3.275 \(\int \frac{1}{(d+e x^2)^2 (a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=1077 \[ \text{result too large to display} \]

[Out]

(e^4*x)/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) + (x*(a*b*c*e*(2*c*d - b*e) + (b^2 - 2*a*c)*(c^2*d^2 + b^2
*e^2 - c*e*(2*b*d + a*e)) - c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^2))/(2*a*(b^2 -
4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2 + b*(b + Sqrt[b^2 - 4*
a*c])*e^2 - c*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]
)/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^3) + (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c*d
- Sqrt[b^2 - 4*a*c]*e) - 4*a*c^2*(3*c*d^2 - e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + b^2*c*(c*d^2 - e*(2*Sqrt[b^2 -
4*a*c]*d + 9*a*e)) - b*c*(3*a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 16*a*e)))*ArcTan[(Sqrt[2]*Sq
rt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b
*d*e + a*e^2)^2) - (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d - 2*Sqrt[b^2 -
4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^
2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^3) - (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(3*a*Sq
rt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 16*a*e)) + b^2*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 9*a*e))
- 4*a*c^2*(3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]]
)/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) + (2*e^(7/2)*(2*c*d
- b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^3) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
)/(2*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^2)

________________________________________________________________________________________

Rubi [A]  time = 12.6389, antiderivative size = 1077, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1238, 199, 205, 1178, 1166} \[ \frac{x e^4}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2-b e d+a e^2\right )^2}+\frac{2 (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\sqrt{d} \left (c d^2-b e d+a e^2\right )^3}+\frac{\sqrt{2} \sqrt{c} \left (3 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) e^2}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^3}-\frac{\sqrt{2} \sqrt{c} \left (3 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right ) e^2}{\sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^3}+\frac{\sqrt{c} \left (e^2 b^4-e \left (2 c d-\sqrt{b^2-4 a c} e\right ) b^3+c \left (c d^2-e \left (2 \sqrt{b^2-4 a c} d+9 a e\right )\right ) b^2-c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+16 a e\right )\right ) b-4 a c^2 \left (3 c d^2-e \left (\sqrt{b^2-4 a c} d+3 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^2}-\frac{\sqrt{c} \left (e^2 b^4-e \left (2 c d+\sqrt{b^2-4 a c} e\right ) b^3+c \left (c d^2+e \left (2 \sqrt{b^2-4 a c} d-9 a e\right )\right ) b^2+c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-16 a e\right )\right ) b-4 a c^2 \left (3 c d^2+e \left (\sqrt{b^2-4 a c} d-3 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b e d+a e^2\right )^2}+\frac{x \left (-c \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right ) x^2+a b c e (2 c d-b e)+\left (b^2-2 a c\right ) \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

(e^4*x)/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^2)) + (x*(a*b*c*e*(2*c*d - b*e) + (b^2 - 2*a*c)*(c^2*d^2 + b^2
*e^2 - c*e*(2*b*d + a*e)) - c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^2))/(2*a*(b^2 -
4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2 + b*(b + Sqrt[b^2 - 4*
a*c])*e^2 - c*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]
)/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^3) + (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c*d
- Sqrt[b^2 - 4*a*c]*e) - 4*a*c^2*(3*c*d^2 - e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + b^2*c*(c*d^2 - e*(2*Sqrt[b^2 -
4*a*c]*d + 9*a*e)) - b*c*(3*a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 16*a*e)))*ArcTan[(Sqrt[2]*Sq
rt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b
*d*e + a*e^2)^2) - (Sqrt[2]*Sqrt[c]*e^2*(3*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d - 2*Sqrt[b^2 -
4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^
2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^3) - (Sqrt[c]*(b^4*e^2 - b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(3*a*Sq
rt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 16*a*e)) + b^2*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 9*a*e))
- 4*a*c^2*(3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]]
)/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) + (2*e^(7/2)*(2*c*d
- b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^3) + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
)/(2*d^(3/2)*(c*d^2 - b*d*e + a*e^2)^2)

Rule 1238

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 1166

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

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2} \, dx &=\int \left (\frac{e^4}{\left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )^2}-\frac{2 e^4 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^3 \left (d+e x^2\right )}+\frac{c^2 d^2+b^2 e^2-c e (2 b d+a e)-c e (2 c d-b e) x^2}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )^2}+\frac{e^2 \left (3 c^2 d^2+2 b^2 e^2-c e (5 b d+a e)-2 c e (2 c d-b e) x^2\right )}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac{e^2 \int \frac{3 c^2 d^2+2 b^2 e^2-c e (5 b d+a e)-2 c e (2 c d-b e) x^2}{a+b x^2+c x^4} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (2 e^4 (2 c d-b e)\right ) \int \frac{1}{d+e x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\int \frac{c^2 d^2+b^2 e^2-c e (2 b d+a e)-c e (2 c d-b e) x^2}{\left (a+b x^2+c x^4\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac{e^4 \int \frac{1}{\left (d+e x^2\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac{x \left (a b c e (2 c d-b e)+\left (b^2-2 a c\right ) \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )}+\frac{2 e^{7/2} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c d^2-b d e+a e^2\right )^3}-\frac{\int \frac{2 b^3 c d e-10 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-6 a e^2\right )+6 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac{e^4 \int \frac{1}{d+e x^2} \, dx}{2 d \left (c d^2-b d e+a e^2\right )^2}-\frac{\left (c e^2 \left (3 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt{b^2-4 a c} d+a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}+\frac{\left (c e^2 \left (3 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt{b^2-4 a c} d+a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{e^4 x}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac{x \left (a b c e (2 c d-b e)+\left (b^2-2 a c\right ) \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} e^2 \left (3 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^3}-\frac{\sqrt{2} \sqrt{c} e^2 \left (3 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^3}+\frac{2 e^{7/2} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c d^2-b d e+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2-b d e+a e^2\right )^2}-\frac{\left (c \left (b^4 e^2-b^3 e \left (2 c d+\sqrt{b^2-4 a c} e\right )+b c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-16 a e\right )\right )+b^2 c \left (c d^2+e \left (2 \sqrt{b^2-4 a c} d-9 a e\right )\right )-4 a c^2 \left (3 c d^2+e \left (\sqrt{b^2-4 a c} d-3 a e\right )\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2}+\frac{\left (c \left (b^4 e^2-b^3 e \left (2 c d-\sqrt{b^2-4 a c} e\right )-4 a c^2 \left (3 c d^2-e \left (\sqrt{b^2-4 a c} d+3 a e\right )\right )+b^2 c \left (c d^2-e \left (2 \sqrt{b^2-4 a c} d+9 a e\right )\right )-b c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+16 a e\right )\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{e^4 x}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac{x \left (a b c e (2 c d-b e)+\left (b^2-2 a c\right ) \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} e^2 \left (3 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d+2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^3}+\frac{\sqrt{c} \left (b^4 e^2-b^3 e \left (2 c d-\sqrt{b^2-4 a c} e\right )-4 a c^2 \left (3 c d^2-e \left (\sqrt{b^2-4 a c} d+3 a e\right )\right )+b^2 c \left (c d^2-e \left (2 \sqrt{b^2-4 a c} d+9 a e\right )\right )-b c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+16 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}-\frac{\sqrt{2} \sqrt{c} e^2 \left (3 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-c e \left (3 b d-2 \sqrt{b^2-4 a c} d+a e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^3}-\frac{\sqrt{c} \left (b^4 e^2-b^3 e \left (2 c d+\sqrt{b^2-4 a c} e\right )+b c \left (3 a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-16 a e\right )\right )+b^2 c \left (c d^2+e \left (2 \sqrt{b^2-4 a c} d-9 a e\right )\right )-4 a c^2 \left (3 c d^2+e \left (\sqrt{b^2-4 a c} d-3 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac{2 e^{7/2} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c d^2-b d e+a e^2\right )^3}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \left (c d^2-b d e+a e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 6.28867, size = 1020, normalized size = 0.95 \[ \frac{1}{4} \left (\frac{2 x e^4}{d \left (c d^2+e (a e-b d)\right )^2 \left (e x^2+d\right )}+\frac{2 \left (9 c d^2+e (a e-5 b d)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{d^{3/2} \left (c d^2+e (a e-b d)\right )^3}-\frac{2 x \left (e^2 b^4+c e \left (e x^2-2 d\right ) b^3+c \left (c d \left (d-2 e x^2\right )-4 a e^2\right ) b^2+c^2 \left (c d^2 x^2-3 a e \left (e x^2-2 d\right )\right ) b+2 a c^2 \left (a e^2-c d \left (d-2 e x^2\right )\right )\right )}{a \left (4 a c-b^2\right ) \left (c d^2+e (a e-b d)\right )^2 \left (c x^4+b x^2+a\right )}+\frac{\sqrt{2} \sqrt{c} \left (d e^3 b^5+e^2 \left (e \left (\sqrt{b^2-4 a c} d-5 a e\right )-3 c d^2\right ) b^4+e \left (c d-\sqrt{b^2-4 a c} e\right ) \left (3 c d^2+5 a e^2\right ) b^3+c \left (-c^2 d^4+3 c e \left (\sqrt{b^2-4 a c} d+4 a e\right ) d^2+a e^3 \left (7 \sqrt{b^2-4 a c} d+29 a e\right )\right ) b^2-c \left (-19 a^2 \sqrt{b^2-4 a c} e^4+2 a c d \left (26 a e-3 \sqrt{b^2-4 a c} d\right ) e^2+c^2 d^3 \left (\sqrt{b^2-4 a c} d+28 a e\right )\right ) b-4 a c^2 \left (-3 c^2 d^4+c e \left (\sqrt{b^2-4 a c} d-12 a e\right ) d^2+a e^3 \left (9 \sqrt{b^2-4 a c} d+7 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (e (b d-a e)-c d^2\right )^3}-\frac{\sqrt{2} \sqrt{c} \left (d e^3 b^5-e^2 \left (3 c d^2+e \left (\sqrt{b^2-4 a c} d+5 a e\right )\right ) b^4+e \left (c d+\sqrt{b^2-4 a c} e\right ) \left (3 c d^2+5 a e^2\right ) b^3-c \left (c^2 d^4+3 c e \left (\sqrt{b^2-4 a c} d-4 a e\right ) d^2+a e^3 \left (7 \sqrt{b^2-4 a c} d-29 a e\right )\right ) b^2+c \left (-19 a^2 \sqrt{b^2-4 a c} e^4-2 a c d \left (3 \sqrt{b^2-4 a c} d+26 a e\right ) e^2+c^2 d^3 \left (\sqrt{b^2-4 a c} d-28 a e\right )\right ) b+4 a c^2 \left (3 c^2 d^4+c e \left (\sqrt{b^2-4 a c} d+12 a e\right ) d^2+a e^3 \left (9 \sqrt{b^2-4 a c} d-7 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}} \left (e (b d-a e)-c d^2\right )^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

((2*e^4*x)/(d*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x^2)) - (2*x*(b^4*e^2 + b^3*c*e*(-2*d + e*x^2) + 2*a*c^2*(a*
e^2 - c*d*(d - 2*e*x^2)) + b^2*c*(-4*a*e^2 + c*d*(d - 2*e*x^2)) + b*c^2*(c*d^2*x^2 - 3*a*e*(-2*d + e*x^2))))/(
a*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^5*d*e^3 + b^3*e*(c*d
- Sqrt[b^2 - 4*a*c]*e)*(3*c*d^2 + 5*a*e^2) + b^4*e^2*(-3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 5*a*e)) - 4*a*c^2*(-
3*c^2*d^4 + c*d^2*e*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^3*(9*Sqrt[b^2 - 4*a*c]*d + 7*a*e)) - b*c*(-19*a^2*Sqr
t[b^2 - 4*a*c]*e^4 + 2*a*c*d*e^2*(-3*Sqrt[b^2 - 4*a*c]*d + 26*a*e) + c^2*d^3*(Sqrt[b^2 - 4*a*c]*d + 28*a*e)) +
b^2*c*(-(c^2*d^4) + 3*c*d^2*e*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*e^3*(7*Sqrt[b^2 - 4*a*c]*d + 29*a*e)))*ArcTan
[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(-(c*d^2
) + e*(b*d - a*e))^3) - (Sqrt[2]*Sqrt[c]*(b^5*d*e^3 + b^3*e*(c*d + Sqrt[b^2 - 4*a*c]*e)*(3*c*d^2 + 5*a*e^2) -
b^2*c*(c^2*d^4 + a*e^3*(7*Sqrt[b^2 - 4*a*c]*d - 29*a*e) + 3*c*d^2*e*(Sqrt[b^2 - 4*a*c]*d - 4*a*e)) - b^4*e^2*(
3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + 4*a*c^2*(3*c^2*d^4 + a*e^3*(9*Sqrt[b^2 - 4*a*c]*d - 7*a*e) + c*d^
2*e*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)) + b*c*(-19*a^2*Sqrt[b^2 - 4*a*c]*e^4 + c^2*d^3*(Sqrt[b^2 - 4*a*c]*d - 28*a
*e) - 2*a*c*d*e^2*(3*Sqrt[b^2 - 4*a*c]*d + 26*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/
(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^(7/2)*(9*c*d^2 + e*(-5
*b*d + a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^3))/4

________________________________________________________________________________________

Maple [B]  time = 0.082, size = 5709, normalized size = 5.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError