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

Optimal. Leaf size=542 \[ \frac{\log \left (d+e x+f x^2\right ) \left (B \left (-f^2 \left (-a^2 f^2+2 a b e f+b^2 \left (-\left (e^2-d f\right )\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )+A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )\right )}{2 f^5}-\frac{\tanh ^{-1}\left (\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right ) \left (A f \left (-f^2 \left (-2 a^2 f^2+2 a b e f+b^2 \left (-\left (e^2-2 d f\right )\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )+c^2 \left (2 d^2 f^2-4 d e^2 f+e^4\right )\right )-B \left (f^2 \left (a^2 e f^2-2 a b f \left (e^2-2 d f\right )+b^2 \left (e^3-3 d e f\right )\right )+2 c f \left (a e f \left (e^2-3 d f\right )-b \left (2 d^2 f^2-4 d e^2 f+e^4\right )\right )+c^2 \left (5 d^2 e f^2-5 d e^3 f+e^5\right )\right )\right )}{f^5 \sqrt{e^2-4 d f}}-\frac{x^2 \left (A c f (c e-2 b f)-B \left (-2 c f (b e-a f)+b^2 f^2+c^2 \left (e^2-d f\right )\right )\right )}{2 f^3}+\frac{x \left (A f \left (-2 c f (b e-a f)+b^2 f^2+c^2 \left (e^2-d f\right )\right )+B (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )\right )}{f^4}-\frac{c x^3 (-A c f-2 b B f+B c e)}{3 f^2}+\frac{B c^2 x^4}{4 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.1025, antiderivative size = 542, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1011, 634, 618, 206, 628} \[ \frac{\log \left (d+e x+f x^2\right ) \left (B \left (-f^2 \left (-a^2 f^2+2 a b e f+b^2 \left (-\left (e^2-d f\right )\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )+A f (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )\right )}{2 f^5}-\frac{\tanh ^{-1}\left (\frac{e+2 f x}{\sqrt{e^2-4 d f}}\right ) \left (A f \left (-f^2 \left (-2 a^2 f^2+2 a b e f+b^2 \left (-\left (e^2-2 d f\right )\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )+c^2 \left (2 d^2 f^2-4 d e^2 f+e^4\right )\right )-B \left (f^2 \left (a^2 e f^2-2 a b f \left (e^2-2 d f\right )+b^2 \left (e^3-3 d e f\right )\right )+2 c f \left (a e f \left (e^2-3 d f\right )-b \left (2 d^2 f^2-4 d e^2 f+e^4\right )\right )+c^2 \left (5 d^2 e f^2-5 d e^3 f+e^5\right )\right )\right )}{f^5 \sqrt{e^2-4 d f}}-\frac{x^2 \left (A c f (c e-2 b f)-B \left (-2 c f (b e-a f)+b^2 f^2+c^2 \left (e^2-d f\right )\right )\right )}{2 f^3}+\frac{x \left (A f \left (-2 c f (b e-a f)+b^2 f^2+c^2 \left (e^2-d f\right )\right )+B (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )\right )}{f^4}-\frac{c x^3 (-A c f-2 b B f+B c e)}{3 f^2}+\frac{B c^2 x^4}{4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1011

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e
, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && IGtQ[p, 0] && IntegerQ[q]

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]

