3.373 \(\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=765 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (c^6 \left (2 a^2 h+3 a b g+b^2 f\right )-c^5 \left (5 a^2 b j+2 a^3 k+4 a b^2 h+b^3 g\right )+c^4 \left (9 a^2 b^2 k+7 a^3 b l+2 a^4 m+5 a b^3 j+b^4 h\right )-b^2 c^3 \left (14 a^2 b l+16 a^3 m+6 a b^2 k+b^3 j\right )+b^4 c^2 \left (20 a^2 m+7 a b l+b^2 k\right )-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right )}{c^8 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-c^4 \left (3 a^2 b k+a^3 l+3 a b^2 j+b^3 h\right )+b c^3 \left (6 a^2 b l+4 a^3 m+4 a b^2 k+b^3 j\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )}{2 c^8}+\frac{x \left (c^4 \left (a^2 k+2 a b j+b^2 h\right )-c^3 \left (3 a^2 b l+a^3 m+3 a b^2 k+b^3 j\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right )}{c^7}+\frac{x^2 \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )}{2 c^6}+\frac{x^3 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )}{3 c^5}+\frac{x^4 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )}{4 c^4}+\frac{x^5 \left (-c (a m+b l)+b^2 m+c^2 k\right )}{5 c^3}+\frac{x^6 (c l-b m)}{6 c^2}+\frac{m x^7}{7 c} \]

[Out]

((c^6*f - c^5*(b*g + a*h) + c^4*(b^2*h + 2*a*b*j + a^2*k) + b^6*m - b^4*c*(b*l + 5*a*m) + b^2*c^2*(b^2*k + 4*a
*b*l + 6*a^2*m) - c^3*(b^3*j + 3*a*b^2*k + 3*a^2*b*l + a^3*m))*x)/c^7 + ((c^5*g - c^4*(b*h + a*j) + c^3*(b^2*j
+ 2*a*b*k + a^2*l) - b^5*m + b^3*c*(b*l + 4*a*m) - b*c^2*(b^2*k + 3*a*b*l + 3*a^2*m))*x^2)/(2*c^6) + ((c^4*h
- c^3*(b*j + a*k) + b^4*m - b^2*c*(b*l + 3*a*m) + c^2*(b^2*k + 2*a*b*l + a^2*m))*x^3)/(3*c^5) + ((c^3*j - c^2*
(b*k + a*l) - b^3*m + b*c*(b*l + 2*a*m))*x^4)/(4*c^4) + ((c^2*k + b^2*m - c*(b*l + a*m))*x^5)/(5*c^3) + ((c*l
- b*m)*x^6)/(6*c^2) + (m*x^7)/(7*c) - ((2*c^8*d - c^7*(b*e + 2*a*f) + c^6*(b^2*f + 3*a*b*g + 2*a^2*h) - c^5*(b
^3*g + 4*a*b^2*h + 5*a^2*b*j + 2*a^3*k) + b^8*m - b^6*c*(b*l + 8*a*m) + b^4*c^2*(b^2*k + 7*a*b*l + 20*a^2*m) -
b^2*c^3*(b^3*j + 6*a*b^2*k + 14*a^2*b*l + 16*a^3*m) + c^4*(b^4*h + 5*a*b^3*j + 9*a^2*b^2*k + 7*a^3*b*l + 2*a^
4*m))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^8*Sqrt[b^2 - 4*a*c]) + ((c^7*e - c^6*(b*f + a*g) + c^5*(b^2*g
+ 2*a*b*h + a^2*j) - c^4*(b^3*h + 3*a*b^2*j + 3*a^2*b*k + a^3*l) - b^7*m + b^5*c*(b*l + 6*a*m) - b^3*c^2*(b^2
*k + 5*a*b*l + 10*a^2*m) + b*c^3*(b^3*j + 4*a*b^2*k + 6*a^2*b*l + 4*a^3*m))*Log[a + b*x + c*x^2])/(2*c^8)

________________________________________________________________________________________

Rubi [A]  time = 5.82519, antiderivative size = 765, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 53, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {1657, 634, 618, 206, 628} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (c^6 \left (2 a^2 h+3 a b g+b^2 f\right )-c^5 \left (5 a^2 b j+2 a^3 k+4 a b^2 h+b^3 g\right )+c^4 \left (9 a^2 b^2 k+7 a^3 b l+2 a^4 m+5 a b^3 j+b^4 h\right )-b^2 c^3 \left (14 a^2 b l+16 a^3 m+6 a b^2 k+b^3 j\right )+b^4 c^2 \left (20 a^2 m+7 a b l+b^2 k\right )-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right )}{c^8 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-c^4 \left (3 a^2 b k+a^3 l+3 a b^2 j+b^3 h\right )+b c^3 \left (6 a^2 b l+4 a^3 m+4 a b^2 k+b^3 j\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )}{2 c^8}+\frac{x \left (c^4 \left (a^2 k+2 a b j+b^2 h\right )-c^3 \left (3 a^2 b l+a^3 m+3 a b^2 k+b^3 j\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right )}{c^7}+\frac{x^2 \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )}{2 c^6}+\frac{x^3 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )}{3 c^5}+\frac{x^4 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )}{4 c^4}+\frac{x^5 \left (-c (a m+b l)+b^2 m+c^2 k\right )}{5 c^3}+\frac{x^6 (c l-b m)}{6 c^2}+\frac{m x^7}{7 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x + c*x^2),x]

[Out]

((c^6*f - c^5*(b*g + a*h) + c^4*(b^2*h + 2*a*b*j + a^2*k) + b^6*m - b^4*c*(b*l + 5*a*m) + b^2*c^2*(b^2*k + 4*a
*b*l + 6*a^2*m) - c^3*(b^3*j + 3*a*b^2*k + 3*a^2*b*l + a^3*m))*x)/c^7 + ((c^5*g - c^4*(b*h + a*j) + c^3*(b^2*j
+ 2*a*b*k + a^2*l) - b^5*m + b^3*c*(b*l + 4*a*m) - b*c^2*(b^2*k + 3*a*b*l + 3*a^2*m))*x^2)/(2*c^6) + ((c^4*h
- c^3*(b*j + a*k) + b^4*m - b^2*c*(b*l + 3*a*m) + c^2*(b^2*k + 2*a*b*l + a^2*m))*x^3)/(3*c^5) + ((c^3*j - c^2*
(b*k + a*l) - b^3*m + b*c*(b*l + 2*a*m))*x^4)/(4*c^4) + ((c^2*k + b^2*m - c*(b*l + a*m))*x^5)/(5*c^3) + ((c*l
- b*m)*x^6)/(6*c^2) + (m*x^7)/(7*c) - ((2*c^8*d - c^7*(b*e + 2*a*f) + c^6*(b^2*f + 3*a*b*g + 2*a^2*h) - c^5*(b
^3*g + 4*a*b^2*h + 5*a^2*b*j + 2*a^3*k) + b^8*m - b^6*c*(b*l + 8*a*m) + b^4*c^2*(b^2*k + 7*a*b*l + 20*a^2*m) -
b^2*c^3*(b^3*j + 6*a*b^2*k + 14*a^2*b*l + 16*a^3*m) + c^4*(b^4*h + 5*a*b^3*j + 9*a^2*b^2*k + 7*a^3*b*l + 2*a^
4*m))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^8*Sqrt[b^2 - 4*a*c]) + ((c^7*e - c^6*(b*f + a*g) + c^5*(b^2*g
+ 2*a*b*h + a^2*j) - c^4*(b^3*h + 3*a*b^2*j + 3*a^2*b*k + a^3*l) - b^7*m + b^5*c*(b*l + 6*a*m) - b^3*c^2*(b^2
*k + 5*a*b*l + 10*a^2*m) + b*c^3*(b^3*j + 4*a*b^2*k + 6*a^2*b*l + 4*a^3*m))*Log[a + b*x + c*x^2])/(2*c^8)

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx &=\int \left (\frac{c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )}{c^7}+\frac{\left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x}{c^6}+\frac{\left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^2}{c^5}+\frac{\left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^3}{c^4}+\frac{\left (c^2 k+b^2 m-c (b l+a m)\right ) x^4}{c^3}+\frac{(c l-b m) x^5}{c^2}+\frac{m x^6}{c}+\frac{c^7 d-a c^6 f+a c^5 (b g+a h)-a c^4 \left (b^2 h+2 a b j+a^2 k\right )-a b^6 m+a b^4 c (b l+5 a m)-a b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )+a c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )+\left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) x}{c^7 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\left (c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )\right ) x}{c^7}+\frac{\left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x^2}{2 c^6}+\frac{\left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^3}{3 c^5}+\frac{\left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^4}{4 c^4}+\frac{\left (c^2 k+b^2 m-c (b l+a m)\right ) x^5}{5 c^3}+\frac{(c l-b m) x^6}{6 c^2}+\frac{m x^7}{7 c}+\frac{\int \frac{c^7 d-a c^6 f+a c^5 (b g+a h)-a c^4 \left (b^2 h+2 a b j+a^2 k\right )-a b^6 m+a b^4 c (b l+5 a m)-a b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )+a c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )+\left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) x}{a+b x+c x^2} \, dx}{c^7}\\ &=\frac{\left (c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )\right ) x}{c^7}+\frac{\left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x^2}{2 c^6}+\frac{\left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^3}{3 c^5}+\frac{\left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^4}{4 c^4}+\frac{\left (c^2 k+b^2 m-c (b l+a m)\right ) x^5}{5 c^3}+\frac{(c l-b m) x^6}{6 c^2}+\frac{m x^7}{7 c}+\frac{\left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^8}+\frac{\left (2 c^8 d-c^7 (b e+2 a f)+c^6 \left (b^2 f+3 a b g+2 a^2 h\right )-c^5 \left (b^3 g+4 a b^2 h+5 a^2 b j+2 a^3 k\right )+b^8 m-b^6 c (b l+8 a m)+b^4 c^2 \left (b^2 k+7 a b l+20 a^2 m\right )-b^2 c^3 \left (b^3 j+6 a b^2 k+14 a^2 b l+16 a^3 m\right )+c^4 \left (b^4 h+5 a b^3 j+9 a^2 b^2 k+7 a^3 b l+2 a^4 m\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^8}\\ &=\frac{\left (c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )\right ) x}{c^7}+\frac{\left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x^2}{2 c^6}+\frac{\left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^3}{3 c^5}+\frac{\left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^4}{4 c^4}+\frac{\left (c^2 k+b^2 m-c (b l+a m)\right ) x^5}{5 c^3}+\frac{(c l-b m) x^6}{6 c^2}+\frac{m x^7}{7 c}+\frac{\left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^8}-\frac{\left (2 c^8 d-c^7 (b e+2 a f)+c^6 \left (b^2 f+3 a b g+2 a^2 h\right )-c^5 \left (b^3 g+4 a b^2 h+5 a^2 b j+2 a^3 k\right )+b^8 m-b^6 c (b l+8 a m)+b^4 c^2 \left (b^2 k+7 a b l+20 a^2 m\right )-b^2 c^3 \left (b^3 j+6 a b^2 k+14 a^2 b l+16 a^3 m\right )+c^4 \left (b^4 h+5 a b^3 j+9 a^2 b^2 k+7 a^3 b l+2 a^4 m\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^8}\\ &=\frac{\left (c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )\right ) x}{c^7}+\frac{\left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x^2}{2 c^6}+\frac{\left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^3}{3 c^5}+\frac{\left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^4}{4 c^4}+\frac{\left (c^2 k+b^2 m-c (b l+a m)\right ) x^5}{5 c^3}+\frac{(c l-b m) x^6}{6 c^2}+\frac{m x^7}{7 c}-\frac{\left (2 c^8 d-c^7 (b e+2 a f)+c^6 \left (b^2 f+3 a b g+2 a^2 h\right )-c^5 \left (b^3 g+4 a b^2 h+5 a^2 b j+2 a^3 k\right )+b^8 m-b^6 c (b l+8 a m)+b^4 c^2 \left (b^2 k+7 a b l+20 a^2 m\right )-b^2 c^3 \left (b^3 j+6 a b^2 k+14 a^2 b l+16 a^3 m\right )+c^4 \left (b^4 h+5 a b^3 j+9 a^2 b^2 k+7 a^3 b l+2 a^4 m\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^8 \sqrt{b^2-4 a c}}+\frac{\left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^8}\\ \end{align*}

Mathematica [A]  time = 0.778456, size = 754, normalized size = 0.99 \[ \frac{\frac{420 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (c^6 \left (2 a^2 h+3 a b g+b^2 f\right )-c^5 \left (5 a^2 b j+2 a^3 k+4 a b^2 h+b^3 g\right )+c^4 \left (9 a^2 b^2 k+7 a^3 b l+2 a^4 m+5 a b^3 j+b^4 h\right )-b^2 c^3 \left (14 a^2 b l+16 a^3 m+6 a b^2 k+b^3 j\right )+b^4 c^2 \left (20 a^2 m+7 a b l+b^2 k\right )-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right )}{\sqrt{4 a c-b^2}}+210 \log (a+x (b+c x)) \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-c^4 \left (3 a^2 b k+a^3 l+3 a b^2 j+b^3 h\right )+b c^3 \left (6 a^2 b l+4 a^3 m+4 a b^2 k+b^3 j\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )+420 c x \left (c^4 \left (a^2 k+2 a b j+b^2 h\right )-c^3 \left (3 a^2 b l+a^3 m+3 a b^2 k+b^3 j\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right )+210 c^2 x^2 \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )+140 c^3 x^3 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )+105 c^4 x^4 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )+84 c^5 x^5 \left (-c (a m+b l)+b^2 m+c^2 k\right )+70 c^6 x^6 (c l-b m)+60 c^7 m x^7}{420 c^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x + c*x^2),x]

[Out]

(420*c*(c^6*f - c^5*(b*g + a*h) + c^4*(b^2*h + 2*a*b*j + a^2*k) + b^6*m - b^4*c*(b*l + 5*a*m) + b^2*c^2*(b^2*k
+ 4*a*b*l + 6*a^2*m) - c^3*(b^3*j + 3*a*b^2*k + 3*a^2*b*l + a^3*m))*x + 210*c^2*(c^5*g - c^4*(b*h + a*j) + c^
3*(b^2*j + 2*a*b*k + a^2*l) - b^5*m + b^3*c*(b*l + 4*a*m) - b*c^2*(b^2*k + 3*a*b*l + 3*a^2*m))*x^2 + 140*c^3*(
c^4*h - c^3*(b*j + a*k) + b^4*m - b^2*c*(b*l + 3*a*m) + c^2*(b^2*k + 2*a*b*l + a^2*m))*x^3 + 105*c^4*(c^3*j -
c^2*(b*k + a*l) - b^3*m + b*c*(b*l + 2*a*m))*x^4 + 84*c^5*(c^2*k + b^2*m - c*(b*l + a*m))*x^5 + 70*c^6*(c*l -
b*m)*x^6 + 60*c^7*m*x^7 + (420*(2*c^8*d - c^7*(b*e + 2*a*f) + c^6*(b^2*f + 3*a*b*g + 2*a^2*h) - c^5*(b^3*g + 4
*a*b^2*h + 5*a^2*b*j + 2*a^3*k) + b^8*m - b^6*c*(b*l + 8*a*m) + b^4*c^2*(b^2*k + 7*a*b*l + 20*a^2*m) - b^2*c^3
*(b^3*j + 6*a*b^2*k + 14*a^2*b*l + 16*a^3*m) + c^4*(b^4*h + 5*a*b^3*j + 9*a^2*b^2*k + 7*a^3*b*l + 2*a^4*m))*Ar
cTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 210*(c^7*e - c^6*(b*f + a*g) + c^5*(b^2*g + 2*a*b*h
+ a^2*j) - c^4*(b^3*h + 3*a*b^2*j + 3*a^2*b*k + a^3*l) - b^7*m + b^5*c*(b*l + 6*a*m) - b^3*c^2*(b^2*k + 5*a*b
*l + 10*a^2*m) + b*c^3*(b^3*j + 4*a*b^2*k + 6*a^2*b*l + 4*a^3*m))*Log[a + x*(b + c*x)])/(420*c^8)

________________________________________________________________________________________

Maple [B]  time = 0.192, size = 1960, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a),x)

[Out]

-1/2/c^2*x^2*a*j+1/2/c^3*x^2*a^2*l+1/2/c^5*ln(c*x^2+b*x+a)*b^4*j-1/2/c^4*ln(c*x^2+b*x+a)*b^3*h-1/2/c^8*ln(c*x^
2+b*x+a)*b^7*m+1/2/c^7*ln(c*x^2+b*x+a)*b^6*l-1/2/c^4*ln(c*x^2+b*x+a)*a^3*l+1/2/c^3*ln(c*x^2+b*x+a)*b^2*g-1/2/c
^2*ln(c*x^2+b*x+a)*b*f+1/2/c^3*ln(c*x^2+b*x+a)*a^2*j-1/2/c^2*ln(c*x^2+b*x+a)*a*g-1/2/c^6*x^2*b^5*m+1/c^5*b^4*k
*x-1/3/c^2*x^3*a*k-1/5/c^2*x^5*b*l+1/3/c^3*x^3*a^2*m-1/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*b^3*g+1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*f-2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c
*x+b)/(4*a*c-b^2)^(1/2))*a^3*k-1/c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*j+1/c^4/(4*a*c-
b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*h-1/c^4*a^3*m*x+1/c^3*a^2*k*x-1/c^7/(4*a*c-b^2)^(1/2)*arcta
n((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^7*l-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e+1/c^6/(4*a*
c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6*k+2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*a^2*h-3/2/c^4*ln(c*x^2+b*x+a)*a*b^2*j-5/2/c^6*ln(c*x^2+b*x+a)*a*b^4*l+2/c^5*ln(c*x^2+b*x+a)*a*b^3*k-2/c/(
4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*f-1/c^2*b*g*x-1/c^2*a*h*x-1/c^6*b^5*l*x+3/c^5*ln(c*x^2+
b*x+a)*a^2*b^2*l+2/c^5*ln(c*x^2+b*x+a)*a^3*b*m-1/4/c^2*x^4*b*k-1/c^4*b^3*j*x+1/c^3*b^2*h*x-1/4/c^2*x^4*a*l-1/4
/c^4*x^4*b^3*m+1/4/c^3*x^4*b^2*l-1/2/c^6*ln(c*x^2+b*x+a)*b^5*k+2/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2))*a^4*m+1/c^8/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^8*m-3/2/c^4*ln(c*x^2+b*x+a)*a
^2*b*k-5/c^6*ln(c*x^2+b*x+a)*a^2*b^3*m+1/c^3*x^2*a*b*k+6/c^5*a^2*b^2*m*x+1/c^3*ln(c*x^2+b*x+a)*a*b*h-3/2/c^4*x
^2*a*b^2*l-3/2/c^4*x^2*a^2*b*m+1/2/c*x^2*g+1/3/c*x^3*h+1/5/c*x^5*k+1/4/c*x^4*j+1/6/c*x^6*l+2/(4*a*c-b^2)^(1/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d+1/2/c*ln(c*x^2+b*x+a)*e+1/c*f*x-1/3/c^2*x^3*b*j-1/6/c^2*x^6*b*m-1/5/c^2
*x^5*a*m-1/3/c^4*x^3*b^3*l+1/3/c^3*x^3*b^2*k+1/5/c^3*x^5*b^2*m+1/3/c^5*x^3*b^4*m+1/c^7*b^6*m*x+1/2/c^5*x^2*b^4
*l-1/2/c^4*x^2*b^3*k+1/2/c^3*x^2*b^2*j-1/2/c^2*x^2*b*h-3/c^4*a^2*b*l*x-5/c^6*a*b^4*m*x+2/c^5*x^2*a*b^3*m-1/c^4
*x^3*a*b^2*m+2/3/c^3*x^3*a*b*l+1/2/c^3*x^4*a*b*m+4/c^5*a*b^3*l*x-3/c^4*a*b^2*k*x+2/c^3*a*b*j*x+3/c^7*ln(c*x^2+
b*x+a)*a*b^5*m+1/7*m*x^7/c+7/c^6/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^5*l-6/c^5/(4*a*c-b^
2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^4*k-14/c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/
2))*a^2*b^3*l-8/c^7/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^6*m+5/c^4/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*j-4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*h+9/
c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^2*k-5/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(
4*a*c-b^2)^(1/2))*a^2*b*j+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*g-16/c^5/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3*b^2*m+7/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*a^3*b*l+20/c^6/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^4*m

________________________________________________________________________________________

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((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.61538, size = 5392, normalized size = 7.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

[1/420*(60*(b^2*c^7 - 4*a*c^8)*m*x^7 + 70*((b^2*c^7 - 4*a*c^8)*l - (b^3*c^6 - 4*a*b*c^7)*m)*x^6 + 84*((b^2*c^7
- 4*a*c^8)*k - (b^3*c^6 - 4*a*b*c^7)*l + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7)*m)*x^5 + 105*((b^2*c^7 - 4*a*c^8
)*j - (b^3*c^6 - 4*a*b*c^7)*k + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7)*l - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*
m)*x^4 + 140*((b^2*c^7 - 4*a*c^8)*h - (b^3*c^6 - 4*a*b*c^7)*j + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7)*k - (b^5*c
^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*l + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*c^6)*m)*x^3 + 210*((b^2*c^
7 - 4*a*c^8)*g - (b^3*c^6 - 4*a*b*c^7)*h + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7)*j - (b^5*c^4 - 6*a*b^3*c^5 + 8*
a^2*b*c^6)*k + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*c^6)*l - (b^7*c^2 - 8*a*b^5*c^3 + 19*a^2*b^3*c^
4 - 12*a^3*b*c^5)*m)*x^2 + 210*(2*c^8*d - b*c^7*e + (b^2*c^6 - 2*a*c^7)*f - (b^3*c^5 - 3*a*b*c^6)*g + (b^4*c^4
- 4*a*b^2*c^5 + 2*a^2*c^6)*h - (b^5*c^3 - 5*a*b^3*c^4 + 5*a^2*b*c^5)*j + (b^6*c^2 - 6*a*b^4*c^3 + 9*a^2*b^2*c
^4 - 2*a^3*c^5)*k - (b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*l + (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2
- 16*a^3*b^2*c^3 + 2*a^4*c^4)*m)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)
*(2*c*x + b))/(c*x^2 + b*x + a)) + 420*((b^2*c^7 - 4*a*c^8)*f - (b^3*c^6 - 4*a*b*c^7)*g + (b^4*c^5 - 5*a*b^2*c
^6 + 4*a^2*c^7)*h - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*j + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*
c^6)*k - (b^7*c^2 - 8*a*b^5*c^3 + 19*a^2*b^3*c^4 - 12*a^3*b*c^5)*l + (b^8*c - 9*a*b^6*c^2 + 26*a^2*b^4*c^3 - 2
5*a^3*b^2*c^4 + 4*a^4*c^5)*m)*x + 210*((b^2*c^7 - 4*a*c^8)*e - (b^3*c^6 - 4*a*b*c^7)*f + (b^4*c^5 - 5*a*b^2*c^
6 + 4*a^2*c^7)*g - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*h + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*c
^6)*j - (b^7*c^2 - 8*a*b^5*c^3 + 19*a^2*b^3*c^4 - 12*a^3*b*c^5)*k + (b^8*c - 9*a*b^6*c^2 + 26*a^2*b^4*c^3 - 25
*a^3*b^2*c^4 + 4*a^4*c^5)*l - (b^9 - 10*a*b^7*c + 34*a^2*b^5*c^2 - 44*a^3*b^3*c^3 + 16*a^4*b*c^4)*m)*log(c*x^2
+ b*x + a))/(b^2*c^8 - 4*a*c^9), 1/420*(60*(b^2*c^7 - 4*a*c^8)*m*x^7 + 70*((b^2*c^7 - 4*a*c^8)*l - (b^3*c^6 -
4*a*b*c^7)*m)*x^6 + 84*((b^2*c^7 - 4*a*c^8)*k - (b^3*c^6 - 4*a*b*c^7)*l + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7)
*m)*x^5 + 105*((b^2*c^7 - 4*a*c^8)*j - (b^3*c^6 - 4*a*b*c^7)*k + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7)*l - (b^5*
c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*m)*x^4 + 140*((b^2*c^7 - 4*a*c^8)*h - (b^3*c^6 - 4*a*b*c^7)*j + (b^4*c^5 - 5*
a*b^2*c^6 + 4*a^2*c^7)*k - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*l + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 -
4*a^3*c^6)*m)*x^3 + 210*((b^2*c^7 - 4*a*c^8)*g - (b^3*c^6 - 4*a*b*c^7)*h + (b^4*c^5 - 5*a*b^2*c^6 + 4*a^2*c^7
)*j - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*k + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*c^6)*l - (b^7*
c^2 - 8*a*b^5*c^3 + 19*a^2*b^3*c^4 - 12*a^3*b*c^5)*m)*x^2 - 420*(2*c^8*d - b*c^7*e + (b^2*c^6 - 2*a*c^7)*f - (
b^3*c^5 - 3*a*b*c^6)*g + (b^4*c^4 - 4*a*b^2*c^5 + 2*a^2*c^6)*h - (b^5*c^3 - 5*a*b^3*c^4 + 5*a^2*b*c^5)*j + (b^
6*c^2 - 6*a*b^4*c^3 + 9*a^2*b^2*c^4 - 2*a^3*c^5)*k - (b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*l +
(b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*m)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*
c)*(2*c*x + b)/(b^2 - 4*a*c)) + 420*((b^2*c^7 - 4*a*c^8)*f - (b^3*c^6 - 4*a*b*c^7)*g + (b^4*c^5 - 5*a*b^2*c^6
+ 4*a^2*c^7)*h - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*j + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*c^6
)*k - (b^7*c^2 - 8*a*b^5*c^3 + 19*a^2*b^3*c^4 - 12*a^3*b*c^5)*l + (b^8*c - 9*a*b^6*c^2 + 26*a^2*b^4*c^3 - 25*a
^3*b^2*c^4 + 4*a^4*c^5)*m)*x + 210*((b^2*c^7 - 4*a*c^8)*e - (b^3*c^6 - 4*a*b*c^7)*f + (b^4*c^5 - 5*a*b^2*c^6 +
4*a^2*c^7)*g - (b^5*c^4 - 6*a*b^3*c^5 + 8*a^2*b*c^6)*h + (b^6*c^3 - 7*a*b^4*c^4 + 13*a^2*b^2*c^5 - 4*a^3*c^6)
*j - (b^7*c^2 - 8*a*b^5*c^3 + 19*a^2*b^3*c^4 - 12*a^3*b*c^5)*k + (b^8*c - 9*a*b^6*c^2 + 26*a^2*b^4*c^3 - 25*a^
3*b^2*c^4 + 4*a^4*c^5)*l - (b^9 - 10*a*b^7*c + 34*a^2*b^5*c^2 - 44*a^3*b^3*c^3 + 16*a^4*b*c^4)*m)*log(c*x^2 +
b*x + a))/(b^2*c^8 - 4*a*c^9)]

________________________________________________________________________________________

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((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**2+b*x+a),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.29237, size = 1326, normalized size = 1.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/420*(60*c^6*m*x^7 + 70*c^6*l*x^6 - 70*b*c^5*m*x^6 + 84*c^6*k*x^5 - 84*b*c^5*l*x^5 + 84*b^2*c^4*m*x^5 - 84*a*
c^5*m*x^5 + 105*c^6*j*x^4 - 105*b*c^5*k*x^4 + 105*b^2*c^4*l*x^4 - 105*a*c^5*l*x^4 - 105*b^3*c^3*m*x^4 + 210*a*
b*c^4*m*x^4 + 140*c^6*h*x^3 - 140*b*c^5*j*x^3 + 140*b^2*c^4*k*x^3 - 140*a*c^5*k*x^3 - 140*b^3*c^3*l*x^3 + 280*
a*b*c^4*l*x^3 + 140*b^4*c^2*m*x^3 - 420*a*b^2*c^3*m*x^3 + 140*a^2*c^4*m*x^3 + 210*c^6*g*x^2 - 210*b*c^5*h*x^2
+ 210*b^2*c^4*j*x^2 - 210*a*c^5*j*x^2 - 210*b^3*c^3*k*x^2 + 420*a*b*c^4*k*x^2 + 210*b^4*c^2*l*x^2 - 630*a*b^2*
c^3*l*x^2 + 210*a^2*c^4*l*x^2 - 210*b^5*c*m*x^2 + 840*a*b^3*c^2*m*x^2 - 630*a^2*b*c^3*m*x^2 + 420*c^6*f*x - 42
0*b*c^5*g*x + 420*b^2*c^4*h*x - 420*a*c^5*h*x - 420*b^3*c^3*j*x + 840*a*b*c^4*j*x + 420*b^4*c^2*k*x - 1260*a*b
^2*c^3*k*x + 420*a^2*c^4*k*x - 420*b^5*c*l*x + 1680*a*b^3*c^2*l*x - 1260*a^2*b*c^3*l*x + 420*b^6*m*x - 2100*a*
b^4*c*m*x + 2520*a^2*b^2*c^2*m*x - 420*a^3*c^3*m*x)/c^7 - 1/2*(b*c^6*f - b^2*c^5*g + a*c^6*g + b^3*c^4*h - 2*a
*b*c^5*h - b^4*c^3*j + 3*a*b^2*c^4*j - a^2*c^5*j + b^5*c^2*k - 4*a*b^3*c^3*k + 3*a^2*b*c^4*k - b^6*c*l + 5*a*b
^4*c^2*l - 6*a^2*b^2*c^3*l + a^3*c^4*l + b^7*m - 6*a*b^5*c*m + 10*a^2*b^3*c^2*m - 4*a^3*b*c^3*m - c^7*e)*log(c
*x^2 + b*x + a)/c^8 + (2*c^8*d + b^2*c^6*f - 2*a*c^7*f - b^3*c^5*g + 3*a*b*c^6*g + b^4*c^4*h - 4*a*b^2*c^5*h +
2*a^2*c^6*h - b^5*c^3*j + 5*a*b^3*c^4*j - 5*a^2*b*c^5*j + b^6*c^2*k - 6*a*b^4*c^3*k + 9*a^2*b^2*c^4*k - 2*a^3
*c^5*k - b^7*c*l + 7*a*b^5*c^2*l - 14*a^2*b^3*c^3*l + 7*a^3*b*c^4*l + b^8*m - 8*a*b^6*c*m + 20*a^2*b^4*c^2*m -
16*a^3*b^2*c^3*m + 2*a^4*c^4*m - b*c^7*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^8)