3.28 \(\int \frac{A+B x^2}{(d+e x^2) (a+b x^2+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=867 \[ \frac{a^{3/4} e (B d-A e) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right )^2}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\left (\frac{\sqrt{c d^2-b e d+a e^2} x}{\sqrt{d} \sqrt{e} \sqrt{c x^4+b x^2+a}}\right )}{2 \sqrt{d} \left (c d^2-b e d+a e^2\right )^{3/2}}-\frac{\sqrt [4]{c} \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c} \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}-\frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right ) \sqrt{c x^4+b x^2+a}}+\frac{\sqrt{c} \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )\right ) x \sqrt{c x^4+b x^2+a}}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}-\frac{x \left (c \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )\right ) x^2+a b c (B d-A e)-\left (b^2-2 a c\right ) (A c d-A b e+a B e)\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.42356, antiderivative size = 1045, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {1720, 1178, 1197, 1103, 1195, 1216, 1706} \[ \frac{a^{3/4} e (B d-A e) \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 \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 ) \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right )^2}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{e^{3/2} (B d-A e) \tan ^{-1}\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 )}{2 \sqrt{d} \left (c d^2-b e d+a e^2\right )^{3/2}}-\frac{\sqrt [4]{c} \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c 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 \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 )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\sqrt [4]{c} \left (a B e-\sqrt{a} \sqrt{c} (B d-A e)+A (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 \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 )}{2 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\sqrt [4]{c} e (B d-A e) \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 \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 )}{2 \sqrt [4]{a} \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{\sqrt{c} \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )\right ) x \sqrt{c x^4+b x^2+a}}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}-\frac{x \left (c \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )\right ) x^2+a b c (B d-A e)-\left (b^2-2 a c\right ) (A c d-A b e+a B e)\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(a*b*c*(B*d - A*e) - (b^2 - 2*a*c)*(A*c*d - A*b*e + a*B*e) + c*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*
a*c*e))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])) + (Sqrt[c]*(a*B*(2*c*d - b*e)
- A*(b*c*d - b^2*e + 2*a*c*e))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(Sqrt[a] +
Sqrt[c]*x^2)) - (e^(3/2)*(B*d - A*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 +
c*x^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^(3/2)) - (c^(1/4)*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*
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])/(a^(3/4)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4
]) - (c^(1/4)*e*(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]*Ellipt
icF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b
*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(a*B*e - Sqrt[a]*Sqrt[c]*(B*d - A*e) + A*(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])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x
^4]) + (a^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sq
rt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^
(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*
x^4])

Rule 1720

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegr
and[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}
, x] && PolyQ[Px, x^2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p + 1/2] && Integer
Q[q]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1197

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

Rule 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

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)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), 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 1216

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Di
st[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(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}, 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]

Rule 1706

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)]*EllipticPi[Cancel[-((B*d - A*e)^2/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2 - (b*A)/(4*a*B)])/(4*d*e*A*q*Sqrt[
a + b*x^2 + c*x^4]), 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]

Rubi steps

\begin{align*} \int \frac{A+B x^2}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \left (\frac{A c d-A b e+a B e+c (B d-A e) x^2}{\left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac{e (-B d+A e)}{\left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}}\right ) \, dx\\ &=\frac{\int \frac{A c d-A b e+a B e+c (B d-A e) x^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx}{c d^2-b d e+a e^2}-\frac{(e (B d-A e)) \int \frac{1}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac{x \left (a b c (B d-A e)-\left (b^2-2 a c\right ) (A c d-A b e+a B e)+c \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\int \frac{-a c (b B d-2 A c d+A b e-2 a B e)-c \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}-\frac{\left (\sqrt{c} e (B d-A e)\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b d e+a e^2\right )}+\frac{\left (\sqrt{a} e^2 (B d-A e)\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}{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{x \left (a b c (B d-A e)-\left (b^2-2 a c\right ) (A c d-A b e+a B e)+c \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{e^{3/2} (B d-A e) \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 )}{2 \sqrt{d} \left (c d^2-b d e+a e^2\right )^{3/2}}-\frac{\sqrt [4]{c} e (B d-A e) \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 )}{2 \sqrt [4]{a} \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{\sqrt [4]{a} e \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) (B d-A e) \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 )}{4 \sqrt [4]{c} d \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} \left (a B e-\sqrt{a} \sqrt{c} (B d-A e)+A (c d-b e)\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a} \left (b-2 \sqrt{a} \sqrt{c}\right ) \left (c d^2-b d e+a e^2\right )}-\frac{\left (\sqrt{c} \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{x \left (a b c (B d-A e)-\left (b^2-2 a c\right ) (A c d-A b e+a B e)+c \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}+\frac{\sqrt{c} \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x \sqrt{a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{e^{3/2} (B d-A e) \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 )}{2 \sqrt{d} \left (c d^2-b d e+a e^2\right )^{3/2}}-\frac{\sqrt [4]{c} \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c 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 )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{c} e (B d-A e) \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 )}{2 \sqrt [4]{a} \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{\sqrt [4]{c} \left (a B e-\sqrt{a} \sqrt{c} (B d-A e)+A (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 )}{2 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} e \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) (B d-A e) \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 )}{4 \sqrt [4]{c} d \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}}\\ \end{align*}

Mathematica [C]  time = 5.24307, size = 1736, normalized size = 2. \[ \frac{4 A \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x b^3+4 A c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x^3 b^2-4 A c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x b^2-4 a B \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x b^2+2 i a A e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \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 ) b^2-2 i a B d e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \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 ) b^2-4 A c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x^3 b-4 a B c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x^3 b+4 a B c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x b-12 a A c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x b+8 a B c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x^3-8 a A c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x^3+8 a A c^2 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d^2 x+8 a^2 B c \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d e x-i \left (\sqrt{b^2-4 a c}-b\right ) d \left (a B (2 c d-b e)+A \left (e b^2-c d b-2 a c e\right )\right ) \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} 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 )+i d \left (a B \left (b \left (b-\sqrt{b^2-4 a c}\right ) e+2 c \left (\sqrt{b^2-4 a c} d-2 a e\right )\right )+A \left (-e b^3+\left (c d+\sqrt{b^2-4 a c} e\right ) b^2+c \left (4 a e-\sqrt{b^2-4 a c} d\right ) b-2 a c \left (2 c d+\sqrt{b^2-4 a c} e\right )\right )\right ) \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\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 )-8 i a^2 A c e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \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 )+8 i a^2 B c d e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{4 c x^2+2 b-2 \sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}} \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 )}{4 a \left (4 a c-b^2\right ) \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d \left (c d^2+e (a e-b d)\right ) \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*A*b^2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 4*a*b*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 8*a*A*c^
2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 4*A*b^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x - 4*a*b^2*B*Sqrt[c/(b
+ Sqrt[b^2 - 4*a*c])]*d*e*x - 12*a*A*b*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x + 8*a^2*B*c*Sqrt[c/(b + Sqrt[b^
2 - 4*a*c])]*d*e*x - 4*A*b*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x^3 + 8*a*B*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*
c])]*d^2*x^3 + 4*A*b^2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - 4*a*b*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d
*e*x^3 - 8*a*A*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - I*(-b + Sqrt[b^2 - 4*a*c])*d*(a*B*(2*c*d - b*e) +
A*(-(b*c*d) + b^2*e - 2*a*c*e))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2
*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*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])] + I*d*(a*B*(b*(b - Sqrt[b^2 - 4*a*c])*e + 2*c*(Sqrt[
b^2 - 4*a*c]*d - 2*a*e)) + A*(-(b^3*e) + b*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e
) - 2*a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt
[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[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])] - (2*I)*a*b^2*B*d*e*Sqrt[(b + Sqrt[b^2 - 4*
a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*E
llipticPi[((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])] + (8*I)*a^2*B*c*d*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b
^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*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])] + (2*I)*a*A*b^2*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2
*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSi
nh[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])] - (8*I)*a^2*A*
c*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^
2)/(b - Sqrt[b^2 - 4*a*c])]*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])])/(4*a*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2
- 4*a*c])]*d*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + b*x^2 + c*x^4])

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 3241, normalized size = 3.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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