Rubi steps

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

Mathematica [A]  time = 0.626175, size = 535, normalized size = 0.99 \[ \frac{6 \log (d+x (e+f x)) \left (B \left (f^2 \left (a^2 f^2-2 a b e f+b^2 \left (e^2-d f\right )\right )-2 c f \left (a f \left (d f-e^2\right )+b \left (e^3-2 d e f\right )\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )+A f (b f-c e) \left (f (2 a f-b e)+c \left (e^2-2 d f\right )\right )\right )-\frac{12 \tan ^{-1}\left (\frac{e+2 f x}{\sqrt{4 d f-e^2}}\right ) \left (B \left (f^2 \left (a^2 e f^2+2 a b f \left (2 d f-e^2\right )+b^2 \left (e^3-3 d e f\right )\right )-2 c f \left (b \left (2 d^2 f^2-4 d e^2 f+e^4\right )-a e f \left (e^2-3 d f\right )\right )+c^2 \left (5 d^2 e f^2-5 d e^3 f+e^5\right )\right )-A f \left (f^2 \left (2 a^2 f^2-2 a b e f+b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )+c^2 \left (2 d^2 f^2-4 d e^2 f+e^4\right )\right )\right )}{\sqrt{4 d f-e^2}}+6 f^2 x^2 \left (B \left (2 c f (a f-b e)+b^2 f^2+c^2 \left (e^2-d f\right )\right )+A c f (2 b f-c e)\right )+12 f x \left (A f \left (2 c f (a f-b e)+b^2 f^2+c^2 \left (e^2-d f\right )\right )-B (c e-b f) \left (f (2 a f-b e)+c \left (e^2-2 d f\right )\right )\right )+4 c f^3 x^3 (A c f+2 b B f-B c e)+3 B c^2 f^4 x^4}{12 f^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*f*(-(B*(c*e - b*f)*(f*(-(b*e) + 2*a*f) + c*(e^2 - 2*d*f))) + A*f*(b^2*f^2 + 2*c*f*(-(b*e) + a*f) + c^2*(e^
2 - d*f)))*x + 6*f^2*(A*c*f*(-(c*e) + 2*b*f) + B*(b^2*f^2 + 2*c*f*(-(b*e) + a*f) + c^2*(e^2 - d*f)))*x^2 + 4*c
*f^3*(-(B*c*e) + 2*b*B*f + A*c*f)*x^3 + 3*B*c^2*f^4*x^4 - (12*(-(A*f*(c^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2) + f^2*
(-2*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*(e^3 - 3*d*e*f)))) + B*(c^2*(e^5 -
5*d*e^3*f + 5*d^2*e*f^2) + f^2*(a^2*e*f^2 + 2*a*b*f*(-e^2 + 2*d*f) + b^2*(e^3 - 3*d*e*f)) - 2*c*f*(-(a*e*f*(e
^2 - 3*d*f)) + b*(e^4 - 4*d*e^2*f + 2*d^2*f^2))))*ArcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/Sqrt[-e^2 + 4*d*f] +
6*(A*f*(-(c*e) + b*f)*(f*(-(b*e) + 2*a*f) + c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) + f^2*(-2*a
*b*e*f + a^2*f^2 + b^2*(e^2 - d*f)) - 2*c*f*(a*f*(-e^2 + d*f) + b*(e^3 - 2*d*e*f))))*Log[d + x*(e + f*x)])/(12
*f^5)

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Maple [B]  time = 0.167, size = 1672, normalized size = 3.1 \begin{align*} \text{result 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+e*x+d),x)

[Out]

-4/f^3/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*A*c^2*d*e^2-2/f/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e
)/(4*d*f-e^2)^(1/2))*e*A*a*b+1/4*B*c^2*x^4/f+1/3/f*A*x^3*c^2+1/2/f*B*x^2*b^2+1/f*A*b^2*x-8/f^3/(4*d*f-e^2)^(1/
2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*B*b*c*d*e^2+6/f^2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))
*B*a*c*d*e+6/f^2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*A*b*c*d*e-1/f^2*A*c^2*d*x+2/f*a*b*B*x+1
/f*B*x^2*a*c-1/2/f^2*B*x^2*c^2*d+2/f*a*c*A*x+2/3/f*B*x^3*b*c-2/f^2*B*b*c*d*x+1/2/f^3*ln(f*x^2+e*x+d)*B*c^2*d^2
-1/2/f^4*ln(f*x^2+e*x+d)*A*c^2*e^3-1/2/f^2*ln(f*x^2+e*x+d)*B*b^2*d+1/2/f^3*ln(f*x^2+e*x+d)*B*b^2*e^2+1/f*ln(f*
x^2+e*x+d)*A*a*b+1/2/f^5*ln(f*x^2+e*x+d)*B*c^2*e^4+1/f*A*x^2*b*c-1/2/f^2*ln(f*x^2+e*x+d)*A*b^2*e-1/3/f^2*B*x^3
*c^2*e-1/2/f^2*A*x^2*c^2*e+1/2/f^3*B*x^2*c^2*e^2+1/f^3*c^2*A*e^2*x-1/f^2*b^2*e*B*x-1/f^4*c^2*e^3*B*x+5/f^4/(4*
d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*B*c^2*d*e^3-5/f^3/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f
-e^2)^(1/2))*B*c^2*d^2*e-2/f^3/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^3*B*a*c+2/f^4/(4*d*f-e^
2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^4*B*b*c-4/f/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2
))*B*a*b*d+2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*A*a^2+1/2/f*ln(f*x^2+e*x+d)*B*a^2+2/f^2/(4*
d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^2*B*a*b+3/f^2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e
^2)^(1/2))*B*b^2*d*e+2/f^3*ln(f*x^2+e*x+d)*B*b*c*d*e+4/f^2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2
))*B*b*c*d^2+2/f^2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^2*A*a*c-2/f^3/(4*d*f-e^2)^(1/2)*arc
tan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^3*A*b*c-4/f/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*A*a*c*d+2
/f^3*b*c*e^2*B*x+2/f^3*B*c^2*d*e*x-1/f^2*ln(f*x^2+e*x+d)*B*a*b*e-1/f^2*ln(f*x^2+e*x+d)*B*a*c*d+1/f^3*ln(f*x^2+
e*x+d)*B*a*c*e^2-1/f^4*ln(f*x^2+e*x+d)*B*b*c*e^3-3/2/f^4*ln(f*x^2+e*x+d)*B*c^2*d*e^2+1/f^4/(4*d*f-e^2)^(1/2)*a
rctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^4*A*c^2+1/f^3*ln(f*x^2+e*x+d)*A*c^2*d*e-1/f^2*ln(f*x^2+e*x+d)*A*a*c*e-1/f
^2*ln(f*x^2+e*x+d)*A*b*c*d+1/f^3*ln(f*x^2+e*x+d)*A*b*c*e^2-2/f/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^
(1/2))*A*b^2*d+2/f^2/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*A*c^2*d^2+1/f^2/(4*d*f-e^2)^(1/2)*a
rctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^2*A*b^2-1/f/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e*B*a^2
-1/f^3/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e)/(4*d*f-e^2)^(1/2))*e^3*B*b^2-1/f^5/(4*d*f-e^2)^(1/2)*arctan((2*f*x+e
)/(4*d*f-e^2)^(1/2))*e^5*B*c^2-1/f^2*B*x^2*b*c*e-2/f^2*b*c*A*e*x-2/f^2*c*a*e*B*x

