\(\int \frac {A+B x^2}{(d-e x^2) \sqrt {a d^2+b d^2 x^2+(b d e-a e^2) x^4}} \, dx\) [132]

Optimal result

Integrand size = 51, antiderivative size = 292 \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2+b d^2 x^2+\left (b d e-a e^2\right ) x^4}} \, dx=-\frac {\sqrt {a} B \sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sqrt {1+\frac {(b d-a e) x^2}{a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b d+a e} x}{\sqrt {a} \sqrt {d}}\right ),\frac {a e}{b d-a e}\right )}{e \sqrt {-b d+a e} \sqrt {a d^2+b d^2 x^2+e (b d-a e) x^4}}+\frac {\sqrt {a} (B d+A e) \sqrt {1+\frac {e x^2}{d}} \sqrt {1+\frac {(b d-a e) x^2}{a d}} \operatorname {EllipticPi}\left (-\frac {a e}{b d-a e},\arcsin \left (\frac {\sqrt {-b d+a e} x}{\sqrt {a} \sqrt {d}}\right ),\frac {a e}{b d-a e}\right )}{\sqrt {d} e \sqrt {-b d+a e} \sqrt {a d^2+b d^2 x^2+e (b d-a e) x^4}} \] Output:

-a^(1/2)*B*d^(1/2)*(1+e*x^2/d)^(1/2)*(1+(-a*e+b*d)*x^2/a/d)^(1/2)*Elliptic 
F((a*e-b*d)^(1/2)*x/a^(1/2)/d^(1/2),(a*e/(-a*e+b*d))^(1/2))/e/(a*e-b*d)^(1 
/2)/(a*d^2+b*d^2*x^2+e*(-a*e+b*d)*x^4)^(1/2)+a^(1/2)*(A*e+B*d)*(1+e*x^2/d) 
^(1/2)*(1+(-a*e+b*d)*x^2/a/d)^(1/2)*EllipticPi((a*e-b*d)^(1/2)*x/a^(1/2)/d 
^(1/2),-a*e/(-a*e+b*d),(a*e/(-a*e+b*d))^(1/2))/d^(1/2)/e/(a*e-b*d)^(1/2)/( 
a*d^2+b*d^2*x^2+e*(-a*e+b*d)*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.69 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2+b d^2 x^2+\left (b d e-a e^2\right ) x^4}} \, dx=\frac {i \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \left (B d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )-(B d+A e) \operatorname {EllipticPi}\left (-1,i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )\right )}{d^2 \left (\frac {e}{d}\right )^{3/2} \sqrt {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}} \] Input:

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

Output:

(I*Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*(B*d*EllipticF[I*Ar 
cSinh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)] - (B*d + A*e)*EllipticPi[-1, I*ArcSi 
nh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)]))/(d^2*(e/d)^(3/2)*Sqrt[(d + e*x^2)*(b* 
d*x^2 + a*(d - e*x^2))])
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(721\) vs. \(2(292)=584\).

Time = 1.26 (sec) , antiderivative size = 721, normalized size of antiderivative = 2.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {2226, 27, 1416, 2222}

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.

