Integrand size = 20, antiderivative size = 414 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac {c^3 (3 c d-2 b e) x^2}{e^7}+\frac {c^4 x^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^9 (d+e x)}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \] Output:
c^2*(21*c^2*d^2+6*b^2*e^2-4*c*e*(-a*e+6*b*d))*x/e^8-c^3*(-2*b*e+3*c*d)*x^2 /e^7+1/3*c^4*x^3/e^6-1/5*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^5+(-b*e+2*c*d)* (a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^4-2/3*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+ 3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^9/(e*x+d)^3+2*(-b*e+2*c*d)*(a*e^2-b*d*e+c* d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))/e^9/(e*x+d)^2-(70*c^4*d^4+b^4* e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2* e^2-10*a*b*d*e+15*b^2*d^2))/e^9/(e*x+d)-4*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^ 2-c*e*(-3*a*e+7*b*d))*ln(e*x+d)/e^9
Time = 0.15 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\frac {15 c^2 e \left (21 c^2 d^2+6 b^2 e^2+4 c e (-6 b d+a e)\right ) x+15 c^3 e^2 (-3 c d+2 b e) x^2+5 c^4 e^3 x^3-\frac {3 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^5}+\frac {15 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^4}-\frac {10 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^3}+\frac {30 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{(d+e x)^2}-\frac {15 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right )}{d+e x}-60 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) \log (d+e x)}{15 e^9} \] Input:
Integrate[(a + b*x + c*x^2)^4/(d + e*x)^6,x]
Output:
(15*c^2*e*(21*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-6*b*d + a*e))*x + 15*c^3*e^2*( -3*c*d + 2*b*e)*x^2 + 5*c^4*e^3*x^3 - (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^5 + (15*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 - ( 10*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a* e))^2)/(d + e*x)^3 + (30*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5 *a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2 )))/(d + e*x)^2 - (15*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2* e^2)))/(d + e*x) - 60*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*Log[d + e*x])/(15*e^9)
Time = 0.87 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8 (d+e x)}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8 (d+e x)^3}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^4}+\frac {c^2 \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^5}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^6}-\frac {2 c^3 x (3 c d-2 b e)}{e^7}+\frac {c^4 x^2}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac {4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac {c^3 x^2 (3 c d-2 b e)}{e^7}+\frac {c^4 x^3}{3 e^6}\) |
Input:
Int[(a + b*x + c*x^2)^4/(d + e*x)^6,x]
Output:
(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e)*x^2)/e^7 + (c^4*x^3)/(3*e^6) - (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4) - (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e )))/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2* d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(e^9*(d + e*x)^2) - (70*c^4*d^4 + b^ 4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2 *e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))/(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*Log[d + e*x])/e^9
