\(\int \frac {(A+B x^2) \sqrt {a+b x^2+c x^4}}{\sqrt {d+e x^2}} \, dx\) [226]

Optimal result

Integrand size = 35, antiderivative size = 805 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\sqrt {d+e x^2}} \, dx=-\frac {(3 B c d-b B e-4 A c e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{8 c e^2 x}+\frac {B x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{4 e}+\frac {\sqrt {b^2-4 a c} (3 B c d-b B e-4 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{8 \sqrt {2} c e^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} (B c d-b B e+4 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{4 \sqrt {2} c e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} \left (4 A c e (c d-b e)-B \left (3 c^2 d^2-b^2 e^2-2 c e (b d-2 a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} c \left (b+\sqrt {b^2-4 a c}\right ) e^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

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

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

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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