3.1.5 \(\int \frac {A+B x}{(a+b x+c x^2)^2 (d+f x^2)} \, dx\) [5]

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

[Out]

(A*b*c*(a*f+c*d)-(A*b-B*a)*(-2*a*c*f+b^2*f+2*c^2*d)-c*(A*b^2*f+2*A*c*(-a*f+c*d)-b*B*(a*f+c*d))*x)/(-4*a*c+b^2)
/(b^2*d*f+(-a*f+c*d)^2)/(c*x^2+b*x+a)-(b^5*B*d*f^2-2*A*b^4*f^2*(-a*f+c*d)-4*A*c^2*(-3*a*f+c*d)*(-a*f+c*d)^2+b^
3*B*f*(-a^2*f^2-4*a*c*d*f+5*c^2*d^2)-4*A*b^2*c*f*(3*a^2*f^2-3*a*c*d*f+2*c^2*d^2)+2*b*B*c*(3*a^3*f^3+3*a^2*c*d*
f^2-7*a*c^2*d^2*f+c^3*d^3))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(c^2*d^2-2*a*c*d*f+f*(a^2
*f+b^2*d))^2-1/2*f*(2*A*b*f*(-a*f+c*d)+B*(c^2*d^2-2*a*c*d*f-f*(-a^2*f+b^2*d)))*ln(c*x^2+b*x+a)/(c^2*d^2-2*a*c*
d*f+f*(a^2*f+b^2*d))^2+1/2*f*(2*A*b*f*(-a*f+c*d)+B*(c^2*d^2-2*a*c*d*f-f*(-a^2*f+b^2*d)))*ln(f*x^2+d)/(c^2*d^2-
2*a*c*d*f+f*(a^2*f+b^2*d))^2-f^(3/2)*(A*b^2*d*f+2*b*B*d*(-a*f+c*d)-A*(-a*f+c*d)^2)*arctan(x*f^(1/2)/d^(1/2))/(
c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))^2/d^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.03, antiderivative size = 596, normalized size of antiderivative = 1.00, 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 = {1032, 1088, 648, 632, 212, 642, 649, 211, 266} \begin {gather*} -\frac {f^{3/2} \text {ArcTan}\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 {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 {\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^3 f^3+3 a^2 c d f^2-7 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 {-(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 )} \end {gather*}

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 211

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

Rule 212

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

Rule 266

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]

Rule 632

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 642

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 648

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 649

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 1032

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)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 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), 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 1088

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]

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 ) \text {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*}

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Mathematica [A]
time = 1.21, size = 523, normalized size = 0.88 \begin {gather*} \frac {-\frac {2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right ) \left (A \left (b^3 f+b c (c d-3 a f)+b^2 c f x+2 c^2 (c d-a f) x\right )+B \left (2 a^2 c f-b c^2 d x-a \left (2 c^2 d+b^2 f+b c f x\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 f^{3/2} \left (-A b^2 d f+A (c d-a f)^2+2 b B d (-c d+a f)\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \left (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 \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 ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+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 )+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 (a+x (b+c x))}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1253\) vs. \(2(580)=1160\).
time = 0.95, size = 1254, normalized size = 2.10