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(886\) vs. \(2(408)=816\).
Time = 0.90 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.14
| method | result | size |
| norman | \(\frac {-\frac {3 a^{4} e^{8}+3 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}-411 a b \,c^{2} d^{5} e^{3}+822 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}-137 b^{3} c \,d^{5} e^{3}+1233 b^{2} c^{2} d^{6} e^{2}-2877 b \,c^{3} d^{7} e +1918 c^{4} d^{8}}{15 e^{9}}+\frac {c^{4} x^{8}}{3 e}-\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+120 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-20 d \,e^{3} b^{3} c +180 d^{2} e^{2} b^{2} c^{2}-420 d^{3} e b \,c^{3}+280 d^{4} c^{4}\right ) x^{4}}{e^{5}}-\frac {2 \left (3 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}-90 a b \,c^{2} d^{2} e^{3}+180 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}-30 b^{3} c \,d^{2} e^{3}+270 b^{2} c^{2} d^{3} e^{2}-630 b \,c^{3} d^{4} e +420 c^{4} d^{5}\right ) x^{3}}{e^{6}}-\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-330 a b \,c^{2} d^{3} e^{3}+660 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}-110 b^{3} c \,d^{3} e^{3}+990 b^{2} c^{2} d^{4} e^{2}-2310 b \,c^{3} d^{5} e +1540 d^{6} c^{4}\right ) x^{2}}{3 e^{7}}-\frac {\left (3 a^{3} b \,e^{7}+2 d \,e^{6} c \,a^{3}+3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}-375 a b \,c^{2} d^{4} e^{3}+750 d^{5} e^{2} a \,c^{3}+3 b^{4} d^{3} e^{4}-125 b^{3} c \,d^{4} e^{3}+1125 b^{2} c^{2} d^{5} e^{2}-2625 b \,c^{3} d^{6} e +1750 d^{7} c^{4}\right ) x}{3 e^{8}}+\frac {2 c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-21 b c d e +14 c^{2} d^{2}\right ) x^{6}}{3 e^{3}}+\frac {2 c^{3} \left (3 b e -2 c d \right ) x^{7}}{3 e^{2}}}{\left (e x +d \right )^{5}}+\frac {4 c \left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-9 d \,e^{2} b^{2} c +21 d^{2} e b \,c^{2}-14 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(887\) |
| default | \(\frac {c^{2} \left (\frac {1}{3} c^{2} e^{2} x^{3}+2 b c \,e^{2} x^{2}-3 c^{2} d e \,x^{2}+4 a c \,e^{2} x +6 b^{2} e^{2} x -24 b c d e x +21 c^{2} d^{2} x \right )}{e^{8}}-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{3 e^{9} \left (e x +d \right )^{3}}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} a \,c^{3}-4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{4 e^{9} \left (e x +d \right )^{4}}+\frac {4 c \left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}+b^{3} e^{3}-9 d \,e^{2} b^{2} c +21 d^{2} e b \,c^{2}-14 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{9}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-20 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}-140 d^{3} e b \,c^{3}+70 d^{4} c^{4}}{e^{9} \left (e x +d \right )}-\frac {12 a^{2} b c \,e^{5}-24 a^{2} c^{2} d \,e^{4}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 a \,c^{3} d^{3} e^{2}-4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{2 e^{9} \left (e x +d \right )^{2}}\) | \(902\) |
| risch | \(\frac {c^{4} x^{3}}{3 e^{6}}+\frac {2 c^{3} b \,x^{2}}{e^{6}}-\frac {3 c^{4} d \,x^{2}}{e^{7}}+\frac {4 c^{3} a x}{e^{6}}+\frac {6 c^{2} b^{2} x}{e^{6}}-\frac {24 c^{3} b d x}{e^{7}}+\frac {21 c^{4} d^{2} x}{e^{8}}+\frac {\left (-6 a^{2} c^{2} e^{7}-12 a \,b^{2} c \,e^{7}+60 a b \,c^{2} d \,e^{6}-60 a \,c^{3} d^{2} e^{5}-b^{4} e^{7}+20 b^{3} c d \,e^{6}-90 b^{2} c^{2} d^{2} e^{5}+140 b \,c^{3} d^{3} e^{4}-70 c^{4} d^{4} e^{3}\right ) x^{4}-2 e^{2} \left (3 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}-90 a b \,c^{2} d^{2} e^{3}+100 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}-30 b^{3} c \,d^{2} e^{3}+150 b^{2} c^{2} d^{3} e^{2}-245 b \,c^{3} d^{4} e +126 c^{4} d^{5}\right ) x^{3}-\frac {2 e \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-330 a b \,c^{2} d^{3} e^{3}+390 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}-110 b^{3} c \,d^{3} e^{3}+585 b^{2} c^{2} d^{4} e^{2}-987 b \,c^{3} d^{5} e +518 d^{6} c^{4}\right ) x^{2}}{3}+\left (-a^{3} b \,e^{7}-\frac {2}{3} d \,e^{6} c \,a^{3}-a^{2} b^{2} d \,e^{6}-3 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}-a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+125 a b \,c^{2} d^{4} e^{3}-154 d^{5} e^{2} a \,c^{3}-b^{4} d^{3} e^{4}+\frac {125}{3} b^{3} c \,d^{4} e^{3}-231 b^{2} c^{2} d^{5} e^{2}+399 b \,c^{3} d^{6} e -\frac {638}{3} d^{7} c^{4}\right ) x -\frac {3 a^{4} e^{8}+3 a^{3} b d \,e^{7}+2 a^{3} c \,d^{2} e^{6}+3 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}-411 a b \,c^{2} d^{5} e^{3}+522 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}-137 b^{3} c \,d^{5} e^{3}+783 b^{2} c^{2} d^{6} e^{2}-1377 b \,c^{3} d^{7} e +743 c^{4} d^{8}}{15 e}}{e^{8} \left (e x +d \right )^{5}}+\frac {12 c^{2} \ln \left (e x +d \right ) a b}{e^{6}}-\frac {24 c^{3} \ln \left (e x +d \right ) d a}{e^{7}}+\frac {4 c \ln \left (e x +d \right ) b^{3}}{e^{6}}-\frac {36 c^{2} \ln \left (e x +d \right ) d \,b^{2}}{e^{7}}+\frac {84 c^{3} \ln \left (e x +d \right ) d^{2} b}{e^{8}}-\frac {56 c^{4} \ln \left (e x +d \right ) d^{3}}{e^{9}}\) | \(928\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1651\) |
Input:
int((c*x^2+b*x+a)^4/(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
(-1/15*(3*a^4*e^8+3*a^3*b*d*e^7+2*a^3*c*d^2*e^6+3*a^2*b^2*d^2*e^6+9*a^2*b* c*d^3*e^5+18*a^2*c^2*d^4*e^4+3*a*b^3*d^3*e^5+36*a*b^2*c*d^4*e^4-411*a*b*c^ 2*d^5*e^3+822*a*c^3*d^6*e^2+3*b^4*d^4*e^4-137*b^3*c*d^5*e^3+1233*b^2*c^2*d ^6*e^2-2877*b*c^3*d^7*e+1918*c^4*d^8)/e^9+1/3*c^4/e*x^8-(6*a^2*c^2*e^4+12* a*b^2*c*e^4-60*a*b*c^2*d*e^3+120*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+180* b^2*c^2*d^2*e^2-420*b*c^3*d^3*e+280*c^4*d^4)/e^5*x^4-2*(3*a^2*b*c*e^5+6*a^ 2*c^2*d*e^4+a*b^3*e^5+12*a*b^2*c*d*e^4-90*a*b*c^2*d^2*e^3+180*a*c^3*d^3*e^ 2+b^4*d*e^4-30*b^3*c*d^2*e^3+270*b^2*c^2*d^3*e^2-630*b*c^3*d^4*e+420*c^4*d ^5)/e^6*x^3-2/3*(2*a^3*c*e^6+3*a^2*b^2*e^6+9*a^2*b*c*d*e^5+18*a^2*c^2*d^2* e^4+3*a*b^3*d*e^5+36*a*b^2*c*d^2*e^4-330*a*b*c^2*d^3*e^3+660*a*c^3*d^4*e^2 +3*b^4*d^2*e^4-110*b^3*c*d^3*e^3+990*b^2*c^2*d^4*e^2-2310*b*c^3*d^5*e+1540 *c^4*d^6)/e^7*x^2-1/3*(3*a^3*b*e^7+2*a^3*c*d*e^6+3*a^2*b^2*d*e^6+9*a^2*b*c *d^2*e^5+18*a^2*c^2*d^3*e^4+3*a*b^3*d^2*e^5+36*a*b^2*c*d^3*e^4-375*a*b*c^2 *d^4*e^3+750*a*c^3*d^5*e^2+3*b^4*d^3*e^4-125*b^3*c*d^4*e^3+1125*b^2*c^2*d^ 5*e^2-2625*b*c^3*d^6*e+1750*c^4*d^7)/e^8*x+2/3*c^2*(6*a*c*e^2+9*b^2*e^2-21 *b*c*d*e+14*c^2*d^2)/e^3*x^6+2/3*c^3*(3*b*e-2*c*d)/e^2*x^7)/(e*x+d)^5+4*c/ e^9*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3 *d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1262 vs. \(2 (408) = 816\).
Time = 0.09 (sec) , antiderivative size = 1262, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="fricas")
Output:
1/15*(5*c^4*e^8*x^8 - 743*c^4*d^8 + 1377*b*c^3*d^7*e - 3*a^3*b*d*e^7 - 3*a ^4*e^8 - 261*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 137*(b^3*c + 3*a*b*c^2)*d^5*e ^3 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^3* e^5 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 10*(2*c^4*d*e^7 - 3*b*c^3*e^8)*x^7 + 10*(14*c^4*d^2*e^6 - 21*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 25*(47*c^4*d^3*e^5 - 60*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7)*x^5 + 5*(335*c^4*d^4*e^4 - 240*b*c^3*d^3*e^5 - 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e ^6 + 60*(b^3*c + 3*a*b*c^2)*d*e^7 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)* x^4 - 10*(85*c^4*d^5*e^3 - 390*b*c^3*d^4*e^4 + 120*(3*b^2*c^2 + 2*a*c^3)*d ^3*e^5 - 90*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2) *d*e^7 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 10*(365*c^4*d^6*e^2 - 810*b*c^3* d^5*e^3 + 180*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(b^3*c + 3*a*b*c^2)*d^3* e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 3*(a*b^3 + 3*a^2*b*c)*d*e ^7 + (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 - 5*(575*c^4*d^7*e - 1125*b*c^3*d^6*e^ 2 + 3*a^3*b*e^8 + 225*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 125*(b^3*c + 3*a*b*c ^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 3*(a*b^3 + 3*a^2* b*c)*d^2*e^6 + (3*a^2*b^2 + 2*a^3*c)*d*e^7)*x - 60*(14*c^4*d^8 - 21*b*c^3* d^7*e + 3*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - (b^3*c + 3*a*b*c^2)*d^5*e^3 + (1 4*c^4*d^3*e^5 - 21*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 5*(14*c^4*d^4*e^4 - 21*b*c^3*d^3*e^5 + 3*(3*b^2...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=\text {Timed out} \] Input:
integrate((c*x**2+b*x+a)**4/(e*x+d)**6,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (408) = 816\).
Time = 0.06 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="maxima")
Output:
-1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 3*a^3*b*d*e^7 + 3*a^4*e^8 + 261*(3 *b^2*c^2 + 2*a*c^3)*d^6*e^2 - 137*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(b^4 + 1 2*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 3*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + (3*a^2*b^ 2 + 2*a^3*c)*d^2*e^6 + 15*(70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 30*(3*b^2* c^2 + 2*a*c^3)*d^2*e^6 - 20*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 30*(126*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 50*(3*b^ 2*c^2 + 2*a*c^3)*d^3*e^5 - 30*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^ 2*c + 6*a^2*c^2)*d*e^7 + (a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 10*(518*c^4*d^6*e^ 2 - 987*b*c^3*d^5*e^3 + 195*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(b^3*c + 3 *a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 5*(638*c^4*d^7*e - 119 7*b*c^3*d^6*e^2 + 3*a^3*b*e^8 + 231*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 125*(b ^3*c + 3*a*b*c^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 3*( a*b^3 + 3*a^2*b*c)*d^2*e^6 + (3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^14*x^5 + 5 *d*e^13*x^4 + 10*d^2*e^12*x^3 + 10*d^3*e^11*x^2 + 5*d^4*e^10*x + d^5*e^9) + 1/3*(c^4*e^2*x^3 - 3*(3*c^4*d*e - 2*b*c^3*e^2)*x^2 + 3*(21*c^4*d^2 - 24* b*c^3*d*e + 2*(3*b^2*c^2 + 2*a*c^3)*e^2)*x)/e^8 - 4*(14*c^4*d^3 - 21*b*c^3 *d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*log(e*x + d)/e^9
Leaf count of result is larger than twice the leaf count of optimal. 909 vs. \(2 (408) = 816\).
Time = 0.37 (sec) , antiderivative size = 909, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="giac")
Output:
-4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*a*c^3*d*e^2 - b^3*c* e^3 - 3*a*b*c^2*e^3)*log(abs(e*x + d))/e^9 - 1/15*(743*c^4*d^8 - 1377*b*c^ 3*d^7*e + 783*b^2*c^2*d^6*e^2 + 522*a*c^3*d^6*e^2 - 137*b^3*c*d^5*e^3 - 41 1*a*b*c^2*d^5*e^3 + 3*b^4*d^4*e^4 + 36*a*b^2*c*d^4*e^4 + 18*a^2*c^2*d^4*e^ 4 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 3*a^2*b^2*d^2*e^6 + 2*a^3*c*d^2* e^6 + 3*a^3*b*d*e^7 + 3*a^4*e^8 + 15*(70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 90*b^2*c^2*d^2*e^6 + 60*a*c^3*d^2*e^6 - 20*b^3*c*d*e^7 - 60*a*b*c^2*d*e^7 + b^4*e^8 + 12*a*b^2*c*e^8 + 6*a^2*c^2*e^8)*x^4 + 30*(126*c^4*d^5*e^3 - 2 45*b*c^3*d^4*e^4 + 150*b^2*c^2*d^3*e^5 + 100*a*c^3*d^3*e^5 - 30*b^3*c*d^2* e^6 - 90*a*b*c^2*d^2*e^6 + b^4*d*e^7 + 12*a*b^2*c*d*e^7 + 6*a^2*c^2*d*e^7 + a*b^3*e^8 + 3*a^2*b*c*e^8)*x^3 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 585*b^2*c^2*d^4*e^4 + 390*a*c^3*d^4*e^4 - 110*b^3*c*d^3*e^5 - 330*a*b*c ^2*d^3*e^5 + 3*b^4*d^2*e^6 + 36*a*b^2*c*d^2*e^6 + 18*a^2*c^2*d^2*e^6 + 3*a *b^3*d*e^7 + 9*a^2*b*c*d*e^7 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8)*x^2 + 5*(638*c ^4*d^7*e - 1197*b*c^3*d^6*e^2 + 693*b^2*c^2*d^5*e^3 + 462*a*c^3*d^5*e^3 - 125*b^3*c*d^4*e^4 - 375*a*b*c^2*d^4*e^4 + 3*b^4*d^3*e^5 + 36*a*b^2*c*d^3*e ^5 + 18*a^2*c^2*d^3*e^5 + 3*a*b^3*d^2*e^6 + 9*a^2*b*c*d^2*e^6 + 3*a^2*b^2* d*e^7 + 2*a^3*c*d*e^7 + 3*a^3*b*e^8)*x)/((e*x + d)^5*e^9) + 1/3*(c^4*e^12* x^3 - 9*c^4*d*e^11*x^2 + 6*b*c^3*e^12*x^2 + 63*c^4*d^2*e^10*x - 72*b*c^3*d *e^11*x + 18*b^2*c^2*e^12*x + 12*a*c^3*e^12*x)/e^18
Time = 5.62 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx=x^2\,\left (\frac {2\,b\,c^3}{e^6}-\frac {3\,c^4\,d}{e^7}\right )-x\,\left (\frac {6\,d\,\left (\frac {4\,b\,c^3}{e^6}-\frac {6\,c^4\,d}{e^7}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^6}+\frac {15\,c^4\,d^2}{e^8}\right )-\frac {x^3\,\left (6\,a^2\,b\,c\,e^7+12\,a^2\,c^2\,d\,e^6+2\,a\,b^3\,e^7+24\,a\,b^2\,c\,d\,e^6-180\,a\,b\,c^2\,d^2\,e^5+200\,a\,c^3\,d^3\,e^4+2\,b^4\,d\,e^6-60\,b^3\,c\,d^2\,e^5+300\,b^2\,c^2\,d^3\,e^4-490\,b\,c^3\,d^4\,e^3+252\,c^4\,d^5\,e^2\right )+x\,\left (a^3\,b\,e^7+\frac {2\,a^3\,c\,d\,e^6}{3}+a^2\,b^2\,d\,e^6+3\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4-125\,a\,b\,c^2\,d^4\,e^3+154\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4-\frac {125\,b^3\,c\,d^4\,e^3}{3}+231\,b^2\,c^2\,d^5\,e^2-399\,b\,c^3\,d^6\,e+\frac {638\,c^4\,d^7}{3}\right )+x^4\,\left (6\,a^2\,c^2\,e^7+12\,a\,b^2\,c\,e^7-60\,a\,b\,c^2\,d\,e^6+60\,a\,c^3\,d^2\,e^5+b^4\,e^7-20\,b^3\,c\,d\,e^6+90\,b^2\,c^2\,d^2\,e^5-140\,b\,c^3\,d^3\,e^4+70\,c^4\,d^4\,e^3\right )+\frac {3\,a^4\,e^8+3\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+9\,a^2\,b\,c\,d^3\,e^5+18\,a^2\,c^2\,d^4\,e^4+3\,a\,b^3\,d^3\,e^5+36\,a\,b^2\,c\,d^4\,e^4-411\,a\,b\,c^2\,d^5\,e^3+522\,a\,c^3\,d^6\,e^2+3\,b^4\,d^4\,e^4-137\,b^3\,c\,d^5\,e^3+783\,b^2\,c^2\,d^6\,e^2-1377\,b\,c^3\,d^7\,e+743\,c^4\,d^8}{15\,e}+x^2\,\left (\frac {4\,a^3\,c\,e^7}{3}+2\,a^2\,b^2\,e^7+6\,a^2\,b\,c\,d\,e^6+12\,a^2\,c^2\,d^2\,e^5+2\,a\,b^3\,d\,e^6+24\,a\,b^2\,c\,d^2\,e^5-220\,a\,b\,c^2\,d^3\,e^4+260\,a\,c^3\,d^4\,e^3+2\,b^4\,d^2\,e^5-\frac {220\,b^3\,c\,d^3\,e^4}{3}+390\,b^2\,c^2\,d^4\,e^3-658\,b\,c^3\,d^5\,e^2+\frac {1036\,c^4\,d^6\,e}{3}\right )}{d^5\,e^8+5\,d^4\,e^9\,x+10\,d^3\,e^{10}\,x^2+10\,d^2\,e^{11}\,x^3+5\,d\,e^{12}\,x^4+e^{13}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (-4\,b^3\,c\,e^3+36\,b^2\,c^2\,d\,e^2-84\,b\,c^3\,d^2\,e-12\,a\,b\,c^2\,e^3+56\,c^4\,d^3+24\,a\,c^3\,d\,e^2\right )}{e^9}+\frac {c^4\,x^3}{3\,e^6} \] Input:
int((a + b*x + c*x^2)^4/(d + e*x)^6,x)
Output:
x^2*((2*b*c^3)/e^6 - (3*c^4*d)/e^7) - x*((6*d*((4*b*c^3)/e^6 - (6*c^4*d)/e ^7))/e - (4*a*c^3 + 6*b^2*c^2)/e^6 + (15*c^4*d^2)/e^8) - (x^3*(2*a*b^3*e^7 + 2*b^4*d*e^6 + 252*c^4*d^5*e^2 + 200*a*c^3*d^3*e^4 + 12*a^2*c^2*d*e^6 - 490*b*c^3*d^4*e^3 - 60*b^3*c*d^2*e^5 + 300*b^2*c^2*d^3*e^4 + 6*a^2*b*c*e^7 + 24*a*b^2*c*d*e^6 - 180*a*b*c^2*d^2*e^5) + x*((638*c^4*d^7)/3 + a^3*b*e^ 7 + b^4*d^3*e^4 + a*b^3*d^2*e^5 + a^2*b^2*d*e^6 + 154*a*c^3*d^5*e^2 - (125 *b^3*c*d^4*e^3)/3 + 6*a^2*c^2*d^3*e^4 + 231*b^2*c^2*d^5*e^2 + (2*a^3*c*d*e ^6)/3 - 399*b*c^3*d^6*e - 125*a*b*c^2*d^4*e^3 + 12*a*b^2*c*d^3*e^4 + 3*a^2 *b*c*d^2*e^5) + x^4*(b^4*e^7 + 6*a^2*c^2*e^7 + 70*c^4*d^4*e^3 + 60*a*c^3*d ^2*e^5 - 140*b*c^3*d^3*e^4 + 90*b^2*c^2*d^2*e^5 + 12*a*b^2*c*e^7 - 20*b^3* c*d*e^6 - 60*a*b*c^2*d*e^6) + (3*a^4*e^8 + 743*c^4*d^8 + 3*b^4*d^4*e^4 + 3 *a*b^3*d^3*e^5 + 522*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 - 137*b^3*c*d^5*e^3 + 3*a^2*b^2*d^2*e^6 + 18*a^2*c^2*d^4*e^4 + 783*b^2*c^2*d^6*e^2 + 3*a^3*b*d* e^7 - 1377*b*c^3*d^7*e - 411*a*b*c^2*d^5*e^3 + 36*a*b^2*c*d^4*e^4 + 9*a^2* b*c*d^3*e^5)/(15*e) + x^2*((4*a^3*c*e^7)/3 + (1036*c^4*d^6*e)/3 + 2*a^2*b^ 2*e^7 + 2*b^4*d^2*e^5 + 260*a*c^3*d^4*e^3 - 658*b*c^3*d^5*e^2 - (220*b^3*c *d^3*e^4)/3 + 12*a^2*c^2*d^2*e^5 + 390*b^2*c^2*d^4*e^3 + 2*a*b^3*d*e^6 + 6 *a^2*b*c*d*e^6 - 220*a*b*c^2*d^3*e^4 + 24*a*b^2*c*d^2*e^5))/(d^5*e^8 + e^1 3*x^5 + 5*d^4*e^9*x + 5*d*e^12*x^4 + 10*d^3*e^10*x^2 + 10*d^2*e^11*x^3) - (log(d + e*x)*(56*c^4*d^3 - 4*b^3*c*e^3 + 36*b^2*c^2*d*e^2 - 12*a*b*c^2...