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

\(\Big \downarrow \) 2226

\(\displaystyle \frac {\left (A \left (-\sqrt {a} e \sqrt {b d-a e}-a e^{3/2}+b d \sqrt {e}\right )-\sqrt {a} B d \sqrt {b d-a e}+a B d \sqrt {e}\right ) \int \frac {1}{\sqrt {e (b d-a e) x^4+b d^2 x^2+a d^2}}dx}{d \sqrt {e} (b d-2 a e)}-\frac {\sqrt {a} \left (\sqrt {a} \sqrt {e}-\sqrt {b d-a e}\right ) (A e+B d) \int \frac {\sqrt {e} \sqrt {b d-a e} x^2+\sqrt {a} d}{\sqrt {a} d \left (d-e x^2\right ) \sqrt {e (b d-a e) x^4+b d^2 x^2+a d^2}}dx}{\sqrt {e} (b d-2 a e)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (A \left (-\sqrt {a} e \sqrt {b d-a e}-a e^{3/2}+b d \sqrt {e}\right )-\sqrt {a} B d \sqrt {b d-a e}+a B d \sqrt {e}\right ) \int \frac {1}{\sqrt {e (b d-a e) x^4+b d^2 x^2+a d^2}}dx}{d \sqrt {e} (b d-2 a e)}-\frac {\left (\sqrt {a} \sqrt {e}-\sqrt {b d-a e}\right ) (A e+B d) \int \frac {\sqrt {e} \sqrt {b d-a e} x^2+\sqrt {a} d}{\left (d-e x^2\right ) \sqrt {e (b d-a e) x^4+b d^2 x^2+a d^2}}dx}{d \sqrt {e} (b d-2 a e)}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {e x^4 (b d-a e)+a d^2+b d^2 x^2}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \left (A \left (-\sqrt {a} e \sqrt {b d-a e}-a e^{3/2}+b d \sqrt {e}\right )-\sqrt {a} B d \sqrt {b d-a e}+a B d \sqrt {e}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt [4]{a} d^{3/2} e^{3/4} (b d-2 a e) \sqrt [4]{b d-a e} \sqrt {e x^4 (b d-a e)+a d^2+b d^2 x^2}}-\frac {\left (\sqrt {a} \sqrt {e}-\sqrt {b d-a e}\right ) (A e+B d) \int \frac {\sqrt {e} \sqrt {b d-a e} x^2+\sqrt {a} d}{\left (d-e x^2\right ) \sqrt {e (b d-a e) x^4+b d^2 x^2+a d^2}}dx}{d \sqrt {e} (b d-2 a e)}\)

\(\Big \downarrow \) 2222

\(\displaystyle \frac {\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {e x^4 (b d-a e)+a d^2+b d^2 x^2}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \left (A \left (-\sqrt {a} e \sqrt {b d-a e}-a e^{3/2}+b d \sqrt {e}\right )-\sqrt {a} B d \sqrt {b d-a e}+a B d \sqrt {e}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt [4]{a} d^{3/2} e^{3/4} (b d-2 a e) \sqrt [4]{b d-a e} \sqrt {e x^4 (b d-a e)+a d^2+b d^2 x^2}}-\frac {\left (\sqrt {a} \sqrt {e}-\sqrt {b d-a e}\right ) (A e+B d) \left (\frac {\left (\sqrt {a} \sqrt {e}-\sqrt {b d-a e}\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {e x^4 (b d-a e)+a d^2+b d^2 x^2}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}+2\right ),2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{4 \sqrt [4]{a} \sqrt {d} e^{3/4} \sqrt [4]{b d-a e} \sqrt {e x^4 (b d-a e)+a d^2+b d^2 x^2}}+\frac {\left (\sqrt {b d-a e}+\sqrt {a} \sqrt {e}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} d x}{\sqrt {e x^4 (b d-a e)+a d^2+b d^2 x^2}}\right )}{2 \sqrt {2} \sqrt {b} d \sqrt {e}}\right )}{d \sqrt {e} (b d-2 a e)}\)

Input:

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

Output:

