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

Optimal. Leaf size=660 \[ \frac{x \left (c x^2 \left (2 a c e+b^2 (-e)+b c d\right )+3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (\frac{8 a b c e-12 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-\frac{8 a b c e-12 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )^2}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^2))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*
d*e + a*e^2)*(a + b*x^2 + c*x^4)) - (Sqrt[c]*e^2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]
*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) + (Sqrt[c]*(
b*c*d - b^2*e + 2*a*c*e + (b^2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c
]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e
^2)) - (Sqrt[c]*e^2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c
]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) + (Sqrt[c]*(b*c*d - b^2*e + 2*a*c*e - (b^
2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*
c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) + (e^(7/2)*ArcTan[(Sqrt[
e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^2)

________________________________________________________________________________________

Rubi [A]  time = 2.87252, antiderivative size = 660, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1238, 205, 1178, 1166} \[ \frac{x \left (c x^2 \left (2 a c e+b^2 (-e)+b c d\right )+3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (\frac{8 a b c e-12 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} \left (-\frac{8 a b c e-12 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+2 a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )^2}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^2))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*
d*e + a*e^2)*(a + b*x^2 + c*x^4)) - (Sqrt[c]*e^2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]
*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) + (Sqrt[c]*(
b*c*d - b^2*e + 2*a*c*e + (b^2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c
]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e
^2)) - (Sqrt[c]*e^2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c
]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) + (Sqrt[c]*(b*c*d - b^2*e + 2*a*c*e - (b^
2*c*d - 12*a*c^2*d - b^3*e + 8*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*
c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) + (e^(7/2)*ArcTan[(Sqrt[
e]*x)/Sqrt[d]])/(Sqrt[d]*(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 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 ) \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 )}+\frac{c d-b e-c e x^2}{\left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )^2}-\frac{e^2 \left (-c d+b e+c e x^2\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=-\frac{e^2 \int \frac{-c d+b e+c e x^2}{a+b x^2+c x^4} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac{e^4 \int \frac{1}{d+e x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac{\int \frac{c d-b e-c e x^2}{\left (a+b x^2+c x^4\right )^2} \, dx}{c d^2-b d e+a e^2}\\ &=\frac{x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c d^2-b d e+a e^2\right )^2}-\frac{\int \frac{-b^2 c d+6 a c^2 d+b^3 e-5 a b c e-c \left (b c d-b^2 e+2 a c e\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 )}-\frac{\left (c e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac{\left (c e^2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}-\frac{\sqrt{c} e^2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c d^2-b d e+a e^2\right )^2}+\frac{\left (c \left (b c d-b^2 e+2 a c e-\frac{b^2 c d-12 a c^2 d-b^3 e+8 a b c e}{\sqrt{b^2-4 a c}}\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 ) \left (c d^2-b d e+a e^2\right )}+\frac{\left (c \left (b c d-b^2 e+2 a c e+\frac{b^2 c d-12 a c^2 d-b^3 e+8 a b c e}{\sqrt{b^2-4 a c}}\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 ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac{x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} e^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac{\sqrt{c} \left (b c d-b^2 e+2 a c e+\frac{b^2 c d-12 a c^2 d-b^3 e+8 a b c e}{\sqrt{b^2-4 a c}}\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 ) \sqrt{b-\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac{\sqrt{c} e^2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac{\sqrt{c} \left (b c d-b^2 e+2 a c e-\frac{b^2 c d-12 a c^2 d-b^3 e+8 a b c e}{\sqrt{b^2-4 a c}}\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 ) \sqrt{b+\sqrt{b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac{e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c d^2-b d e+a e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 3.04297, size = 708, normalized size = 1.07 \[ \frac{\frac{\sqrt{2} \sqrt{c} \left (b^2 \left (-c d e \left (2 d \sqrt{b^2-4 a c}+3 a e\right )-3 a e^3 \sqrt{b^2-4 a c}+c^2 d^3\right )+2 a c \left (c d e \left (d \sqrt{b^2-4 a c}-14 a e\right )+5 a e^3 \sqrt{b^2-4 a c}-6 c^2 d^3\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+20 a e\right )+a e^2 \left (16 a e-d \sqrt{b^2-4 a c}\right )\right )+b^3 e \left (e \left (d \sqrt{b^2-4 a c}-3 a e\right )-2 c d^2\right )+b^4 d e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (-b^2 \left (c d e \left (2 d \sqrt{b^2-4 a c}-3 a e\right )+3 a e^3 \sqrt{b^2-4 a c}+c^2 d^3\right )+2 a c \left (c d e \left (d \sqrt{b^2-4 a c}+14 a e\right )+5 a e^3 \sqrt{b^2-4 a c}+6 c^2 d^3\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-20 a e\right )-a e^2 \left (d \sqrt{b^2-4 a c}+16 a e\right )\right )+b^3 e \left (e \left (d \sqrt{b^2-4 a c}+3 a e\right )+2 c d^2\right )+b^4 (-d) e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 x \left (e (a e-b d)+c d^2\right ) \left (-b c \left (3 a e+c d x^2\right )+2 a c^2 \left (d-e x^2\right )+b^2 c \left (e x^2-d\right )+b^3 e\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{4 e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}}{4 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(c*d^2 + e*(-(b*d) + a*e))*x*(b^3*e - b*c*(3*a*e + c*d*x^2) + 2*a*c^2*(d - e*x^2) + b^2*c*(-d + e*x^2)))/(
a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^4*d*e^2 + 2*a*c*(-6*c^2*d^3 + 5*a*Sqrt[b^2 - 4*a*c
]*e^3 + c*d*e*(Sqrt[b^2 - 4*a*c]*d - 14*a*e)) + b^3*e*(-2*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)) + b^2*(c^2*
d^3 - 3*a*Sqrt[b^2 - 4*a*c]*e^3 - c*d*e*(2*Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + b*c*(a*e^2*(-(Sqrt[b^2 - 4*a*c]*d)
+ 16*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 20*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]]) + (Sqrt[2]*Sqrt[c]*(-(b^4*d*e^2) - b^2*(c^2*d^3 + 3*a*Sqrt[b
^2 - 4*a*c]*e^3 + c*d*e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)) + b^3*e*(2*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) +
2*a*c*(6*c^2*d^3 + 5*a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(Sqrt[b^2 - 4*a*c]*d + 14*a*e)) + b*c*(c*d^2*(Sqrt[b^2 -
4*a*c]*d - 20*a*e) - a*e^2*(Sqrt[b^2 - 4*a*c]*d + 16*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]]) + (4*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d])
/(4*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B]  time = 0.074, size = 3841, normalized size = 5.8 \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)/(c*x^4+b*x^2+a)^2,x)

[Out]

1/4/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(
c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^4*d*e^2+1/4/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c/(-4*a*c+b^2)
^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4*d*e^2
-1/2/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arcta
n(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*d^2*e-1/2/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^2/(-4*a*c+
b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^3*
d^2*e+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)/a/(4*a*c-b^2)*x*b^3*c*d^2*e+1/4/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2
)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*d*e^2+1/
4/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^3*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a
*c+b^2)^(1/2)-b)*c)^(1/2))*b*d^3-1/4/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*d^3-3/4/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c/(-4
*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))
*b^3*e^3-3/4/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a
rctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*e^3-1/4/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^2*2^(1/2)/
(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*d*e^2-1/2/(a*e^2-b*d*
e+c*d^2)^2/(c*x^4+b*x^2+a)*c/a/(4*a*c-b^2)*x^3*b^3*d*e^2+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)*c^2/a/(4*a*c-
b^2)*x^3*b^2*d^2*e+e^4/(a*e^2-b*d*e+c*d^2)^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-7/(a*e^2-b*d*e+c*d^2)^2*a/(4*
a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1
/2)-b)*c)^(1/2))*d*e^2+1/2/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar
ctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*d^2*e+1/4/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c*2^(1/2)
/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^3*d*e^2+1/4/(a*e^2-b
*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2
)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*d^3-1/4/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*d*e^2+1/4/(a*e^2-b*d*e+c*d^2)^2/a/(4
*a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^
(1/2))*c)^(1/2))*b^2*d^3+4/(a*e^2-b*d*e+c*d^2)^2*a/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*e^3-7/(a*e^2-b*d*e+c*d^2)^2*a/(4*a*c-b
^2)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))
*c)^(1/2))*d*e^2-1/2/(a*e^2-b*d*e+c*d^2)^2/a/(4*a*c-b^2)*c^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(
c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*d^2*e-3/4/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^2/(-4*a*c+b^2)
^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*d*e^
2+5/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(
c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*d^2*e+4/(a*e^2-b*d*e+c*d^2)^2*a/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(
1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*e^3-3/4/
(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2
^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*d*e^2+5/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^3/(-4*a*c+b^2)^(1/2)*
2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*d^2*e+3/4/(a*e
^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(
1/2))*c)^(1/2))*b^2*e^3-3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^4/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1
/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d^3+5/2/(a*e^2-b*d*e+c*d^2)^2*a/(4*a*c-b^2)
*c^2*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*e^3-5/2/(a
*e^2-b*d*e+c*d^2)^2*a/(4*a*c-b^2)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+
b^2)^(1/2))*c)^(1/2))*e^3+1/2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^3*b*d*e^2-1/2/(a*e^2-b*d
*e+c*d^2)^2/(c*x^4+b*x^2+a)*c^3/a/(4*a*c-b^2)*x^3*b*d^3-3/2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)*a/(4*a*c-b^2
)*x*b*e^3*c+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*x*c^2*d*e^2+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b
*x^2+a)/(4*a*c-b^2)*x*b^2*c*d*e^2-5/2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b*c^2*d^2*e-1/2/(a*e
^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)/a/(4*a*c-b^2)*x*b^4*d*e^2-1/2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)/a/(4*a*c
-b^2)*x*b^2*c^2*d^3+1/2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^3*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh
(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*d^2*e-1/2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^3*2^(1/2)/((b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d^2*e-3/4/(a*e^2-b*d*e+c*d^2)^2
/(4*a*c-b^2)*c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*
b^2*e^3-3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)*c^4/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*ar
ctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*d^3+1/2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*
x*b^3*e^3+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*c^3*d^3-1/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a
)*c^2*a/(4*a*c-b^2)*x^3*e^3+1/2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^3*b^2*e^3-1/(a*e^2-b*d*e
+c*d^2)^2/(c*x^4+b*x^2+a)*c^3/(4*a*c-b^2)*x^3*d^2*e

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError