\(\int \frac {A+B x^2}{(d+e x^2) \sqrt {a d^2+b d^2 x^2+(b d e-a e^2) x^4}} \, dx\) [131]

Optimal result

Integrand size = 50, antiderivative size = 261 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2+b d^2 x^2+\left (b d e-a e^2\right ) x^4}} \, dx=\frac {a (B d-A e) \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|2-\frac {b d}{a e}\right )}{d^{3/2} \sqrt {e} (b d-2 a e) \sqrt {a d^2+b d^2 x^2+e (b d-a e) x^4}}+\frac {(A b d-a B d-a A e) \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),2-\frac {b d}{a e}\right )}{d^{3/2} \sqrt {e} (b d-2 a e) \sqrt {a d^2+b d^2 x^2+e (b d-a e) x^4}} \] Output:

a*(-A*e+B*d)*(e*x^2+d)*((a*d+(-a*e+b*d)*x^2)/a/(e*x^2+d))^(1/2)*EllipticE( 
e^(1/2)*x/d^(1/2)/(1+e*x^2/d)^(1/2),(2-b*d/a/e)^(1/2))/d^(3/2)/e^(1/2)/(-2 
*a*e+b*d)/(a*d^2+b*d^2*x^2+e*(-a*e+b*d)*x^4)^(1/2)+(-A*a*e+A*b*d-B*a*d)*(e 
*x^2+d)*((a*d+(-a*e+b*d)*x^2)/a/(e*x^2+d))^(1/2)*InverseJacobiAM(arctan(e^ 
(1/2)*x/d^(1/2)),(2-b*d/a/e)^(1/2))/d^(3/2)/e^(1/2)/(-2*a*e+b*d)/(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 = 11.60 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2+b d^2 x^2+\left (b d e-a e^2\right ) x^4}} \, dx=\frac {\sqrt {\frac {e}{d}} (B d-A e) x \left (b d x^2+a \left (d-e x^2\right )\right )+i a d (B d-A e) \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|-1+\frac {b d}{a e}\right )-i A d (b d-2 a e) \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )}{d^2 \sqrt {\frac {e}{d}} (b d-2 a e) \sqrt {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}} \] 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:

(Sqrt[e/d]*(B*d - A*e)*x*(b*d*x^2 + a*(d - e*x^2)) + I*a*d*(B*d - A*e)*Sqr 
t[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*ArcSinh[Sqrt[ 
e/d]*x], -1 + (b*d)/(a*e)] - I*A*d*(b*d - 2*a*e)*Sqrt[1 + (b*x^2)/a - (e*x 
^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[e/d]*x], -1 + (b*d)/(a 
*e)])/(d^2*Sqrt[e/d]*(b*d - 2*a*e)*Sqrt[(d + e*x^2)*(b*d*x^2 + a*(d - e*x^ 
2))])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1395, 400, 313, 320}

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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

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:

(Sqrt[d + e*x^2]*Sqrt[a*d + (b*d - a*e)*x^2]*(((B*d - A*e)*Sqrt[a*d + (b*d 
 - a*e)*x^2]*EllipticE[ArcTan[(Sqrt[e]*x)/Sqrt[d]], 2 - (b*d)/(a*e)])/(d^( 
3/2)*Sqrt[e]*(b*d - 2*a*e)*Sqrt[d + e*x^2]*Sqrt[(a*d + (b*d - a*e)*x^2)/(a 
*(d + e*x^2))]) + ((A*b*d - a*B*d - a*A*e)*Sqrt[a*d + (b*d - a*e)*x^2]*Ell 
ipticF[ArcTan[(Sqrt[e]*x)/Sqrt[d]], 2 - (b*d)/(a*e)])/(a*d^(3/2)*Sqrt[e]*( 
b*d - 2*a*e)*Sqrt[d + e*x^2]*Sqrt[(a*d + (b*d - a*e)*x^2)/(a*(d + e*x^2))] 
)))/Sqrt[a*d^2 + b*d^2*x^2 + e*(b*d - a*e)*x^4]
 

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 400

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[Sqrt[a + b*x^ 
2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & 
& PosQ[d/c]
 

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs. \(2(252)=504\).

