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 \[ \frac {x^3 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )}{3 c^5}+\frac {x^2 \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )}{2 c^6}+\frac {\log \left (a+b x+c x^2\right ) \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )-c^4 \left (a^3 l+3 a^2 b k+3 a b^2 j+b^3 h\right )+b c^3 \left (4 a^3 m+6 a^2 b l+4 a b^2 k+b^3 j\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )}{2 c^8}+\frac {x \left (c^4 \left (a^2 k+2 a b j+b^2 h\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-c^3 \left (a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right )}{c^7}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (c^6 \left (2 a^2 h+3 a b g+b^2 f\right )+b^4 c^2 \left (20 a^2 m+7 a b l+b^2 k\right )-c^5 \left (2 a^3 k+5 a^2 b j+4 a b^2 h+b^3 g\right )-b^2 c^3 \left (16 a^3 m+14 a^2 b l+6 a b^2 k+b^3 j\right )+c^4 \left (2 a^4 m+7 a^3 b l+9 a^2 b^2 k+5 a b^3 j+b^4 h\right )-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right )}{c^8 \sqrt {b^2-4 a c}}+\frac {x^4 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )}{4 c^4}+\frac {x^5 \left (-c (a m+b l)+b^2 m+c^2 k\right )}{5 c^3}+\frac {x^6 (c l-b m)}{6 c^2}+\frac {m x^7}{7 c} \]
[Out]
(c^6*f-c^5*(a*h+b*g)+c^4*(a^2*k+2*a*b*j+b^2*h)+b^6*m-b^4*c*(5*a*m+b*l)+b^2*c^2*(6*a^2*m+4*a*b*l+b^2*k)-c^3*(a^
3*m+3*a^2*b*l+3*a*b^2*k+b^3*j))*x/c^7+1/2*(c^5*g-c^4*(a*j+b*h)+c^3*(a^2*l+2*a*b*k+b^2*j)-b^5*m+b^3*c*(4*a*m+b*
l)-b*c^2*(3*a^2*m+3*a*b*l+b^2*k))*x^2/c^6+1/3*(c^4*h-c^3*(a*k+b*j)+b^4*m-b^2*c*(3*a*m+b*l)+c^2*(a^2*m+2*a*b*l+
b^2*k))*x^3/c^5+1/4*(c^3*j-c^2*(a*l+b*k)-b^3*m+b*c*(2*a*m+b*l))*x^4/c^4+1/5*(c^2*k+b^2*m-c*(a*m+b*l))*x^5/c^3+
1/6*(-b*m+c*l)*x^6/c^2+1/7*m*x^7/c+1/2*(c^7*e-c^6*(a*g+b*f)+c^5*(a^2*j+2*a*b*h+b^2*g)-c^4*(a^3*l+3*a^2*b*k+3*a
*b^2*j+b^3*h)-b^7*m+b^5*c*(6*a*m+b*l)-b^3*c^2*(10*a^2*m+5*a*b*l+b^2*k)+b*c^3*(4*a^3*m+6*a^2*b*l+4*a*b^2*k+b^3*
j))*ln(c*x^2+b*x+a)/c^8-(2*c^8*d-c^7*(2*a*f+b*e)+c^6*(2*a^2*h+3*a*b*g+b^2*f)-c^5*(2*a^3*k+5*a^2*b*j+4*a*b^2*h+
b^3*g)+b^8*m-b^6*c*(8*a*m+b*l)+b^4*c^2*(20*a^2*m+7*a*b*l+b^2*k)-b^2*c^3*(16*a^3*m+14*a^2*b*l+6*a*b^2*k+b^3*j)+
c^4*(2*a^4*m+7*a^3*b*l+9*a^2*b^2*k+5*a*b^3*j+b^4*h))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^8/(-4*a*c+b^2)^(1
/2)
________________________________________________________________________________________
Rubi [A] time = 5.83, antiderivative size = 765, normalized size of antiderivative = 1.00,
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 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 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 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 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 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]
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*}
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Mathematica [A] time = 0.73, size = 754, normalized size = 0.99 \[ \frac {140 c^3 x^3 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )+210 c^2 x^2 \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )+210 \log (a+x (b+c x)) \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )-c^4 \left (a^3 l+3 a^2 b k+3 a b^2 j+b^3 h\right )+b c^3 \left (4 a^3 m+6 a^2 b l+4 a b^2 k+b^3 j\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )+420 c x \left (c^4 \left (a^2 k+2 a b j+b^2 h\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-c^3 \left (a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right )+\frac {420 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (c^6 \left (2 a^2 h+3 a b g+b^2 f\right )+b^4 c^2 \left (20 a^2 m+7 a b l+b^2 k\right )-c^5 \left (2 a^3 k+5 a^2 b j+4 a b^2 h+b^3 g\right )-b^2 c^3 \left (16 a^3 m+14 a^2 b l+6 a b^2 k+b^3 j\right )+c^4 \left (2 a^4 m+7 a^3 b l+9 a^2 b^2 k+5 a b^3 j+b^4 h\right )-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right )}{\sqrt {4 a c-b^2}}+105 c^4 x^4 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )+84 c^5 x^5 \left (-c (a m+b l)+b^2 m+c^2 k\right )+70 c^6 x^6 (c l-b m)+60 c^7 m x^7}{420 c^8} \]