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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+e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.73957, size = 3933, normalized size = 7.26 \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+e*x+d),x, algorithm="fricas")

[Out]

[1/12*(3*(B*c^2*e^2*f^4 - 4*B*c^2*d*f^5)*x^4 - 4*(B*c^2*e^3*f^3 + 4*(2*B*b*c + A*c^2)*d*f^5 - (4*B*c^2*d*e + (
2*B*b*c + A*c^2)*e^2)*f^4)*x^3 + 6*(B*c^2*e^4*f^2 - 4*(B*b^2 + 2*(B*a + A*b)*c)*d*f^5 + (4*B*c^2*d^2 + 4*(2*B*
b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*f^4 - (5*B*c^2*d*e^2 + (2*B*b*c + A*c^2)*e^3)*f^3)*x^2 - 6*(
B*c^2*e^5 - 2*A*a^2*f^5 + (2*(2*B*a*b + A*b^2 + 2*A*a*c)*d + (B*a^2 + 2*A*a*b)*e)*f^4 - (2*(2*B*b*c + A*c^2)*d
^2 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e + (2*B*a*b + A*b^2 + 2*A*a*c)*e^2)*f^3 + (5*B*c^2*d^2*e + 4*(2*B*b*c + A*
c^2)*d*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*e^3)*f^2 - (5*B*c^2*d*e^3 + (2*B*b*c + A*c^2)*e^4)*f)*sqrt(e^2 - 4*d*f)
*log((2*f^2*x^2 + 2*e*f*x + e^2 - 2*d*f - sqrt(e^2 - 4*d*f)*(2*f*x + e))/(f*x^2 + e*x + d)) - 12*(B*c^2*e^5*f
+ 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*f^5 - (4*(2*B*b*c + A*c^2)*d^2 + 4*(B*b^2 + 2*(B*a + A*b)*c)*d*e + (2*B*a*b
+ A*b^2 + 2*A*a*c)*e^2)*f^4 + (8*B*c^2*d^2*e + 5*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*e^3)*f^3
- (6*B*c^2*d*e^3 + (2*B*b*c + A*c^2)*e^4)*f^2)*x + 6*(B*c^2*e^6 - 4*(B*a^2 + 2*A*a*b)*d*f^5 + (4*(B*b^2 + 2*(B
*a + A*b)*c)*d^2 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e + (B*a^2 + 2*A*a*b)*e^2)*f^4 - (4*B*c^2*d^3 + 8*(2*B*b*c
+ A*c^2)*d^2*e + 5*(B*b^2 + 2*(B*a + A*b)*c)*d*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^3)*f^3 + (13*B*c^2*d^2*e^2
+ 6*(2*B*b*c + A*c^2)*d*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*e^4)*f^2 - (7*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*f)*
log(f*x^2 + e*x + d))/(e^2*f^5 - 4*d*f^6), 1/12*(3*(B*c^2*e^2*f^4 - 4*B*c^2*d*f^5)*x^4 - 4*(B*c^2*e^3*f^3 + 4*
(2*B*b*c + A*c^2)*d*f^5 - (4*B*c^2*d*e + (2*B*b*c + A*c^2)*e^2)*f^4)*x^3 + 6*(B*c^2*e^4*f^2 - 4*(B*b^2 + 2*(B*
a + A*b)*c)*d*f^5 + (4*B*c^2*d^2 + 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*f^4 - (5*B*c^2*d*e
^2 + (2*B*b*c + A*c^2)*e^3)*f^3)*x^2 + 12*(B*c^2*e^5 - 2*A*a^2*f^5 + (2*(2*B*a*b + A*b^2 + 2*A*a*c)*d + (B*a^2
+ 2*A*a*b)*e)*f^4 - (2*(2*B*b*c + A*c^2)*d^2 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e + (2*B*a*b + A*b^2 + 2*A*a*c)*
e^2)*f^3 + (5*B*c^2*d^2*e + 4*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*e^3)*f^2 - (5*B*c^2*d*e^3 +
(2*B*b*c + A*c^2)*e^4)*f)*sqrt(-e^2 + 4*d*f)*arctan(-sqrt(-e^2 + 4*d*f)*(2*f*x + e)/(e^2 - 4*d*f)) - 12*(B*c^2
*e^5*f + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*f^5 - (4*(2*B*b*c + A*c^2)*d^2 + 4*(B*b^2 + 2*(B*a + A*b)*c)*d*e + (2
