3.53 \(\int \frac{d+e x+f x^2}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=621 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d-25 a b^2 c d+a b^3 f+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (-\frac{-52 a^2 b c f+168 a^2 c^2 d-30 a b^2 c d+a b^3 f+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )-a b^2 \left (30 c d-f \sqrt{b^2-4 a c}\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

-(e*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2
))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*c*e*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) +
(x*(3*b^4*d - 25*a*b^2*c*d + 28*a^2*c^2*d + a*b^3*f + 8*a^2*b*c*f + c*(3*b^3*d - 24*a*b*c*d + a*b^2*f + 20*a^
2*c*f)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*
f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) - a*b^2*(30*c*d - Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqr
t[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)
*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d - 24*a*b*c*d + a*b^2*f + 20*a^2*c*f - (3*b^4*d - 30*a*b^2*c*
d + 168*a^2*c^2*d + a*b^3*f - 52*a^2*b*c*f)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 -
4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (6*c^2*e*ArcTanh[(b + 2*c*x^2)/Sqrt[b^
2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 4.51193, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {1673, 1178, 1166, 205, 12, 1107, 614, 618, 206} \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d-25 a b^2 c d+a b^3 f+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (-\frac{-52 a^2 b c f+168 a^2 c^2 d-30 a b^2 c d+a b^3 f+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )-a b^2 \left (30 c d-f \sqrt{b^2-4 a c}\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2
))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*c*e*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) +
(x*(3*b^4*d - 25*a*b^2*c*d + 28*a^2*c^2*d + a*b^3*f + 8*a^2*b*c*f + c*(3*b^3*d - 24*a*b*c*d + a*b^2*f + 20*a^
2*c*f)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d + a*
f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) - a*b^2*(30*c*d - Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(42*c*d + 5*Sqr
t[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)
*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(3*b^3*d - 24*a*b*c*d + a*b^2*f + 20*a^2*c*f - (3*b^4*d - 30*a*b^2*c*
d + 168*a^2*c^2*d + a*b^3*f - 52*a^2*b*c*f)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 -
4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (6*c^2*e*ArcTanh[(b + 2*c*x^2)/Sqrt[b^
2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

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]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 614

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2}{\left (a+b x^2+c x^4\right )^3} \, dx &=\int \frac{e x}{\left (a+b x^2+c x^4\right )^3} \, dx+\int \frac{d+f x^2}{\left (a+b x^2+c x^4\right )^3} \, dx\\ &=\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{\int \frac{-3 b^2 d+14 a c d-a b f-5 c (b d-2 a f) x^2}{\left (a+b x^2+c x^4\right )^2} \, dx}{4 a \left (b^2-4 a c\right )}+e \int \frac{x}{\left (a+b x^2+c x^4\right )^3} \, dx\\ &=\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\int \frac{3 b^4 d-27 a b^2 c d+84 a^2 c^2 d+a b^3 f-16 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2}{a+b x^2+c x^4} \, dx}{8 a^2 \left (b^2-4 a c\right )^2}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{(3 c e) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}+\frac{\left (c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f-\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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}{16 a^2 \left (b^2-4 a c\right )^2}+\frac{\left (c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f+\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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}{16 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f+\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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 )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f-\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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 )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{\left (3 c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f+\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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 )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f-\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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 )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{\left (6 c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (3 b^4 d-25 a b^2 c d+28 a^2 c^2 d+a b^3 f+8 a^2 b c f+c \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f+\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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 )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (3 b^3 d-24 a b c d+a b^2 f+20 a^2 c f-\frac{3 b^4 d-30 a b^2 c d+168 a^2 c^2 d+a b^3 f-52 a^2 b c f}{\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 )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 4.4632, size = 625, normalized size = 1.01 \[ \frac{1}{16} \left (\frac{8 a^2 c \left (b (3 e+2 f x)+c x \left (7 d+6 e x+5 f x^2\right )\right )+2 a b x \left (b^2 f-25 b c d+b c f x^2-24 c^2 d x^2\right )+6 b^3 d x \left (b+c x^2\right )}{a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )+a b^2 \left (f \sqrt{b^2-4 a c}-30 c d\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}-42 c d\right )+b^3 \left (3 d \sqrt{b^2-4 a c}-a f\right )+a b^2 \left (f \sqrt{b^2-4 a c}+30 c d\right )+4 a b c \left (13 a f-6 d \sqrt{b^2-4 a c}\right )-3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{48 c^2 e \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{48 c^2 e \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{4 a b (e+f x)+8 a c x (d+x (e+f x))-4 b d x \left (b+c x^2\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a*b*(e + f*x) - 4*b*d*x*(b + c*x^2) + 8*a*c*x*(d + x*(e + f*x)))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2)
+ (6*b^3*d*x*(b + c*x^2) + 2*a*b*x*(-25*b*c*d + b^2*f - 24*c^2*d*x^2 + b*c*f*x^2) + 8*a^2*c*(b*(3*e + 2*f*x) +
c*x*(7*d + 6*e*x + 5*f*x^2)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(3*b^4*d + b^3*(3*
Sqrt[b^2 - 4*a*c]*d + a*f) - 4*a*b*c*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(-30*c*d + Sqrt[b^2 - 4*a*c]*f)
+ 4*a^2*c*(42*c*d + 5*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2
- 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-3*b^4*d + b^3*(3*Sqrt[b^2 - 4*a*c]*d - a*f)
+ 4*a*b*c*(-6*Sqrt[b^2 - 4*a*c]*d + 13*a*f) + a*b^2*(30*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(-42*c*d + 5*Sqrt
[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + S
qrt[b^2 - 4*a*c]]) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) - (48*c^2*e*Log[b +
Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/16

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Maple [B]  time = 0.28, size = 7858, normalized size = 12.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [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((f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError