3.1.23 \(\int \frac {A+B x^2}{(d+e x^2)^3 \sqrt {a+b x^2+c x^4}} \, dx\) [23]

Optimal. Leaf size=1125 \[ -\frac {\sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}-\frac {\left (B d \left (3 c^2 d^4-10 a c d^2 e^2+a e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{16 d^{5/2} \sqrt {e} \left (c d^2-b d e+a e^2\right )^{5/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (B d \left (3 c^2 d^4-10 a c d^2 e^2+a e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}} \]

[Out]

-1/16*(B*d*(3*c^2*d^4-10*a*c*d^2*e^2+a*e^3*(-a*e+4*b*d))-A*e*(15*c^2*d^4-2*c*d^2*e*(-3*a*e+10*b*d)+e^2*(3*a^2*
e^2-8*a*b*d*e+8*b^2*d^2)))*arctan(x*(a*e^2-b*d*e+c*d^2)^(1/2)/d^(1/2)/e^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^(5/2)/(
a*e^2-b*d*e+c*d^2)^(5/2)/e^(1/2)-1/4*e*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^2+1/
8*e*(3*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))*x*(c*x^4+b*x^2+a)^(1/2)/d^2/(a*e^2-b*d*e+c*d^
2)^2/(e*x^2+d)-1/8*(3*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))*x*c^(1/2)*(c*x^4+b*x^2+a)^(1/2
)/d^2/(a*e^2-b*d*e+c*d^2)^2/(a^(1/2)+x^2*c^(1/2))+1/8*a^(1/4)*c^(1/4)*(3*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(5*c
*d^2-e*(a*e+2*b*d)))*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin
(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)
+x^2*c^(1/2))^2)^(1/2)/d^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)^(1/2)+1/32*(B*d*(3*c^2*d^4-10*a*c*d^2*e^2+a*e
^3*(-a*e+4*b*d))-A*e*(15*c^2*d^4-2*c*d^2*e*(-3*a*e+10*b*d)+e^2*(3*a^2*e^2-8*a*b*d*e+8*b^2*d^2)))*(cos(2*arctan
(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/
4*(-e*a^(1/2)+d*c^(1/2))^2/d/e/a^(1/2)/c^(1/2),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(e*a^(1/2)+d*c^(1/2))*(a^(1/2)
+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^(1/4)/d^3/e/(a*e^2-b*d*e+c*d^2)^2/(-e*
a^(1/2)+d*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)+1/8*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c
^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^
(1/2))*(a*e*(3*A*e+B*d)+4*A*d*(-b*e+c*d)+d*(-A*e+B*d)*a^(1/2)*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^
2)^(1/2)/a^(1/4)/d^2/(a*e^2-b*d*e+c*d^2)/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.27, antiderivative size = 1125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1710, 1728, 1209, 1722, 1117, 1720} \begin {gather*} -\frac {\sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \sqrt {c x^4+b x^2+a} x}{8 d^2 \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \sqrt {c x^4+b x^2+a} x}{8 d^2 \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}-\frac {e (B d-A e) \sqrt {c x^4+b x^2+a} x}{4 d \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )^2}-\frac {\left (B \left (3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right )-A e \left (15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left (8 b^2 d^2-8 a b e d+3 a^2 e^2\right )\right )\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right )}{16 d^{5/2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 d^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt [4]{c} \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^4+b x^2+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (B \left (3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right )-A e \left (15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left (8 b^2 d^2-8 a b e d+3 a^2 e^2\right )\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/((d + e*x^2)^3*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

-1/8*(Sqrt[c]*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*x*Sqrt[a + b*x^2 + c*x^4])
/(d^2*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(4*d*(c*d
^2 - b*d*e + a*e^2)*(d + e*x^2)^2) + (e*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a*e)))*
x*Sqrt[a + b*x^2 + c*x^4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) - ((B*(3*c^2*d^5 - 10*a*c*d^3*e^2 + a
*d*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)
))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(16*d^(5/2)*Sqrt[e]*(c*d
^2 - b*d*e + a*e^2)^(5/2)) + (a^(1/4)*c^(1/4)*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e*(2*b*d + a
*e)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*
x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*d^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4
)*(Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e) + 4*A*d*(c*d - b*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*
x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])
/(8*a^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[c]*d + Sqrt[
a]*e)*(B*(3*c^2*d^5 - 10*a*c*d^3*e^2 + a*d*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2*e*(10*b*d - 3*a*e) +
e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]
*x^2)^2]*EllipticPi[-1/4*(Sqrt[c]*d - Sqrt[a]*e)^2/(Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 -
b/(Sqrt[a]*Sqrt[c]))/4])/(32*a^(1/4)*c^(1/4)*d^3*e*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a +
b*x^2 + c*x^4])

Rule 1117

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 1209

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

Rule 1710

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{A = Coeff
[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1
)*(Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b*d*e + a
*e^2)), Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d
- b*e)*(q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 1)))*x^2 + c*(C*d^2 - B*d*
e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]

Rule 1720

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))*(ArcTan[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)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1722

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]}, Dist[(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] + Dist[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[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2
- a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]

Rule 1728

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]
, A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/(e*q), Int[(1 - q*x^2)/Sqrt[a + b
*x^2 + c*x^4], x], x] + Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[
a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[
c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] &&  !GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \sqrt {a+b x^2+c x^4}} \, dx &=-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}-\frac {\int \frac {-4 A c d^2-a B d e+A e (4 b d-3 a e)-2 (B d-A e) (2 c d-b e) x^2+c e (B d-A e) x^4}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}} \, dx}{4 d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {\int \frac {a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B d e-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right ) x^2-c e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x^4}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{8 d^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {\int \frac {c e \left (a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B d e-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )\right )-\sqrt {a} c^{3/2} d e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )+\left (c e \left (c d-\sqrt {a} \sqrt {c} e\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )-2 c^2 d e \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{8 c d^2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (\sqrt {a} \sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{8 d^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{4 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )}+\frac {\left (\sqrt {a} \left (B \left (3 c^2 d^5-10 a c d^3 e^2+a d e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{8 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {\sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{4 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2}+\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}-\frac {\left (B \left (3 c^2 d^5-10 a c d^3 e^2+a d e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{16 d^{5/2} \sqrt {e} \left (c d^2-b d e+a e^2\right )^{5/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 d^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)+4 A d (c d-b e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (B \left (3 c^2 d^5-10 a c d^3 e^2+a d e^3 (4 b d-a e)\right )-A e \left (15 c^2 d^4-2 c d^2 e (10 b d-3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 13.31, size = 781, normalized size = 0.69 \begin {gather*} \frac {-\frac {4 d e^2 x \left (a+b x^2+c x^4\right ) \left (2 d (B d-A e) \left (c d^2+e (-b d+a e)\right )+\left (-3 A e \left (3 c d^2+e (-2 b d+a e)\right )+B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (d+e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\left (-b+\sqrt {b^2-4 a c}\right ) d e \left (3 A e \left (3 c d^2+e (-2 b d+a e)\right )+B \left (-5 c d^3+d e (2 b d+a e)\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+d \left (B d \left (6 c^2 d^3+c d e \left (-5 b d+5 \sqrt {b^2-4 a c} d-6 a e\right )-\left (-b+\sqrt {b^2-4 a c}\right ) e^2 (2 b d+a e)\right )-A e \left (14 c^2 d^3-3 \left (-b+\sqrt {b^2-4 a c}\right ) e^2 (2 b d-a e)+c d e \left (-17 b d+9 \sqrt {b^2-4 a c} d+2 a e\right )\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+2 \left (B \left (-3 c^2 d^5+10 a c d^3 e^2+a d e^3 (-4 b d+a e)\right )+A e \left (15 c^2 d^4+2 c d^2 e (-10 b d+3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right ) \Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{32 d^3 e \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/((d + e*x^2)^3*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