*B*a*b + A*b^2 + 2*A*a*c)*e^2)*f^4 + (8*B*c^2*d^2*e + 5*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*e^
3)*f^3 - (6*B*c^2*d*e^3 + (2*B*b*c + A*c^2)*e^4)*f^2)*x + 6*(B*c^2*e^6 - 4*(B*a^2 + 2*A*a*b)*d*f^5 + (4*(B*b^2
+ 2*(B*a + A*b)*c)*d^2 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e + (B*a^2 + 2*A*a*b)*e^2)*f^4 - (4*B*c^2*d^3 + 8*(2
*B*b*c + A*c^2)*d^2*e + 5*(B*b^2 + 2*(B*a + A*b)*c)*d*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^3)*f^3 + (13*B*c^2*d
^2*e^2 + 6*(2*B*b*c + A*c^2)*d*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*e^4)*f^2 - (7*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e
^5)*f)*log(f*x^2 + e*x + d))/(e^2*f^5 - 4*d*f^6)]

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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+e*x+d),x)

[Out]

Timed out

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Giac [A]  time = 1.15544, size = 996, normalized size = 1.84 \begin{align*} \frac{3 \, B c^{2} f^{3} x^{4} + 8 \, B b c f^{3} x^{3} + 4 \, A c^{2} f^{3} x^{3} - 4 \, B c^{2} f^{2} x^{3} e - 6 \, B c^{2} d f^{2} x^{2} + 6 \, B b^{2} f^{3} x^{2} + 12 \, B a c f^{3} x^{2} + 12 \, A b c f^{3} x^{2} - 12 \, B b c f^{2} x^{2} e - 6 \, A c^{2} f^{2} x^{2} e - 24 \, B b c d f^{2} x - 12 \, A c^{2} d f^{2} x + 24 \, B a b f^{3} x + 12 \, A b^{2} f^{3} x + 24 \, A a c f^{3} x + 6 \, B c^{2} f x^{2} e^{2} + 24 \, B c^{2} d f x e - 12 \, B b^{2} f^{2} x e - 24 \, B a c f^{2} x e - 24 \, A b c f^{2} x e + 24 \, B b c f x e^{2} + 12 \, A c^{2} f x e^{2} - 12 \, B c^{2} x e^{3}}{12 \, f^{4}} + \frac{{\left (B c^{2} d^{2} f^{2} - B b^{2} d f^{3} - 2 \, B a c d f^{3} - 2 \, A b c d f^{3} + B a^{2} f^{4} + 2 \, A a b f^{4} + 4 \, B b c d f^{2} e + 2 \, A c^{2} d f^{2} e - 2 \, B a b f^{3} e - A b^{2} f^{3} e - 2 \, A a c f^{3} e - 3 \, B c^{2} d f e^{2} + B b^{2} f^{2} e^{2} + 2 \, B a c f^{2} e^{2} + 2 \, A b c f^{2} e^{2} - 2 \, B b c f e^{3} - A c^{2} f e^{3} + B c^{2} e^{4}\right )} \log \left (f x^{2} + x e + d\right )}{2 \, f^{5}} + \frac{{\left (4 \, B b c d^{2} f^{3} + 2 \, A c^{2} d^{2} f^{3} - 4 \, B a b d f^{4} - 2 \, A b^{2} d f^{4} - 4 \, A a c d f^{4} + 2 \, A a^{2} f^{5} - 5 \, B c^{2} d^{2} f^{2} e + 3 \, B b^{2} d f^{3} e + 6 \, B a c d f^{3} e + 6 \, A b c d f^{3} e - B a^{2} f^{4} e - 2 \, A a b f^{4} e - 8 \, B b c d f^{2} e^{2} - 4 \, A c^{2} d f^{2} e^{2} + 2 \, B a b f^{3} e^{2} + A b^{2} f^{3} e^{2} + 2 \, A a c f^{3} e^{2} + 5 \, B c^{2} d f e^{3} - B b^{2} f^{2} e^{3} - 2 \, B a c f^{2} e^{3} - 2 \, A b c f^{2} e^{3} + 2 \, B b c f e^{4} + A c^{2} f e^{4} - B c^{2} e^{5}\right )} \arctan \left (\frac{2 \, f x + e}{\sqrt{4 \, d f - e^{2}}}\right )}{\sqrt{4 \, d f - e^{2}} f^{5}} \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+e*x+d),x, algorithm="giac")

