\(\int \frac {x^2 (A+B x^2+C x^4)}{\sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\) [122]

Optimal result

Integrand size = 43, antiderivative size = 806 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=-\frac {(3 c C d-4 B c e+3 b C e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{8 c^2 e^2 x}+\frac {C x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{4 c e}+\frac {\sqrt {b^2-4 a c} (3 c C d-4 B c e+3 b C e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{8 \sqrt {2} c^2 e^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} (c C d-4 B c e+3 b C e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{4 \sqrt {2} c^2 e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} (4 c e (b C d-2 A c e+a C e)-(c d+b e) (3 c C d-4 B c e+3 b C e)) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} c^2 \left (b+\sqrt {b^2-4 a c}\right ) e^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/8*(-4*B*c*e+3*C*b*e+3*C*c*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/ 
e^2/x+1/4*C*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/e+1/16*(-4*a*c+b^2)^ 
(1/2)*(-4*B*c*e+3*C*b*e+3*C*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)/c^2/e^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)-1/8*(-4*a*c+b^2)^(1/2)*(-4*B*c*e+3*C*b*e+C*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))*2^(1/2)/c^2/e/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/4*(-4*a*c+b^2)^ 
(1/2)*(4*c*e*(-2*A*c*e+C*a*e+C*b*d)-(b*e+c*d)*(-4*B*c*e+3*C*b*e+3*C*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*EllipticPi(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^ 
(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^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)/c^2/(b+(-4* 
a*c+b^2)^(1/2))/e^2/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(x^2*(A + B*x^2 + C*x^4))/(Sqrt[d + e*x^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.

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

\(\Big \downarrow \) 2250

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ 
q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol 
yQ[Px, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

integrate(x^2*(C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^2*(C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^2*(C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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