((-4*d*e^2*x*(a + b*x^2 + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e)) + (-3*A*e*(3*c*d^2 + e*(-2*b*d +
a*e)) + B*(5*c*d^3 - d*e*(2*b*d + a*e)))*(d + e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[2]*Sqrt[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*((-b + Sqrt[b^2 - 4*a*c])*d*e*
(3*A*e*(3*c*d^2 + e*(-2*b*d + a*e)) + B*(-5*c*d^3 + d*e*(2*b*d + a*e)))*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b
+ Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + d*(B*d*(6*c^2*d^3 + c*d*e*(-5*b*d
+ 5*Sqrt[b^2 - 4*a*c]*d - 6*a*e) - (-b + Sqrt[b^2 - 4*a*c])*e^2*(2*b*d + a*e)) - A*e*(14*c^2*d^3 - 3*(-b + Sq
rt[b^2 - 4*a*c])*e^2*(2*b*d - a*e) + c*d*e*(-17*b*d + 9*Sqrt[b^2 - 4*a*c]*d + 2*a*e)))*EllipticF[I*ArcSinh[Sqr
t[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + 2*(B*(-3*c^2*d^5 +
10*a*c*d^3*e^2 + a*d*e^3*(-4*b*d + a*e)) + A*e*(15*c^2*d^4 + 2*c*d^2*e*(-10*b*d + 3*a*e) + e^2*(8*b^2*d^2 - 8
*a*b*d*e + 3*a^2*e^2)))*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2
- 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(32*d^3*e*
(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + b*x^2 + c*x^4])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4475\) vs. \(2(1080)=2160\).
time = 0.15, size = 4476, normalized size = 3.98

method	result	size
default	\(\text {Expression too large to display}\)	\(4476\)
elliptic	\(\text {Expression too large to display}\)	\(5671\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(A*e-B*d)/e*(1/4*e^2/d/(a*e^2-b*d*e+c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^2+3/8*e^2*(a*e^2-2*b*d*e+3*c*d^2)
/(a*e^2-b*d*e+c*d^2)^2/d^2*x*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)-3/8*e^2*c/(a*e^2-b*d*e+c*d^2)^2/d*a*2^(1/2)/(-b/a
+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2
)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a
)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b+3/8*e^2*c/(a*e^2-b*d*e+c*d^2)^2/d*a*2^(1/2)/(-b/a+1/a
*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2)^(1
/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1
/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*b-1/32*c/(a*e^2-b*d*e+c*d^2)^2/d*2^(1/2)/(-b/a+1/a*(-4*a*c+
b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2
)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^
(1/2))/a/c)^(1/2))*a*e^2+3/8/(a*e^2-b*d*e+c*d^2)^2/d^3*e^4*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*
b*x^2/a-1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)
^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(-
4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))*a^2+1/(a*e^2-b*d*e+c*d^2)^2/d*e^2*2^(1/2
)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2*x^
2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)
,-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1
/2))*b^2+9/16*e*c^2/(a*e^2-b*d*e+c*d^2)^2*a*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(
-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^
(1/2))*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)
)-3/16*e^3*c/(a*e^2-b*d*e+c*d^2)^2/d^2*a^2*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-
4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(
1/2))*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))
-9/16*e*c^2/(a*e^2-b*d*e+c*d^2)^2*a*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-4*a*c+b
^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*E
llipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-5/2/(a
*e^2-b*d*e+c*d^2)^2*e*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2*x^2/a*(-4*a*c+b^2)^(1/2))
^(1/2)*(1+1/2*b*x^2/a+1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+
(-4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((
-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))*b*c+3/16*e^3*c/(a*e^2-b*d*e+c*d^2)^2/d^2*a^2*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(
1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x
^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b
*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+15/8/(a*e^2-b*d*e+c*d^2)^2*d*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(
1+1/2*b*x^2/a-1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*
x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2
*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))*c^2+1/8*c/(a*e^2-b*d*e+c*d^2)^2*2^
(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2*x^2/a*
(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*
(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))*e*b-1/(a*e^2-b*d*e+c*d^2)^2/d^2*e^3*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^
(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(
1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2
))*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))*a*b+3/4/(a*e^2-b*d*e
+c*d^2)^2/d*e^2*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)
*(1+1/2*b*x^2/a+1/2*x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*
c+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4
*a*c+b^2)^(1/2))/a)^(1/2))*a*c-7/32*c^2/(a*e^2-...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*(x^2*e + d)^3), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x^{2}}{\left (d + e x^{2}\right )^{3} \sqrt {a + b x^{2} + c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/(e*x**2+d)**3/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral((A + B*x**2)/((d + e*x**2)**3*sqrt(a + b*x**2 + c*x**4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*(x^2*e + d)^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {B\,x^2+A}{{\left (e\,x^2+d\right )}^3\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/((d + e*x^2)^3*(a + b*x^2 + c*x^4)^(1/2)),x)

[Out]

int((A + B*x^2)/((d + e*x^2)^3*(a + b*x^2 + c*x^4)^(1/2)), x)

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