[Out]

1/12*(3*B*c^2*f^3*x^4 + 8*B*b*c*f^3*x^3 + 4*A*c^2*f^3*x^3 - 4*B*c^2*f^2*x^3*e - 6*B*c^2*d*f^2*x^2 + 6*B*b^2*f^
3*x^2 + 12*B*a*c*f^3*x^2 + 12*A*b*c*f^3*x^2 - 12*B*b*c*f^2*x^2*e - 6*A*c^2*f^2*x^2*e - 24*B*b*c*d*f^2*x - 12*A
*c^2*d*f^2*x + 24*B*a*b*f^3*x + 12*A*b^2*f^3*x + 24*A*a*c*f^3*x + 6*B*c^2*f*x^2*e^2 + 24*B*c^2*d*f*x*e - 12*B*
b^2*f^2*x*e - 24*B*a*c*f^2*x*e - 24*A*b*c*f^2*x*e + 24*B*b*c*f*x*e^2 + 12*A*c^2*f*x*e^2 - 12*B*c^2*x*e^3)/f^4
+ 1/2*(B*c^2*d^2*f^2 - B*b^2*d*f^3 - 2*B*a*c*d*f^3 - 2*A*b*c*d*f^3 + B*a^2*f^4 + 2*A*a*b*f^4 + 4*B*b*c*d*f^2*e
+ 2*A*c^2*d*f^2*e - 2*B*a*b*f^3*e - A*b^2*f^3*e - 2*A*a*c*f^3*e - 3*B*c^2*d*f*e^2 + B*b^2*f^2*e^2 + 2*B*a*c*f
^2*e^2 + 2*A*b*c*f^2*e^2 - 2*B*b*c*f*e^3 - A*c^2*f*e^3 + B*c^2*e^4)*log(f*x^2 + x*e + d)/f^5 + (4*B*b*c*d^2*f^
3 + 2*A*c^2*d^2*f^3 - 4*B*a*b*d*f^4 - 2*A*b^2*d*f^4 - 4*A*a*c*d*f^4 + 2*A*a^2*f^5 - 5*B*c^2*d^2*f^2*e + 3*B*b^
2*d*f^3*e + 6*B*a*c*d*f^3*e + 6*A*b*c*d*f^3*e - B*a^2*f^4*e - 2*A*a*b*f^4*e - 8*B*b*c*d*f^2*e^2 - 4*A*c^2*d*f^
2*e^2 + 2*B*a*b*f^3*e^2 + A*b^2*f^3*e^2 + 2*A*a*c*f^3*e^2 + 5*B*c^2*d*f*e^3 - B*b^2*f^2*e^3 - 2*B*a*c*f^2*e^3
- 2*A*b*c*f^2*e^3 + 2*B*b*c*f*e^4 + A*c^2*f*e^4 - B*c^2*e^5)*arctan((2*f*x + e)/sqrt(4*d*f - e^2))/(sqrt(4*d*f
- e^2)*f^5)