Integrand size = 53, antiderivative size = 765 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx=\frac {\left (c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )\right ) x}{c^7}+\frac {\left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x^2}{2 c^6}+\frac {\left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^3}{3 c^5}+\frac {\left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^4}{4 c^4}+\frac {\left (c^2 k+b^2 m-c (b l+a m)\right ) x^5}{5 c^3}+\frac {(c l-b m) x^6}{6 c^2}+\frac {m x^7}{7 c}-\frac {\left (2 c^8 d-c^7 (b e+2 a f)+c^6 \left (b^2 f+3 a b g+2 a^2 h\right )-c^5 \left (b^3 g+4 a b^2 h+5 a^2 b j+2 a^3 k\right )+b^8 m-b^6 c (b l+8 a m)+b^4 c^2 \left (b^2 k+7 a b l+20 a^2 m\right )-b^2 c^3 \left (b^3 j+6 a b^2 k+14 a^2 b l+16 a^3 m\right )+c^4 \left (b^4 h+5 a b^3 j+9 a^2 b^2 k+7 a^3 b l+2 a^4 m\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^8 \sqrt {b^2-4 a c}}+\frac {\left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^8} \] Output:
(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-(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)+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
Time = 0.72 (sec) , antiderivative size = 754, normalized size of antiderivative = 0.99 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx=\frac {420 c \left (c^6 f-c^5 (b g+a h)+c^4 \left (b^2 h+2 a b j+a^2 k\right )+b^6 m-b^4 c (b l+5 a m)+b^2 c^2 \left (b^2 k+4 a b l+6 a^2 m\right )-c^3 \left (b^3 j+3 a b^2 k+3 a^2 b l+a^3 m\right )\right ) x+210 c^2 \left (c^5 g-c^4 (b h+a j)+c^3 \left (b^2 j+2 a b k+a^2 l\right )-b^5 m+b^3 c (b l+4 a m)-b c^2 \left (b^2 k+3 a b l+3 a^2 m\right )\right ) x^2+140 c^3 \left (c^4 h-c^3 (b j+a k)+b^4 m-b^2 c (b l+3 a m)+c^2 \left (b^2 k+2 a b l+a^2 m\right )\right ) x^3+105 c^4 \left (c^3 j-c^2 (b k+a l)-b^3 m+b c (b l+2 a m)\right ) x^4+84 c^5 \left (c^2 k+b^2 m-c (b l+a m)\right ) x^5+70 c^6 (c l-b m) x^6+60 c^7 m x^7+\frac {420 \left (2 c^8 d-c^7 (b e+2 a f)+c^6 \left (b^2 f+3 a b g+2 a^2 h\right )-c^5 \left (b^3 g+4 a b^2 h+5 a^2 b j+2 a^3 k\right )+b^8 m-b^6 c (b l+8 a m)+b^4 c^2 \left (b^2 k+7 a b l+20 a^2 m\right )-b^2 c^3 \left (b^3 j+6 a b^2 k+14 a^2 b l+16 a^3 m\right )+c^4 \left (b^4 h+5 a b^3 j+9 a^2 b^2 k+7 a^3 b l+2 a^4 m\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+210 \left (c^7 e-c^6 (b f+a g)+c^5 \left (b^2 g+2 a b h+a^2 j\right )-c^4 \left (b^3 h+3 a b^2 j+3 a^2 b k+a^3 l\right )-b^7 m+b^5 c (b l+6 a m)-b^3 c^2 \left (b^2 k+5 a b l+10 a^2 m\right )+b c^3 \left (b^3 j+4 a b^2 k+6 a^2 b l+4 a^3 m\right )\right ) \log (a+x (b+c x))}{420 c^8} \] Input:
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]
Output:
(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))*ArcTan[(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)
Time = 4.20 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2188, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2188 |
| \(\displaystyle \int \left (\frac {x^2 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )}{c^5}+\frac {x \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )}{c^6}+\frac {c^4 \left (a^2 k+2 a b j+b^2 h\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-c^3 \left (a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f}{c^7}+\frac {-a c^4 \left (a^2 k+2 a b j+b^2 h\right )-a b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )+a c^3 \left (a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right )+x \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )-c^4 \left (a^3 l+3 a^2 b k+3 a b^2 j+b^3 h\right )+b c^3 \left (4 a^3 m+6 a^2 b l+4 a b^2 k+b^3 j\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )-a b^6 m+a b^4 c (5 a m+b l)+a c^5 (a h+b g)-a c^6 f+c^7 d}{c^7 \left (a+b x+c x^2\right )}+\frac {x^3 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )}{c^4}+\frac {x^4 \left (-c (a m+b l)+b^2 m+c^2 k\right )}{c^3}+\frac {x^5 (c l-b m)}{c^2}+\frac {m x^6}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^3 \left (c^2 \left (a^2 m+2 a b l+b^2 k\right )-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right )}{3 c^5}+\frac {x^2 \left (c^3 \left (a^2 l+2 a b k+b^2 j\right )-b c^2 \left (3 a^2 m+3 a b l+b^2 k\right )+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right )}{2 c^6}+\frac {\log \left (a+b x+c x^2\right ) \left (c^5 \left (a^2 j+2 a b h+b^2 g\right )-b^3 c^2 \left (10 a^2 m+5 a b l+b^2 k\right )-c^4 \left (a^3 l+3 a^2 b k+3 a b^2 j+b^3 h\right )+b c^3 \left (4 a^3 m+6 a^2 b l+4 a b^2 k+b^3 j\right )+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right )}{2 c^8}+\frac {x \left (c^4 \left (a^2 k+2 a b j+b^2 h\right )+b^2 c^2 \left (6 a^2 m+4 a b l+b^2 k\right )-c^3 \left (a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right )-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right )}{c^7}-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (c^6 \left (2 a^2 h+3 a b g+b^2 f\right )+b^4 c^2 \left (20 a^2 m+7 a b l+b^2 k\right )-c^5 \left (2 a^3 k+5 a^2 b j+4 a b^2 h+b^3 g\right )-b^2 c^3 \left (16 a^3 m+14 a^2 b l+6 a b^2 k+b^3 j\right )+c^4 \left (2 a^4 m+7 a^3 b l+9 a^2 b^2 k+5 a b^3 j+b^4 h\right )-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right )}{c^8 \sqrt {b^2-4 a c}}+\frac {x^4 \left (-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right )}{4 c^4}+\frac {x^5 \left (-c (a m+b l)+b^2 m+c^2 k\right )}{5 c^3}+\frac {x^6 (c l-b m)}{6 c^2}+\frac {m x^7}{7 c}\) |
Input:
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]
Output:
((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 + 1 6*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 cTanh[(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)
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 2.21 (sec) , antiderivative size = 1086, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1086\) |
| risch | \(\text {Expression too large to display}\) | \(49911\) |
Input:
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,meth od=_RETURNVERBOSE)
Output:
-1/c^7*(-1/3*b^4*c^2*m*x^3+1/3*b^3*c^3*l*x^3-1/3*b^2*c^4*k*x^3+a^3*c^3*m*x -a^2*c^4*k*x+3/2*a*b^2*c^3*l*x^2+1/3*a*c^5*k*x^3+3*a*b^2*c^3*k*x-2*a*b*c^4 *j*x-1/4*b^2*c^4*l*x^4+1/4*b*c^5*k*x^4-1/2*a*b*c^4*m*x^4+a*b^2*c^3*m*x^3+1 /6*b*c^5*m*x^6-a*b*c^4*k*x^2+5*a*b^4*c*m*x-2*a*b^3*c^2*m*x^2-2/3*a*b*c^4*l *x^3+3/2*a^2*b*c^3*m*x^2-4*a*b^3*c^2*l*x-6*a^2*b^2*c^2*m*x+3*a^2*b*c^3*l*x -1/2*a^2*c^4*l*x^2+1/2*a*c^5*j*x^2+1/2*b^5*c*m*x^2+1/2*b^3*c^3*k*x^2-1/2*b ^2*c^4*j*x^2+1/2*b*c^5*h*x^2-1/3*a^2*c^4*m*x^3-1/5*b^2*c^4*m*x^5+1/5*b*c^5 *l*x^5+1/4*a*c^5*l*x^4+1/4*b^3*c^3*m*x^4+b^3*c^3*j*x+1/3*b*c^5*j*x^3+1/5*a *c^5*m*x^5-b^2*c^4*h*x+b*c^5*g*x-1/2*b^4*c^2*l*x^2+a*c^5*h*x+b^5*c*l*x-b^4 *c^2*k*x-b^6*m*x-1/6*c^6*l*x^6-c^6*f*x-1/7*m*x^7*c^6-1/4*c^6*j*x^4-1/2*c^6 *g*x^2-1/3*c^6*h*x^3-1/5*c^6*k*x^5)+1/c^7*(1/2*(4*a^3*b*c^3*m-a^3*c^4*l-10 *a^2*b^3*c^2*m+6*a^2*b^2*c^3*l-3*a^2*b*c^4*k+a^2*c^5*j+6*a*b^5*c*m-5*a*b^4 *c^2*l+4*a*b^3*c^3*k-3*a*b^2*c^4*j+2*a*b*c^5*h-a*c^6*g-b^7*m+b^6*c*l-b^5*c ^2*k+b^4*c^3*j-b^3*c^4*h+b^2*c^5*g-b*c^6*f+c^7*e)/c*ln(c*x^2+b*x+a)+2*(a^4 *c^3*m-6*a^3*b^2*c^2*m+3*a^3*b*c^3*l-a^3*c^4*k+5*a^2*b^4*c*m-4*a^2*b^3*c^2 *l+3*a^2*b^2*c^3*k-2*a^2*b*c^4*j+a^2*c^5*h-a*b^6*m+a*b^5*c*l-a*b^4*c^2*k+a *b^3*c^3*j-a*b^2*c^4*h+a*b*c^5*g-a*c^6*f+c^7*d-1/2*(4*a^3*b*c^3*m-a^3*c^4* l-10*a^2*b^3*c^2*m+6*a^2*b^2*c^3*l-3*a^2*b*c^4*k+a^2*c^5*j+6*a*b^5*c*m-5*a *b^4*c^2*l+4*a*b^3*c^3*k-3*a*b^2*c^4*j+2*a*b*c^5*h-a*c^6*g-b^7*m+b^6*c*l-b ^5*c^2*k+b^4*c^3*j-b^3*c^4*h+b^2*c^5*g-b*c^6*f+c^7*e)*b/c)/(4*a*c-b^2)^...
Time = 0.40 (sec) , antiderivative size = 2643, normalized size of antiderivative = 3.45 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
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")
Output:
[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^...
Leaf count of result is larger than twice the leaf count of optimal. 4306 vs. \(2 (785) = 1570\).
Time = 135.84 (sec) , antiderivative size = 4306, normalized size of antiderivative = 5.63 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
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)
Output:
x**6*(-b*m/(6*c**2) + l/(6*c)) + x**5*(-a*m/(5*c**2) + b**2*m/(5*c**3) - b *l/(5*c**2) + k/(5*c)) + x**4*(a*b*m/(2*c**3) - a*l/(4*c**2) - b**3*m/(4*c **4) + b**2*l/(4*c**3) - b*k/(4*c**2) + j/(4*c)) + x**3*(a**2*m/(3*c**3) - a*b**2*m/c**4 + 2*a*b*l/(3*c**3) - a*k/(3*c**2) + b**4*m/(3*c**5) - b**3* l/(3*c**4) + b**2*k/(3*c**3) - b*j/(3*c**2) + h/(3*c)) + x**2*(-3*a**2*b*m /(2*c**4) + a**2*l/(2*c**3) + 2*a*b**3*m/c**5 - 3*a*b**2*l/(2*c**4) + a*b* k/c**3 - a*j/(2*c**2) - b**5*m/(2*c**6) + b**4*l/(2*c**5) - b**3*k/(2*c**4 ) + b**2*j/(2*c**3) - b*h/(2*c**2) + g/(2*c)) + x*(-a**3*m/c**4 + 6*a**2*b **2*m/c**5 - 3*a**2*b*l/c**4 + a**2*k/c**3 - 5*a*b**4*m/c**6 + 4*a*b**3*l/ c**5 - 3*a*b**2*k/c**4 + 2*a*b*j/c**3 - a*h/c**2 + b**6*m/c**7 - b**5*l/c* *6 + b**4*k/c**5 - b**3*j/c**4 + b**2*h/c**3 - b*g/c**2 + f/c) + (-sqrt(-4 *a*c + b**2)*(2*a**4*c**4*m - 16*a**3*b**2*c**3*m + 7*a**3*b*c**4*l - 2*a* *3*c**5*k + 20*a**2*b**4*c**2*m - 14*a**2*b**3*c**3*l + 9*a**2*b**2*c**4*k - 5*a**2*b*c**5*j + 2*a**2*c**6*h - 8*a*b**6*c*m + 7*a*b**5*c**2*l - 6*a* b**4*c**3*k + 5*a*b**3*c**4*j - 4*a*b**2*c**5*h + 3*a*b*c**6*g - 2*a*c**7* f + b**8*m - b**7*c*l + b**6*c**2*k - b**5*c**3*j + b**4*c**4*h - b**3*c** 5*g + b**2*c**6*f - b*c**7*e + 2*c**8*d)/(2*c**8*(4*a*c - b**2)) + (4*a**3 *b*c**3*m - a**3*c**4*l - 10*a**2*b**3*c**2*m + 6*a**2*b**2*c**3*l - 3*a** 2*b*c**4*k + a**2*c**5*j + 6*a*b**5*c*m - 5*a*b**4*c**2*l + 4*a*b**3*c**3* k - 3*a*b**2*c**4*j + 2*a*b*c**5*h - a*c**6*g - b**7*m + b**6*c*l - b**...
Exception generated. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
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")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.30 (sec) , antiderivative size = 981, normalized size of antiderivative = 1.28 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
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")
Output:
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 + 21 0*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)/c^7 + 1/2*(c^7*e - 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)*log(c*x^2 + b*x + a)/c^8 + (2*c^8*d - b*c^7*e + 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...
Time = 22.67 (sec) , antiderivative size = 2779, normalized size of antiderivative = 3.63 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
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)
Output:
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 - 1 4*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 - ...
Time = 0.19 (sec) , antiderivative size = 2534, normalized size of antiderivative = 3.31 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
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)
Output:
(840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*c**4*m - 6720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c* *3*m + 2940*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b *c**4*l - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3 *c**5*k + 8400*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a** 2*b**4*c**2*m - 5880*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 ))*a**2*b**3*c**3*l + 3780*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**4*k - 2100*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 4*a*c - b**2))*a**2*b*c**5*j + 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqr t(4*a*c - b**2))*a**2*c**6*h - 3360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq rt(4*a*c - b**2))*a*b**6*c*m + 2940*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq rt(4*a*c - b**2))*a*b**5*c**2*l - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x) /sqrt(4*a*c - b**2))*a*b**4*c**3*k + 2100*sqrt(4*a*c - b**2)*atan((b + 2*c *x)/sqrt(4*a*c - b**2))*a*b**3*c**4*j - 1680*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**5*h + 1260*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**6*g - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**7*f + 420*sqrt(4*a*c - b**2)*atan((b + 2 *c*x)/sqrt(4*a*c - b**2))*b**8*m - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x) /sqrt(4*a*c - b**2))*b**7*c*l + 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq rt(4*a*c - b**2))*b**6*c**2*k - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)...