\(\int (A+B x^2) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx\) [106]

Optimal result

Integrand size = 35, antiderivative size = 964 \[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx=\frac {\left (6 A c e (c d+b e)-B \left (3 c^2 d^2+3 b^2 e^2-2 c e (b d+4 a e)\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{48 c^2 e^2 x}+\frac {(B c d+b B e+6 A c e) x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{24 c e}+\frac {1}{6} B x^3 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}-\frac {\sqrt {b^2-4 a c} \left (6 A c e (c d+b e)-B \left (3 c^2 d^2+3 b^2 e^2-2 c e (b d+4 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 )}{48 \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} \left (6 A c e (5 c d-b e)+B \left (c^2 d^2+3 b^2 e^2-4 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 {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 )}{24 \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} \left (B (c d-b e) \left (c^2 d^2-b^2 e^2+4 a c e^2\right )-2 A c e \left (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 )}{4 \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:

1/48*(6*A*c*e*(b*e+c*d)-B*(3*c^2*d^2+3*b^2*e^2-2*c*e*(4*a*e+b*d)))*(e*x^2+ 
d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e^2/x+1/24*(6*A*c*e+B*b*e+B*c*d)*x*(e*x 
^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/e+1/6*B*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x 
^2+a)^(1/2)-1/96*(-4*a*c+b^2)^(1/2)*(6*A*c*e*(b*e+c*d)-B*(3*c^2*d^2+3*b^2* 
e^2-2*c*e*(4*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 
)/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)-1/48*(-4*a*c+b^2)^(1/2)*(6*A*c*e*(-b*e+5*c*d)+B*(c^2*d^2+3*b^2 
*e^2-4*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*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)+1/8*(-4*a*c+b^2)^(1/2)*(B*(-b*e+c*d)*(4*a*c*e^2-b^2*e^2+c^2*d 
^2)-2*A*c*e*(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*E 
llipticPi(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]

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

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(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), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(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. \[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx=\int \left (B\,x^2+A\right )\,\sqrt {e\,x^2+d}\,\sqrt {c\,x^4+b\,x^2+a} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(6*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c*e*x + sqrt(d + e*x**2)*s 
qrt(a + b*x**2 + c*x**4)*b**2*e*x + sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x 
**4)*b*c*d*x + 4*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c*e*x**3 + 1 
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*b*c*e**2 + 6*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**2*d*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**3*e**2 + 2*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*c*d*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**2*d**2 + 1 
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**2*c*e**2 - 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*b**2*e**2 + 22*int((sqrt(d + e*x**2)*sqr 
t(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*c*d*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**3*d*e - 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**2)/(...