((a*B*d*Sqrt[e] - Sqrt[a]*B*d*Sqrt[b*d - a*e] + A*(b*d*Sqrt[e] - a*e^(3/2) 
 - Sqrt[a]*e*Sqrt[b*d - a*e]))*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)*S 
qrt[(a*d^2 + b*d^2*x^2 + e*(b*d - a*e)*x^4)/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d 
- a*e]*x^2)^2]*EllipticF[2*ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^(1/4)*S 
qrt[d])], (2 - (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(2*a^(1/4)*d^( 
3/2)*e^(3/4)*(b*d - 2*a*e)*(b*d - a*e)^(1/4)*Sqrt[a*d^2 + b*d^2*x^2 + e*(b 
*d - a*e)*x^4]) - ((B*d + A*e)*(Sqrt[a]*Sqrt[e] - Sqrt[b*d - a*e])*(((Sqrt 
[a]*Sqrt[e] + Sqrt[b*d - a*e])*ArcTanh[(Sqrt[2]*Sqrt[b]*d*x)/Sqrt[a*d^2 + 
b*d^2*x^2 + e*(b*d - a*e)*x^4]])/(2*Sqrt[2]*Sqrt[b]*d*Sqrt[e]) + ((Sqrt[a] 
*Sqrt[e] - Sqrt[b*d - a*e])*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)*Sqrt 
[(a*d^2 + b*d^2*x^2 + e*(b*d - a*e)*x^4)/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a 
*e]*x^2)^2]*EllipticPi[(2 + (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/4, 2* 
ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^(1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt[ 
a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(4*a^(1/4)*Sqrt[d]*e^(3/4)*(b*d - a*e)^(1 
/4)*Sqrt[a*d^2 + b*d^2*x^2 + e*(b*d - a*e)*x^4])))/(d*Sqrt[e]*(b*d - 2*a*e 
))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 2226

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) 
- a*B*(e + d*q))/(c*d^2 - a*e^2)   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + 
Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2))   Int[(1 + q*x^2)/((d + e*x^ 
2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
 

Maple [A] (verified)

Time = 1.88 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.98

method	result	size
default	\(\frac {\left (A e +B d \right ) \sqrt {-\frac {e \,x^{2}}{d}+1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {a e -b d}{d a}}, \frac {a e}{a e -b d}, \frac {\sqrt {-\frac {e}{d}}}{\sqrt {\frac {a e -b d}{d a}}}\right )}{e d \sqrt {\frac {e}{d}-\frac {b}{a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}-\frac {B \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{d a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{e \sqrt {\frac {a e -b d}{d a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}\)	\(285\)
elliptic	\(-\frac {B \sqrt {-\frac {e \,x^{2}}{d}+1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{e \sqrt {\frac {e}{d}-\frac {b}{a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}+\frac {\sqrt {-\frac {e \,x^{2}}{d}+1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {a e -b d}{d a}}, \frac {a e}{a e -b d}, \frac {\sqrt {-\frac {e}{d}}}{\sqrt {\frac {a e -b d}{d a}}}\right ) A}{d \sqrt {\frac {e}{d}-\frac {b}{a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}+\frac {\sqrt {-\frac {e \,x^{2}}{d}+1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {a e -b d}{d a}}, \frac {a e}{a e -b d}, \frac {\sqrt {-\frac {e}{d}}}{\sqrt {\frac {a e -b d}{d a}}}\right ) B}{e \sqrt {\frac {e}{d}-\frac {b}{a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}\)	\(414\)

Input:

int((B*x^2+A)/(-e*x^2+d)/(a*d^2+b*d^2*x^2+(-a*e^2+b*d*e)*x^4)^(1/2),x,meth 
od=_RETURNVERBOSE)
 

Output:

(A*e+B*d)/e/d/(1/d*e-b/a)^(1/2)*(-e*x^2/d+1+b/a*x^2)^(1/2)*(1+e*x^2/d)^(1/ 
2)/(-a*e^2*x^4+b*d*e*x^4+b*d^2*x^2+a*d^2)^(1/2)*EllipticPi(x*(1/d*(a*e-b*d 
)/a)^(1/2),a*e/(a*e-b*d),(-1/d*e)^(1/2)/(1/d*(a*e-b*d)/a)^(1/2))-B/e/(1/d* 
(a*e-b*d)/a)^(1/2)*(1-1/d*(a*e-b*d)/a*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(-a*e^2 
*x^4+b*d*e*x^4+b*d^2*x^2+a*d^2)^(1/2)*EllipticF(x*(1/d*(a*e-b*d)/a)^(1/2), 
(-1+b*d*e/(-a*e^2+b*d*e))^(1/2))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2+b d^2 x^2+\left (b d e-a e^2\right ) x^4}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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