method	result	size
default	\(\frac {f^{2} \left (\frac {\left (-2 A a b \,f^{2}+2 A b c d f +B \,a^{2} f^{2}-2 B a c d f -B \,b^{2} d f +B \,c^{2} d^{2}\right ) \ln \left (f \,x^{2}+d \right )}{2 f}+\frac {\left (A \,a^{2} f^{2}-2 A a c d f -A \,b^{2} d f +A \,c^{2} d^{2}+2 B a b d f -2 B b c \,d^{2}\right ) \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}}\right )}{a^{4} f^{4}-4 a^{3} c d \,f^{3}+2 a^{2} b^{2} d \,f^{3}+6 a^{2} c^{2} d^{2} f^{2}-4 a \,b^{2} c \,d^{2} f^{2}-4 a \,c^{3} d^{3} f +b^{4} d^{2} f^{2}+2 b^{2} c^{2} d^{3} f +c^{4} d^{4}}-\frac {\frac {\frac {c \left (2 A \,a^{3} c \,f^{3}-A \,a^{2} b^{2} f^{3}-6 A \,a^{2} c^{2} d \,f^{2}+4 A a \,b^{2} c d \,f^{2}+6 A a \,c^{3} d^{2} f -A \,b^{4} d \,f^{2}-3 A \,b^{2} c^{2} d^{2} f -2 A \,c^{4} d^{3}+B \,a^{3} b \,f^{3}-B \,a^{2} b c d \,f^{2}+B a \,b^{3} d \,f^{2}-B a b \,c^{2} d^{2} f +B \,b^{3} c \,d^{2} f +B b \,c^{3} d^{3}\right ) x}{4 a c -b^{2}}+\frac {3 A \,a^{3} b c \,f^{3}-A \,a^{2} b^{3} f^{3}-7 A \,a^{2} b \,c^{2} d \,f^{2}+5 A a \,b^{3} c d \,f^{2}+5 A a b \,c^{3} d^{2} f -A \,b^{5} d \,f^{2}-2 A \,b^{3} c^{2} d^{2} f -A b \,c^{4} d^{3}-2 B \,a^{4} c \,f^{3}+B \,a^{3} b^{2} f^{3}+6 B \,a^{3} c^{2} d \,f^{2}-4 B \,a^{2} b^{2} c d \,f^{2}-6 B \,a^{2} c^{3} d^{2} f +B a \,b^{4} d \,f^{2}+3 B a \,b^{2} c^{2} d^{2} f +2 B a \,c^{4} d^{3}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 A \,a^{2} b \,c^{2} f^{3}+2 A a \,b^{3} c \,f^{3}+8 A a b \,c^{3} d \,f^{2}-2 A \,b^{3} c^{2} d \,f^{2}+4 B \,a^{3} c^{2} f^{3}-B \,a^{2} b^{2} c \,f^{3}-8 B \,a^{2} c^{3} d \,f^{2}-2 B a \,b^{2} c^{2} d \,f^{2}+4 B a \,c^{4} d^{2} f +B \,b^{4} c d \,f^{2}-B \,b^{2} c^{3} d^{2} f \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (6 A \,a^{3} c^{2} f^{3}-10 A \,a^{2} b^{2} c \,f^{3}-14 A \,a^{2} c^{3} d \,f^{2}+2 A a \,b^{4} f^{3}+10 A a \,b^{2} c^{2} d \,f^{2}+10 A a \,c^{4} d^{2} f -2 A \,b^{4} c d \,f^{2}-4 A \,b^{2} c^{3} d^{2} f -2 A \,c^{5} d^{3}+5 B \,a^{3} b c \,f^{3}-B \,a^{2} b^{3} f^{3}-B \,a^{2} b \,c^{2} d \,f^{2}-3 B a \,b^{3} c d \,f^{2}-5 B a b \,c^{3} d^{2} f +b^{5} B d \,f^{2}+2 B \,b^{3} c^{2} d^{2} f +B b \,c^{4} d^{3}-\frac {\left (-8 A \,a^{2} b \,c^{2} f^{3}+2 A a \,b^{3} c \,f^{3}+8 A a b \,c^{3} d \,f^{2}-2 A \,b^{3} c^{2} d \,f^{2}+4 B \,a^{3} c^{2} f^{3}-B \,a^{2} b^{2} c \,f^{3}-8 B \,a^{2} c^{3} d \,f^{2}-2 B a \,b^{2} c^{2} d \,f^{2}+4 B a \,c^{4} d^{2} f +B \,b^{4} c d \,f^{2}-B \,b^{2} c^{3} d^{2} f \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{4} f^{4}-4 a^{3} c d \,f^{3}+2 a^{2} b^{2} d \,f^{3}+6 a^{2} c^{2} d^{2} f^{2}-4 a \,b^{2} c \,d^{2} f^{2}-4 a \,c^{3} d^{3} f +b^{4} d^{2} f^{2}+2 b^{2} c^{2} d^{3} f +c^{4} d^{4}}\)	\(1254\)
risch	\(\text {Expression too large to display}\)	\(3364134\)

