(1/2)+1/3*c*(a*B*(7*b^2*d*e+4*a*c* 
d*e-8*b*(a*e^2+c*d^2))-A*(b^2*c*d^2-b^3*d*e+20*a*b*c*d*e-4*a*c*(4*a*e^2+5* 
c*d^2)))*x*(e*x^2+d)^(1/2)/a/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/(c*x^4+b*x 
^2+a)^(1/2)-1/6*(a*B*(b^3*d*e+12*a*b*c*d*e-b^2*(2*a*e^2+c*d^2)-4*a*c*(2*a* 
e^2+3*c*d^2))+A*(2*b^4*d*e-17*a*b^2*c*d*e+4*a^2*c^2*d*e-b^3*(a*e^2+2*c*d^2 
)+4*a*b*c*(3*a*e^2+4*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)*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^2/(-4*a*c+b^2)^(3/2)/(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)+1/3*2^(1/2)*(2*a*B*(- 
4*a*c*e-b^2*e+4*b*c*d)+A*(12*a*b*c*e-20*a*c^2*d-b^3*e+b^2*c*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]

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

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

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^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.

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

\(\Big \downarrow \) 2260

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

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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