Time = 2.37 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.14

method	result	size
elliptic	\(\frac {\left (-a \,e^{2} x^{2}+b d e \,x^{2}+a d e \right ) x \left (A e -B d \right )}{d^{2} e \left (2 a e -b d \right ) \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-a \,e^{2} x^{2}+b d e \,x^{2}+a d e \right )}}+\frac {\left (\frac {B}{e}+\frac {A e -B d}{d e}-\frac {a \left (A e -B d \right )}{d \left (2 a e -b d \right )}\right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{d a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{d a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}-\frac {2 \left (-\frac {\left (a e -b d \right ) \left (A e -B d \right )}{d^{2} \left (2 a e -b d \right )}-\frac {\left (-2 a \,e^{2}+2 b d e \right ) \left (A e -B d \right )}{2 d^{2} e \left (2 a e -b d \right )}-\frac {\left (-a \,e^{2}+b d e \right ) \left (A e -B d \right )}{e \,d^{2} \left (2 a e -b d \right )}\right ) a \,d^{2} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{d a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )\right )}{\sqrt {\frac {a e -b d}{d a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}\, \left (b \,d^{2}+d \left (2 a e -b d \right )\right )}\)	\(559\)
default	\(\frac {B \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{d a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{e \sqrt {\frac {a e -b d}{d a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}+\frac {\left (A e -B d \right ) \left (\frac {\left (-a \,e^{2} x^{2}+b d e \,x^{2}+a d e \right ) x}{d^{2} \left (2 a e -b d \right ) \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-a \,e^{2} x^{2}+b d e \,x^{2}+a d e \right )}}+\frac {\left (\frac {1}{d}-\frac {a e}{d \left (2 a e -b d \right )}\right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{d a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{d a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}-\frac {2 \left (-\frac {e \left (a e -b d \right )}{d^{2} \left (2 a e -b d \right )}-\frac {-2 a \,e^{2}+2 b d e}{2 d^{2} \left (2 a e -b d \right )}-\frac {-a \,e^{2}+b d e}{d^{2} \left (2 a e -b d \right )}\right ) a \,d^{2} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{d a}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{d a}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )\right )}{\sqrt {\frac {a e -b d}{d a}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}\, \left (b \,d^{2}+d \left (2 a e -b d \right )\right )}\right )}{e}\)	\(638\)

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,metho 
d=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (252) = 504\).

Time = 0.10 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.98 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2+b d^2 x^2+\left (b d e-a e^2\right ) x^4}} \, dx=-\frac {{\left (B b^{2} d^{4} - A a^{2} d e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + {\left (B b^{2} d^{3} e - A a^{2} e^{4} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{2}\right )} \sqrt {a d^{2}} \sqrt {-\frac {b d - a e}{a d}} E(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e}) - {\left (B b^{2} d^{4} - A a^{2} d e^{3} + {\left (B a^{2} - {\left (A + 2 \, B\right )} a b - A b^{2}\right )} d^{3} e + {\left ({\left (A + B\right )} a^{2} + 2 \, A a b\right )} d^{2} e^{2} + {\left (B b^{2} d^{3} e - A a^{2} e^{4} + {\left (B a^{2} - {\left (A + 2 \, B\right )} a b - A b^{2}\right )} d^{2} e^{2} + {\left ({\left (A + B\right )} a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{2}\right )} \sqrt {a d^{2}} \sqrt {-\frac {b d - a e}{a d}} F(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e}) - {\left (B a b d^{3} e + A a^{2} d e^{3} - {\left (B a^{2} + A a b\right )} d^{2} e^{2}\right )} \sqrt {b d^{2} x^{2} + {\left (b d e - a e^{2}\right )} x^{4} + a d^{2}} x}{a b^{2} d^{6} e - 3 \, a^{2} b d^{5} e^{2} + 2 \, a^{3} d^{4} e^{3} + {\left (a b^{2} d^{5} e^{2} - 3 \, a^{2} b d^{4} e^{3} + 2 \, a^{3} d^{3} e^{4}\right )} x^{2}} \] 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:

-((B*b^2*d^4 - A*a^2*d*e^3 - (2*B*a*b + A*b^2)*d^3*e + (B*a^2 + 2*A*a*b)*d 
^2*e^2 + (B*b^2*d^3*e - A*a^2*e^4 - (2*B*a*b + A*b^2)*d^2*e^2 + (B*a^2 + 2 
*A*a*b)*d*e^3)*x^2)*sqrt(a*d^2)*sqrt(-(b*d - a*e)/(a*d))*elliptic_e(arcsin 
(x*sqrt(-(b*d - a*e)/(a*d))), a*e/(b*d - a*e)) - (B*b^2*d^4 - A*a^2*d*e^3 
+ (B*a^2 - (A + 2*B)*a*b - A*b^2)*d^3*e + ((A + B)*a^2 + 2*A*a*b)*d^2*e^2 
+ (B*b^2*d^3*e - A*a^2*e^4 + (B*a^2 - (A + 2*B)*a*b - A*b^2)*d^2*e^2 + ((A 
 + B)*a^2 + 2*A*a*b)*d*e^3)*x^2)*sqrt(a*d^2)*sqrt(-(b*d - a*e)/(a*d))*elli 
ptic_f(arcsin(x*sqrt(-(b*d - a*e)/(a*d))), a*e/(b*d - a*e)) - (B*a*b*d^3*e 
 + A*a^2*d*e^3 - (B*a^2 + A*a*b)*d^2*e^2)*sqrt(b*d^2*x^2 + (b*d*e - a*e^2) 
*x^4 + a*d^2)*x)/(a*b^2*d^6*e - 3*a^2*b*d^5*e^2 + 2*a^3*d^4*e^3 + (a*b^2*d 
^5*e^2 - 3*a^2*b*d^4*e^3 + 2*a^3*d^3*e^4)*x^2)
 

Sympy [F]

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

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