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,method=_RETURNVERBOSE)

[Out]

f^2/(a^4*f^4-4*a^3*c*d*f^3+2*a^2*b^2*d*f^3+6*a^2*c^2*d^2*f^2-4*a*b^2*c*d^2*f^2-4*a*c^3*d^3*f+b^4*d^2*f^2+2*b^2
*c^2*d^3*f+c^4*d^4)*(1/2*(-2*A*a*b*f^2+2*A*b*c*d*f+B*a^2*f^2-2*B*a*c*d*f-B*b^2*d*f+B*c^2*d^2)/f*ln(f*x^2+d)+(A
*a^2*f^2-2*A*a*c*d*f-A*b^2*d*f+A*c^2*d^2+2*B*a*b*d*f-2*B*b*c*d^2)/(d*f)^(1/2)*arctan(f*x/(d*f)^(1/2)))-1/(a^4*
f^4-4*a^3*c*d*f^3+2*a^2*b^2*d*f^3+6*a^2*c^2*d^2*f^2-4*a*b^2*c*d^2*f^2-4*a*c^3*d^3*f+b^4*d^2*f^2+2*b^2*c^2*d^3*
f+c^4*d^4)*((c*(2*A*a^3*c*f^3-A*a^2*b^2*f^3-6*A*a^2*c^2*d*f^2+4*A*a*b^2*c*d*f^2+6*A*a*c^3*d^2*f-A*b^4*d*f^2-3*
A*b^2*c^2*d^2*f-2*A*c^4*d^3+B*a^3*b*f^3-B*a^2*b*c*d*f^2+B*a*b^3*d*f^2-B*a*b*c^2*d^2*f+B*b^3*c*d^2*f+B*b*c^3*d^
3)/(4*a*c-b^2)*x+(3*A*a^3*b*c*f^3-A*a^2*b^3*f^3-7*A*a^2*b*c^2*d*f^2+5*A*a*b^3*c*d*f^2+5*A*a*b*c^3*d^2*f-A*b^5*
d*f^2-2*A*b^3*c^2*d^2*f-A*b*c^4*d^3-2*B*a^4*c*f^3+B*a^3*b^2*f^3+6*B*a^3*c^2*d*f^2-4*B*a^2*b^2*c*d*f^2-6*B*a^2*
c^3*d^2*f+B*a*b^4*d*f^2+3*B*a*b^2*c^2*d^2*f+2*B*a*c^4*d^3)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)*(1/2*(-8*A
*a^2*b*c^2*f^3+2*A*a*b^3*c*f^3+8*A*a*b*c^3*d*f^2-2*A*b^3*c^2*d*f^2+4*B*a^3*c^2*f^3-B*a^2*b^2*c*f^3-8*B*a^2*c^3
*d*f^2-2*B*a*b^2*c^2*d*f^2+4*B*a*c^4*d^2*f+B*b^4*c*d*f^2-B*b^2*c^3*d^2*f)/c*ln(c*x^2+b*x+a)+2*(6*A*a^3*c^2*f^3
-10*A*a^2*b^2*c*f^3-14*A*a^2*c^3*d*f^2+2*A*a*b^4*f^3+10*A*a*b^2*c^2*d*f^2+10*A*a*c^4*d^2*f-2*A*b^4*c*d*f^2-4*A
*b^2*c^3*d^2*f-2*A*c^5*d^3+5*B*a^3*b*c*f^3-B*a^2*b^3*f^3-B*a^2*b*c^2*d*f^2-3*B*a*b^3*c*d*f^2-5*B*a*b*c^3*d^2*f
+b^5*B*d*f^2+2*B*b^3*c^2*d^2*f+B*b*c^4*d^3-1/2*(-8*A*a^2*b*c^2*f^3+2*A*a*b^3*c*f^3+8*A*a*b*c^3*d*f^2-2*A*b^3*c
^2*d*f^2+4*B*a^3*c^2*f^3-B*a^2*b^2*c*f^3-8*B*a^2*c^3*d*f^2-2*B*a*b^2*c^2*d*f^2+4*B*a*c^4*d^2*f+B*b^4*c*d*f^2-B
*b^2*c^3*d^2*f)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1313 vs. \(2 (579) = 1158\).
time = 4.24, size = 1313, normalized size = 2.20 \begin {gather*} -\frac {{\left (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}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (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}\right )}} + \frac {{\left (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}\right )} \log \left (f x^{2} + d\right )}{2 \, {\left (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}\right )}} - \frac {{\left (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}\right )} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{{\left (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}\right )} \sqrt {d f}} + \frac {{\left (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}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (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}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {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} + {\left (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}\right )} x}{{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )}^{2} {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}

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))

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Mupad [B]
time = 7.53, size = 2500, normalized size = 4.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((A*b^3*f + A*b*c^2*d - 2*B*a*c^2*d - B*a*b^2*f + 2*B*a^2*c*f - 3*A*a*b*c*f)/((4*a*c - b^2)*(a^2*f^2 + c^2*d^2
+ b^2*d*f - 2*a*c*d*f)) - (x*(2*A*a*c^2*f - 2*A*c^3*d + B*b*c^2*d - A*b^2*c*f + B*a*b*c*f))/((4*a*c - b^2)*(a
^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)))/(a + b*x + c*x^2) + symsum(log((x*(4*A^3*b^3*c^4*f^6 + 16*B^3*a^3*c^
4*f^6 - 3*B^3*a^2*b^2*c^3*f^6 + B^3*b^2*c^5*d^2*f^4 - 16*A^3*a*b*c^5*f^6 + 20*A^2*B*a^2*c^5*f^6 - 3*A^2*B*b^4*
c^3*f^6 + 4*A^2*B*c^7*d^2*f^4 - 16*B^3*a^2*c^5*d*f^5 + 6*B^3*a*b^2*c^4*d*f^5 - 24*A^2*B*a*c^6*d*f^5 + 6*A*B^2*
a*b^3*c^3*f^6 - 28*A*B^2*a^2*b*c^4*f^6 + 8*A^2*B*a*b^2*c^4*f^6 - 4*A*B^2*b*c^6*d^2*f^4 - 6*A*B^2*b^3*c^4*d*f^5
+ 8*A^2*B*b^2*c^5*d*f^5 + 16*A*B^2*a*b*c^5*d*f^5))/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f
^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 +
2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f
- 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3) - root(2560*a^3*b^2*c
^9*d^8*f*z^4 - 1152*a^2*b^4*c^8*d^8*f*z^4 + 384*a^5*b^8*c*d^3*f^6*z^4 + 384*a*b^8*c^5*d^7*f^2*z^4 + 288*a^3*b^
10*c*d^4*f^5*z^4 + 288*a*b^10*c^3*d^6*f^3*z^4 + 224*a^7*b^6*c*d^2*f^7*z^4 - 192*a^10*b^2*c^2*d*f^8*z^4 + 224*a
*b^6*c^7*d^8*f*z^4 + 80*a*b^12*c*d^5*f^4*z^4 + 48*a^9*b^4*c*d*f^8*z^4 - 33920*a^6*b^2*c^6*d^5*f^4*z^4 + 27936*
a^5*b^4*c^5*d^5*f^4*z^4 + 26112*a^7*b^2*c^5*d^4*f^5*z^4 + 26112*a^5*b^2*c^7*d^6*f^3*z^4 - 20352*a^6*b^4*c^4*d^
4*f^5*z^4 - 20352*a^4*b^4*c^6*d^6*f^3*z^4 - 13080*a^4*b^6*c^4*d^5*f^4*z^4 - 11520*a^8*b^2*c^4*d^3*f^6*z^4 - 11
520*a^4*b^2*c^8*d^7*f^2*z^4 + 8736*a^5*b^6*c^3*d^4*f^5*z^4 + 8736*a^3*b^6*c^5*d^6*f^3*z^4 + 7488*a^7*b^4*c^3*d
^3*f^6*z^4 + 7488*a^3*b^4*c^7*d^7*f^2*z^4 + 3840*a^3*b^8*c^3*d^5*f^4*z^4 + 2560*a^9*b^2*c^3*d^2*f^7*z^4 - 2416
*a^6*b^6*c^2*d^3*f^6*z^4 - 2416*a^2*b^6*c^6*d^7*f^2*z^4 - 2160*a^4*b^8*c^2*d^4*f^5*z^4 - 2160*a^2*b^8*c^4*d^6*
f^3*z^4 - 1152*a^8*b^4*c^2*d^2*f^7*z^4 - 720*a^2*b^10*c^2*d^5*f^4*z^4 - 16*b^8*c^6*d^8*f*z^4 - 2048*a^4*c^10*d
^8*f*z^4 + 256*a^11*c^3*d*f^8*z^4 - 4*a^8*b^6*d*f^8*z^4 + 48*a*b^4*c^9*d^9*z^4 - 24*b^10*c^4*d^7*f^2*z^4 - 16*
b^12*c^2*d^6*f^3*z^4 + 17920*a^7*c^7*d^5*f^4*z^4 - 14336*a^8*c^6*d^4*f^5*z^4 - 14336*a^6*c^8*d^6*f^3*z^4 + 716
8*a^9*c^5*d^3*f^6*z^4 + 7168*a^5*c^9*d^7*f^2*z^4 - 2048*a^10*c^4*d^2*f^7*z^4 - 24*a^4*b^10*d^3*f^6*z^4 - 16*a^
6*b^8*d^2*f^7*z^4 - 16*a^2*b^12*d^4*f^5*z^4 - 192*a^2*b^2*c^10*d^9*z^4 - 4*b^14*d^5*f^4*z^4 - 4*b^6*c^8*d^9*z^
4 + 256*a^3*c^11*d^9*z^4 + 912*A*B*a^6*b*c^3*d*f^6*z^2 + 192*A*B*a^4*b^5*c*d*f^6*z^2 + 920*A*B*a^4*b^3*c^3*d^2
*f^5*z^2 - 480*A*B*a^2*b^5*c^3*d^3*f^4*z^2 - 336*A*B*a^2*b^3*c^5*d^4*f^3*z^2 - 272*A*B*a^3*b^3*c^4*d^3*f^4*z^2
+ 240*A*B*a^3*b^5*c^2*d^2*f^5*z^2 + 192*A*B*a*b*c^8*d^6*f*z^2 - 2496*A*B*a^5*b*c^4*d^2*f^5*z^2 + 1872*A*B*a^4
*b*c^5*d^3*f^4*z^2 - 744*A*B*a^5*b^3*c^2*d*f^6*z^2 - 720*A*B*a^2*b*c^7*d^5*f^2*z^2 + 504*A*B*a*b^3*c^6*d^5*f^2
*z^2 + 256*A*B*a^3*b*c^6*d^4*f^3*z^2 + 168*A*B*a*b^7*c^2*d^3*f^4*z^2 - 144*A*B*a^2*b^7*c*d^2*f^5*z^2 + 144*A*B
*a*b^5*c^4*d^4*f^3*z^2 - 56*B^2*a*b^2*c^7*d^6*f*z^2 - 36*B^2*a^5*b^4*c*d*f^6*z^2 - 16*B^2*a*b^8*c*d^3*f^4*z^2
- 164*A^2*a^3*b^6*c*d*f^6*z^2 - 16*A^2*a*b^8*c*d^2*f^5*z^2 - 96*A*B*b^5*c^5*d^5*f^2*z^2 - 24*A*B*b^7*c^3*d^4*f
