3.5 \(\int \frac{A+B x}{(a+b x+c x^2)^2 (d+f x^2)} \, dx\)

Optimal. Leaf size=596 \[ -\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt{d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^2 c d f^2+3 a^3 f^3-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\left (b^2-4 a c\right )^{3/2} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac{f \log \left (a+b x+c x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac{f \log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac{-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )} \]

[Out]

(A*b*c*(c*d + a*f) - (A*b - a*B)*(2*c^2*d + b^2*f - 2*a*c*f) - c*(A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f
))*x)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(a + b*x + c*x^2)) - (f^(3/2)*(A*b^2*d*f + 2*b*B*d*(c*d - a*f)
- A*(c*d - a*f)^2)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) - ((b^5*
B*d*f^2 - 2*A*b^4*f^2*(c*d - a*f) - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + b^3*B*f*(5*c^2*d^2 - 4*a*c*d*f - a^2
*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3
*a^3*f^3))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*
f))^2) - (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[a + b*x + c*x^2])/(2*(c^2*
d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) + (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)
))*Log[d + f*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)

________________________________________________________________________________________

Rubi [A]  time = 1.77168, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1018, 1074, 634, 618, 206, 628, 635, 205, 260} \[ -\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt{d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^2 c d f^2+3 a^3 f^3-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\left (b^2-4 a c\right )^{3/2} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac{f \log \left (a+b x+c x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac{f \log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac{-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b*c*(c*d + a*f) - (A*b - a*B)*(2*c^2*d + b^2*f - 2*a*c*f) - c*(A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f
))*x)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(a + b*x + c*x^2)) - (f^(3/2)*(A*b^2*d*f + 2*b*B*d*(c*d - a*f)
- A*(c*d - a*f)^2)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) - ((b^5*
B*d*f^2 - 2*A*b^4*f^2*(c*d - a*f) - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + b^3*B*f*(5*c^2*d^2 - 4*a*c*d*f - a^2
*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3
*a^3*f^3))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*
f))^2) - (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[a + b*x + c*x^2])/(2*(c^2*
d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) + (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)
))*Log[d + f*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)

Rule 1018

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(
2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)
*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f
*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*
(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*
(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f
) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}
, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

Rule 1074

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol]
:> With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*b*B*f -
a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 +
b*B*d*f - A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]]
/; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 634

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

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])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx &=\frac{A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+\left (b^2-4 a c\right ) f (B c d+A b f-a B f) x+c f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x^2}{\left (a+b x+c x^2\right ) \left (d+f x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right )}\\ &=\frac{A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{-a b \left (b^2-4 a c\right ) f^2 (B c d+A b f-a B f)-a c^2 d f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+a^2 c f^2 \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+c^2 d \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )+b^2 f \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )-a c f \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )+c \left (c \left (b^2-4 a c\right ) d f (B c d+A b f-a B f)-a \left (b^2-4 a c\right ) f^2 (B c d+A b f-a B f)-b c d f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+b f \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )\right ) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac{\int \frac{b \left (b^2-4 a c\right ) d f^2 (B c d+A b f-a B f)+c^2 d^2 f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )-a c d f^2 \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )-c d f \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )+a f^2 \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )-f \left (c \left (b^2-4 a c\right ) d f (B c d+A b f-a B f)-a \left (b^2-4 a c\right ) f^2 (B c d+A b f-a B f)-b c d f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+b f \left (-(b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )\right ) x}{d+f x^2} \, dx}{\left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ &=\frac{A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac{\left (f^2 \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right )\right ) \int \frac{1}{d+f x^2} \, dx}{\left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac{\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac{\left (f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac{\left (f^2 \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right )\right ) \int \frac{x}{d+f x^2} \, dx}{\left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ &=\frac{A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac{f^{3/2} \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac{f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac{f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac{\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ &=\frac{A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac{f^{3/2} \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right )}{\sqrt{d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac{\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac{f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac{f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ \end{align*}

Mathematica [A]  time = 2.16971, size = 523, normalized size = 0.88 \[ \frac{-\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )-b^3 B f \left (a^2 f^2+4 a c d f-5 c^2 d^2\right )+2 b B c \left (3 a^2 c d f^2+3 a^3 f^3-7 a c^2 d^2 f+c^3 d^3\right )+2 A b^4 f^2 (a f-c d)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\left (4 a c-b^2\right )^{3/2}}+f \log \left (d+f x^2\right ) \left (B \left (f \left (a^2 f-b^2 d\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )-\frac{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right ) \left (B \left (2 a^2 c f-a \left (b^2 f+b c f x+2 c^2 d\right )-b c^2 d x\right )+A \left (b c (c d-3 a f)+2 c^2 x (c d-a f)+b^2 c f x+b^3 f\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+f \log (a+x (b+c x)) \left (B \left (f \left (b^2 d-a^2 f\right )+2 a c d f-c^2 d^2\right )+2 A b f (a f-c d)\right )+\frac{2 f^{3/2} \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{d}}\right ) \left (A (c d-a f)^2+2 b B d (a f-c d)-A b^2 d f\right )}{\sqrt{d}}}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))*(A*(b^3*f + b*c*(c*d - 3*a*f) + b^2*c*f*x + 2*c^2*(c*d - a*f)*x
) + B*(2*a^2*c*f - b*c^2*d*x - a*(2*c^2*d + b^2*f + b*c*f*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*f^(3/2)
*(-(A*b^2*d*f) + A*(c*d - a*f)^2 + 2*b*B*d*(-(c*d) + a*f))*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/Sqrt[d] - (2*(b^5*B*d*
f^2 - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + 2*A*b^4*f^2*(-(c*d) + a*f) - b^3*B*f*(-5*c^2*d^2 + 4*a*c*d*f + a^2
*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3
*a^3*f^3))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2
- 2*a*c*d*f + f*(-(b^2*d) + a^2*f)))*Log[d + f*x^2] + f*(2*A*b*f*(-(c*d) + a*f) + B*(-(c^2*d^2) + 2*a*c*d*f +
f*(b^2*d - a^2*f)))*Log[a + x*(b + c*x)])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)

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Maple [B]  time = 0.204, size = 9311, normalized size = 15.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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((B*x+A)/(c*x^2+b*x+a)^2/(f*x^2+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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+A)/(c*x^2+b*x+a)^2/(f*x^2+d),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+A)/(c*x**2+b*x+a)**2/(f*x**2+d),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.23283, size = 1773, normalized size = 2.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(B*c^2*d^2*f - B*b^2*d*f^2 - 2*B*a*c*d*f^2 + 2*A*b*c*d*f^2 + B*a^2*f^3 - 2*A*a*b*f^3)*log(c*x^2 + b*x + a
)/(c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 + 2*a^2*b^2
*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4) + 1/2*(B*c^2*d^2*f - B*b^2*d*f^2 - 2*B*a*c*d*f^2 + 2*A*b*c*d*f^2 + B*a^2*f^3
- 2*A*a*b*f^3)*log(f*x^2 + d)/(c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 +
6*a^2*c^2*d^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4) - (2*B*b*c*d^2*f^2 - A*c^2*d^2*f^2 - 2*B*a*b*d*
f^3 + A*b^2*d*f^3 + 2*A*a*c*d*f^3 - A*a^2*f^4)*arctan(f*x/sqrt(d*f))/((c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3
*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4)*sqrt(d*f
)) + (2*B*b*c^4*d^3 - 4*A*c^5*d^3 + 5*B*b^3*c^2*d^2*f - 14*B*a*b*c^3*d^2*f - 8*A*b^2*c^3*d^2*f + 20*A*a*c^4*d^
2*f + B*b^5*d*f^2 - 4*B*a*b^3*c*d*f^2 - 2*A*b^4*c*d*f^2 + 6*B*a^2*b*c^2*d*f^2 + 12*A*a*b^2*c^2*d*f^2 - 28*A*a^
2*c^3*d*f^2 - B*a^2*b^3*f^3 + 2*A*a*b^4*f^3 + 6*B*a^3*b*c*f^3 - 12*A*a^2*b^2*c*f^3 + 12*A*a^3*c^2*f^3)*arctan(
(2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4*d^4 - 4*a*c^5*d^4 + 2*b^4*c^2*d^3*f - 12*a*b^2*c^3*d^3*f + 16*a^2*c^
4*d^3*f + b^6*d^2*f^2 - 8*a*b^4*c*d^2*f^2 + 22*a^2*b^2*c^2*d^2*f^2 - 24*a^3*c^3*d^2*f^2 + 2*a^2*b^4*d*f^3 - 12
*a^3*b^2*c*d*f^3 + 16*a^4*c^2*d*f^3 + a^4*b^2*f^4 - 4*a^5*c*f^4)*sqrt(-b^2 + 4*a*c)) + (2*B*a*c^4*d^3 - A*b*c^
4*d^3 + 3*B*a*b^2*c^2*d^2*f - 2*A*b^3*c^2*d^2*f - 6*B*a^2*c^3*d^2*f + 5*A*a*b*c^3*d^2*f + B*a*b^4*d*f^2 - A*b^
5*d*f^2 - 4*B*a^2*b^2*c*d*f^2 + 5*A*a*b^3*c*d*f^2 + 6*B*a^3*c^2*d*f^2 - 7*A*a^2*b*c^2*d*f^2 + B*a^3*b^2*f^3 -
A*a^2*b^3*f^3 - 2*B*a^4*c*f^3 + 3*A*a^3*b*c*f^3 + (B*b*c^4*d^3 - 2*A*c^5*d^3 + B*b^3*c^2*d^2*f - B*a*b*c^3*d^2
*f - 3*A*b^2*c^3*d^2*f + 6*A*a*c^4*d^2*f + B*a*b^3*c*d*f^2 - A*b^4*c*d*f^2 - B*a^2*b*c^2*d*f^2 + 4*A*a*b^2*c^2
*d*f^2 - 6*A*a^2*c^3*d*f^2 + B*a^3*b*c*f^3 - A*a^2*b^2*c*f^3 + 2*A*a^3*c^2*f^3)*x)/((c^2*d^2 + b^2*d*f - 2*a*c
*d*f + a^2*f^2)^2*(c*x^2 + b*x + a)*(b^2 - 4*a*c))