Time = 0.19 (sec) , antiderivative size = 1564, normalized size of antiderivative = 3.78 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^4/(e*x+d)^6,x)
Output:
(180*log(d + e*x)*a*b*c**2*d**6*e**3 + 900*log(d + e*x)*a*b*c**2*d**5*e**4 *x + 1800*log(d + e*x)*a*b*c**2*d**4*e**5*x**2 + 1800*log(d + e*x)*a*b*c** 2*d**3*e**6*x**3 + 900*log(d + e*x)*a*b*c**2*d**2*e**7*x**4 + 180*log(d + e*x)*a*b*c**2*d*e**8*x**5 - 360*log(d + e*x)*a*c**3*d**7*e**2 - 1800*log(d + e*x)*a*c**3*d**6*e**3*x - 3600*log(d + e*x)*a*c**3*d**5*e**4*x**2 - 360 0*log(d + e*x)*a*c**3*d**4*e**5*x**3 - 1800*log(d + e*x)*a*c**3*d**3*e**6* x**4 - 360*log(d + e*x)*a*c**3*d**2*e**7*x**5 + 60*log(d + e*x)*b**3*c*d** 6*e**3 + 300*log(d + e*x)*b**3*c*d**5*e**4*x + 600*log(d + e*x)*b**3*c*d** 4*e**5*x**2 + 600*log(d + e*x)*b**3*c*d**3*e**6*x**3 + 300*log(d + e*x)*b* *3*c*d**2*e**7*x**4 + 60*log(d + e*x)*b**3*c*d*e**8*x**5 - 540*log(d + e*x )*b**2*c**2*d**7*e**2 - 2700*log(d + e*x)*b**2*c**2*d**6*e**3*x - 5400*log (d + e*x)*b**2*c**2*d**5*e**4*x**2 - 5400*log(d + e*x)*b**2*c**2*d**4*e**5 *x**3 - 2700*log(d + e*x)*b**2*c**2*d**3*e**6*x**4 - 540*log(d + e*x)*b**2 *c**2*d**2*e**7*x**5 + 1260*log(d + e*x)*b*c**3*d**8*e + 6300*log(d + e*x) *b*c**3*d**7*e**2*x + 12600*log(d + e*x)*b*c**3*d**6*e**3*x**2 + 12600*log (d + e*x)*b*c**3*d**5*e**4*x**3 + 6300*log(d + e*x)*b*c**3*d**4*e**5*x**4 + 1260*log(d + e*x)*b*c**3*d**3*e**6*x**5 - 840*log(d + e*x)*c**4*d**9 - 4 200*log(d + e*x)*c**4*d**8*e*x - 8400*log(d + e*x)*c**4*d**7*e**2*x**2 - 8 400*log(d + e*x)*c**4*d**6*e**3*x**3 - 4200*log(d + e*x)*c**4*d**5*e**4*x* *4 - 840*log(d + e*x)*c**4*d**4*e**5*x**5 - 3*a**4*d*e**8 - 3*a**3*b*d*...