^3*z^2 - 580*B^2*a^4*b^2*c^4*d^3*f^4*z^2 + 536*B^2*a^3*b^4*c^3*d^3*f^4*z^2 - 348*B^2*a^4*b^4*c^2*d^2*f^5*z^2 +
316*B^2*a^2*b^2*c^6*d^5*f^2*z^2 + 200*B^2*a^5*b^2*c^3*d^2*f^5*z^2 - 120*B^2*a^2*b^4*c^4*d^4*f^3*z^2 - 66*B^2*
a^2*b^6*c^2*d^3*f^4*z^2 - 16*B^2*a^3*b^2*c^5*d^4*f^3*z^2 + 1952*A^2*a^4*b^2*c^4*d^2*f^5*z^2 - 1792*A^2*a^3*b^2
*c^5*d^3*f^4*z^2 - 1272*A^2*a^3*b^4*c^3*d^2*f^5*z^2 + 976*A^2*a^2*b^2*c^6*d^4*f^3*z^2 + 960*A^2*a^2*b^4*c^4*d^
3*f^4*z^2 + 282*A^2*a^2*b^6*c^2*d^2*f^5*z^2 - 72*A*B*b^3*c^7*d^6*f*z^2 - 16*A*B*b^9*c*d^3*f^4*z^2 - 16*A*B*a^3
*b^7*d*f^6*z^2 + 16*A*B*a*b^9*d^2*f^5*z^2 - 180*B^2*a*b^4*c^5*d^5*f^2*z^2 + 132*B^2*a^6*b^2*c^2*d*f^6*z^2 + 10
8*B^2*a^3*b^6*c*d^2*f^5*z^2 + 20*B^2*a*b^6*c^3*d^4*f^3*z^2 - 736*A^2*a^5*b^2*c^3*d*f^6*z^2 + 624*A^2*a^4*b^4*c
^2*d*f^6*z^2 - 416*A^2*a*b^2*c^7*d^5*f^2*z^2 - 276*A^2*a*b^4*c^5*d^4*f^3*z^2 - 196*A^2*a*b^6*c^3*d^3*f^4*z^2 +
31*B^2*b^6*c^4*d^5*f^2*z^2 + 2*B^2*b^8*c^2*d^4*f^3*z^2 - 768*B^2*a^5*c^5*d^3*f^4*z^2 + 512*B^2*a^6*c^4*d^2*f^
5*z^2 + 512*B^2*a^4*c^6*d^4*f^3*z^2 - 128*B^2*a^3*c^7*d^5*f^2*z^2 + 80*A^2*b^4*c^6*d^5*f^2*z^2 + 31*A^2*b^6*c^
4*d^4*f^3*z^2 + 14*A^2*b^8*c^2*d^3*f^4*z^2 - 1152*A^2*a^3*c^7*d^4*f^3*z^2 + 1008*A^2*a^4*c^6*d^3*f^4*z^2 + 624
*A^2*a^2*c^8*d^5*f^2*z^2 - 288*A^2*a^5*c^5*d^2*f^5*z^2 - 10*B^2*a^2*b^8*d^2*f^5*z^2 - 48*A^2*a^6*b^2*c^2*f^7*z
^2 - 16*A*B*b*c^9*d^7*z^2 + 20*B^2*b^4*c^6*d^6*f*z^2 - 128*B^2*a^7*c^3*d*f^6*z^2 + 64*A^2*b^2*c^8*d^6*f*z^2 -
112*A^2*a^6*c^4*d*f^6*z^2 + 3*B^2*a^4*b^6*d*f^6*z^2 + 14*A^2*a^2*b^8*d*f^6*z^2 + 12*A^2*a^5*b^4*c*f^7*z^2 - 16
0*A^2*a*c^9*d^6*f*z^2 + 3*B^2*b^10*d^3*f^4*z^2 - A^2*b^10*d^2*f^5*z^2 + 64*A^2*a^7*c^3*f^7*z^2 + 4*B^2*b^2*c^8
*d^7*z^2 - A^2*a^4*b^6*f^7*z^2 + 16*A^2*c^10*d^...

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