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)
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fricas [A] time = 1.84, size = 2643, normalized size = 3.45 \[ \text {result too large to display} \]
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)]
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giac [A] time = 0.18, size = 982, normalized size = 1.28 \[ \frac {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 - 420 \, 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}{420 \, c^{7}} - \frac {{\left (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\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{8}} + \frac {{\left (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\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{8}} \]
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)
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maple [B] time = 0.01, size = 1960, normalized size = 2.56 \[ \text {result too large to display} \]
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]
1/7*m*x^7/c-1/2/c^2*ln(c*x^2+b*x+a)*b*f-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)*a^2*j-1/2/c^2*ln
(c*x^2+b*x+a)*a*g-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-1/6/c^2*x^6*b*m+1/c^7*b^6*m*x-1/
4/c^2*x^4*a*l-1/4/c^4*x^4*b^3*m+1/5/c^3*x^5*b^2*m-1/5/c^2*x^5*b*l-1/5/c^2*x^5*a*m+1/c^5*b^4*k*x-1/c^4*b^3*j*x+
1/c^3*b^2*h*x-1/c^2*b*g*x+1/2/c^5*x^2*b^4*l-1/2/c^4*x^2*b^3*k-1/2/c^2*x^2*a*j-1/2/c^6*x^2*b^5*m-1/3/c^2*x^3*a*
k+1/3/c^5*x^3*b^4*m-1/3/c^4*x^3*b^3*l+1/3/c^3*x^3*b^2*k-1/3/c^2*x^3*b*j+1/2/c^3*x^2*a^2*l-1/4/c^2*x^4*b*k+1/3/
c^3*x^3*a^2*m+1/4/c^3*x^4*b^2*l+1/2/c^3*x^2*b^2*j-1/2/c^2*x^2*b*h-1/c^4*a^3*m*x+1/c^3*a^2*k*x-1/c^2*a*h*x-1/c^
6*b^5*l*x-1/2/c^6*ln(c*x^2+b*x+a)*b^5*k+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^3*ln
(c*x^2+b*x+a)*b^2*g+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-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+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-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)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3*b*l-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+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+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+1/2/c*x^2*g+1/3/c*x^3*h+1/6/c*x^6*l+1/5/c*x^5*k+1/4/c*x^4*j+3/c^5*ln(c*x^2+b*x+a)*a^2*b^2*l-3/2/c^4*ln(c*
x^2+b*x+a)*a^2*b*k+3/c^7*ln(c*x^2+b*x+a)*a*b^5*m-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
-1/c^7/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^7*l+2/c^5*ln(c*x^2+b*x+a)*a^3*b*m-5/c^6*ln(c*x^
2+b*x+a)*a^2*b^3*m-3/2/c^4*ln(c*x^2+b*x+a)*a*b^2*j+1/c^3*ln(c*x^2+b*x+a)*a*b*h+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-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^8/(4*a*c-
b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^8*m-3/c^4*a*b^2*k*x+2/c^3*a*b*j*x+6/c^5*a^2*b^2*m*x-3/c^4*a^2
*b*l*x+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-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^6/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6*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^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^3*x^2*a*b*k-5/c^6*a*b^4*m*x+4/c^5*a*b
^3*l*x-3/2/c^4*x^2*a^2*b*m+2/c^5*x^2*a*b^3*m-3/2/c^4*x^2*a*b^2*l-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+5/c^4/(4*a*c-b^2)^(1/2)*arctan((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+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-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-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+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
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 >> 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 details)Is 4*a*c-b^2 positive or negative?
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mupad [B] time = 7.26, size = 2779, normalized size = 3.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[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]
x^6*(l/(6*c) - (b*m)/(6*c^2)) + x*(f/c + (b*((a*(j/c - (a*(l/c - (b*m)/c^2))/c + (b*((b*(l/c - (b*m)/c^2))/c -
k/c + (a*m)/c^2))/c))/c - g/c + (b*(h/c - (b*(j/c - (a*(l/c - (b*m)/c^2))/c + (b*((b*(l/c - (b*m)/c^2))/c - k
/c + (a*m)/c^2))/c))/c + (a*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)/c^2))/c))/c))/c - (a*(h/c - (b*(j/c - (a*(l
/c - (b*m)/c^2))/c + (b*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)/c^2))/c))/c + (a*((b*(l/c - (b*m)/c^2))/c - k/c
+ (a*m)/c^2))/c))/c) + x^4*(j/(4*c) - (a*(l/c - (b*m)/c^2))/(4*c) + (b*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)
/c^2))/(4*c)) - x^2*((a*(j/c - (a*(l/c - (b*m)/c^2))/c + (b*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)/c^2))/c))/(
2*c) - g/(2*c) + (b*(h/c - (b*(j/c - (a*(l/c - (b*m)/c^2))/c + (b*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)/c^2))
/c))/c + (a*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)/c^2))/c))/(2*c)) + x^3*(h/(3*c) - (b*(j/c - (a*(l/c - (b*m)
/c^2))/c + (b*((b*(l/c - (b*m)/c^2))/c - k/c + (a*m)/c^2))/c))/(3*c) + (a*((b*(l/c - (b*m)/c^2))/c - k/c + (a*
m)/c^2))/(3*c)) - x^5*((b*(l/c - (b*m)/c^2))/(5*c) - k/(5*c) + (a*m)/(5*c^2)) + (log((2*c^9*x*(-(2*c^8*d + b^8
*m + b^2*c^6*f + 2*a^2*c^6*h - b^3*c^5*g + b^4*c^4*h - 2*a^3*c^5*k - b^5*c^3*j + b^6*c^2*k + 2*a^4*c^4*m - 2*a
*c^7*f - b*c^7*e - b^7*c*l + 9*a^2*b^2*c^4*k - 14*a^2*b^3*c^3*l + 20*a^2*b^4*c^2*m - 16*a^3*b^2*c^3*m + 3*a*b*
c^6*g - 8*a*b^6*c*m - 4*a*b^2*c^5*h + 5*a*b^3*c^4*j - 5*a^2*b*c^5*j - 6*a*b^4*c^3*k + 7*a*b^5*c^2*l + 7*a^3*b*
c^4*l)^2/(c^16*(4*a*c - b^2)))^(1/2) - b^8*m - 2*c^8*d - b^2*c^6*f - 2*a^2*c^6*h + b^3*c^5*g - b^4*c^4*h + 2*a
^3*c^5*k + b^5*c^3*j - b^6*c^2*k - 2*a^4*c^4*m + 2*a*c^7*f + b*c^7*e + b^7*c*l + b*c^8*(-(2*c^8*d + b^8*m + b^
2*c^6*f + 2*a^2*c^6*h - b^3*c^5*g + b^4*c^4*h - 2*a^3*c^5*k - b^5*c^3*j + b^6*c^2*k + 2*a^4*c^4*m - 2*a*c^7*f
- b*c^7*e - b^7*c*l + 9*a^2*b^2*c^4*k - 14*a^2*b^3*c^3*l + 20*a^2*b^4*c^2*m - 16*a^3*b^2*c^3*m + 3*a*b*c^6*g -
8*a*b^6*c*m - 4*a*b^2*c^5*h + 5*a*b^3*c^4*j - 5*a^2*b*c^5*j - 6*a*b^4*c^3*k + 7*a*b^5*c^2*l + 7*a^3*b*c^4*l)^
2/(c^16*(4*a*c - b^2)))^(1/2) - 9*a^2*b^2*c^4*k + 14*a^2*b^3*c^3*l - 20*a^2*b^4*c^2*m + 16*a^3*b^2*c^3*m - 3*a
*b*c^6*g + 8*a*b^6*c*m + 4*a*b^2*c^5*h - 5*a*b^3*c^4*j + 5*a^2*b*c^5*j + 6*a*b^4*c^3*k - 7*a*b^5*c^2*l - 7*a^3
*b*c^4*l)*(2*c^8*d + b^8*m + 2*c^9*x*(-(2*c^8*d + b^8*m + b^2*c^6*f + 2*a^2*c^6*h - b^3*c^5*g + b^4*c^4*h - 2*
a^3*c^5*k - b^5*c^3*j + b^6*c^2*k + 2*a^4*c^4*m - 2*a*c^7*f - b*c^7*e - b^7*c*l + 9*a^2*b^2*c^4*k - 14*a^2*b^3
*c^3*l + 20*a^2*b^4*c^2*m - 16*a^3*b^2*c^3*m + 3*a*b*c^6*g - 8*a*b^6*c*m - 4*a*b^2*c^5*h + 5*a*b^3*c^4*j - 5*a
^2*b*c^5*j - 6*a*b^4*c^3*k + 7*a*b^5*c^2*l + 7*a^3*b*c^4*l)^2/(c^16*(4*a*c - b^2)))^(1/2) + b^2*c^6*f + 2*a^2*
c^6*h - b^3*c^5*g + b^4*c^4*h - 2*a^3*c^5*k - b^5*c^3*j + b^6*c^2*k + 2*a^4*c^4*m - 2*a*c^7*f - b*c^7*e - b^7*
c*l + b*c^8*(-(2*c^8*d + b^8*m + b^2*c^6*f + 2*a^2*c^6*h - b^3*c^5*g + b^4*c^4*h - 2*a^3*c^5*k - b^5*c^3*j + b
^6*c^2*k + 2*a^4*c^4*m - 2*a*c^7*f - b*c^7*e - b^7*c*l + 9*a^2*b^2*c^4*k - 14*a^2*b^3*c^3*l + 20*a^2*b^4*c^2*m
- 16*a^3*b^2*c^3*m + 3*a*b*c^6*g - 8*a*b^6*c*m - 4*a*b^2*c^5*h + 5*a*b^3*c^4*j - 5*a^2*b*c^5*j - 6*a*b^4*c^3*
k + 7*a*b^5*c^2*l + 7*a^3*b*c^4*l)^2/(c^16*(4*a*c - b^2)))^(1/2) + 9*a^2*b^2*c^4*k - 14*a^2*b^3*c^3*l + 20*a^2
*b^4*c^2*m - 16*a^3*b^2*c^3*m + 3*a*b*c^6*g - 8*a*b^6*c*m - 4*a*b^2*c^5*h + 5*a*b^3*c^4*j - 5*a^2*b*c^5*j - 6*
a*b^4*c^3*k + 7*a*b^5*c^2*l + 7*a^3*b*c^4*l))*(b^9*m - b^2*c^7*e - 4*a^2*c^7*g + b^3*c^6*f - b^4*c^5*g + b^5*c
^4*h + 4*a^3*c^6*j - b^6*c^3*j - 4*a^4*c^5*l + b^7*c^2*k + 4*a*c^8*e - b^8*c*l - 13*a^2*b^2*c^5*j + 19*a^2*b^3
*c^4*k - 26*a^2*b^4*c^3*l + 25*a^3*b^2*c^4*l + 34*a^2*b^5*c^2*m - 44*a^3*b^3*c^3*m - 4*a*b*c^7*f - 10*a*b^7*c*
m + 5*a*b^2*c^6*g - 6*a*b^3*c^5*h + 8*a^2*b*c^6*h + 7*a*b^4*c^4*j - 8*a*b^5*c^3*k - 12*a^3*b*c^5*k + 9*a*b^6*c
^2*l + 16*a^4*b*c^4*m))/(2*(4*a*c^9 - b^2*c^8)) + (m*x^7)/(7*c) + (atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c
- b^2)^(1/2))*(2*c^8*d + b^8*m + b^2*c^6*f + 2*a^2*c^6*h - b^3*c^5*g + b^4*c^4*h - 2*a^3*c^5*k - b^5*c^3*j +
b^6*c^2*k + 2*a^4*c^4*m - 2*a*c^7*f - b*c^7*e - b^7*c*l + 9*a^2*b^2*c^4*k - 14*a^2*b^3*c^3*l + 20*a^2*b^4*c^2*
m - 16*a^3*b^2*c^3*m + 3*a*b*c^6*g - 8*a*b^6*c*m - 4*a*b^2*c^5*h + 5*a*b^3*c^4*j - 5*a^2*b*c^5*j - 6*a*b^4*c^3
*k + 7*a*b^5*c^2*l + 7*a^3*b*c^4*l))/(c^8*(4*a*c - b^2)^(1/2))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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
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