\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^8} \, dx\) [456]

Optimal result

Integrand size = 20, antiderivative size = 424 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\frac {c^4 x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{7 e^9 (d+e x)^7}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^6}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^4}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{3 e^9 (d+e x)^3}+\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9} \] Output:

c^4*x/e^8-1/7*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^7+2/3*(-b*e+2*c*d)*(a*e^2- 
b*d*e+c*d^2)^3/e^9/(e*x+d)^6-2/5*(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)^5+(-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)^4-1/3*(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^2-1 
0*a*b*d*e+15*b^2*d^2))/e^9/(e*x+d)^3+2*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c 
*e*(-3*a*e+7*b*d))/e^9/(e*x+d)^2-2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7 
*b*d))/e^9/(e*x+d)-4*c^3*(-b*e+2*c*d)*ln(e*x+d)/e^9
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=-\frac {c^4 \left (1443 d^8+9261 d^7 e x+24843 d^6 e^2 x^2+35525 d^5 e^3 x^3+28175 d^4 e^4 x^4+11025 d^3 e^5 x^5+735 d^2 e^6 x^6-735 d e^7 x^7-105 e^8 x^8\right )+e^4 \left (15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b^3 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+c e^3 \left (4 a^3 e^3 \left (d^2+7 d e x+21 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+12 a b^2 e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b^3 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+6 c^2 e^2 \left (a^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 a b e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+15 b^2 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+c^3 e \left (60 a e \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )-b d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )+420 c^3 (2 c d-b e) (d+e x)^7 \log (d+e x)}{105 e^9 (d+e x)^7} \] Input:

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^8,x]
 

Output:

-1/105*(c^4*(1443*d^8 + 9261*d^7*e*x + 24843*d^6*e^2*x^2 + 35525*d^5*e^3*x 
^3 + 28175*d^4*e^4*x^4 + 11025*d^3*e^5*x^5 + 735*d^2*e^6*x^6 - 735*d*e^7*x 
^7 - 105*e^8*x^8) + e^4*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2 
*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2* 
x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 
 35*e^4*x^4)) + c*e^3*(4*a^3*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 9*a^2*b*e^ 
2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 12*a*b^2*e*(d^4 + 7*d^3* 
e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 10*b^3*(d^5 + 7*d^4*e* 
x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + 6*c^2* 
e^2*(a^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4 
) + 5*a*b*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4* 
x^4 + 21*e^5*x^5) + 15*b^2*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3* 
x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + c^3*e*(60*a*e*(d^6 + 7 
*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 
 + 7*e^6*x^6) - b*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e^2*x^2 + 30625*d 
^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) + 420*c^ 
3*(2*c*d - b*e)*(d + e*x)^7*Log[d + e*x])/(e^9*(d + e*x)^7)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 424, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)^4}+\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8 (d+e x)^3}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8 (d+e x)^5}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^6}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^7}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^8}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)}+\frac {c^4}{e^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{3 e^9 (d+e x)^3}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}+\frac {2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^4}{7 e^9 (d+e x)^7}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac {c^4 x}{e^8}\)

Input:

Int[(a + b*x + c*x^2)^4/(d + e*x)^8,x]
 

Output:

(c^4*x)/e^8 - (c*d^2 - b*d*e + a*e^2)^4/(7*e^9*(d + e*x)^7) + (2*(2*c*d - 
b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^9*(d + e*x)^6) - (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)))/(5*e^9*(d + e*x)^ 
5) + ((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)^4) - (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))/(3*e^9*(d + e*x)^3) + (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b 
^2*e^2 - c*e*(7*b*d - 3*a*e)))/(e^9*(d + e*x)^2) - (2*c^2*(14*c^2*d^2 + 3* 
b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^9*(d + e*x)) - (4*c^3*(2*c*d - b*e)*Log 
[d + e*x])/e^9
 

Defintions of rubi rules used

rule 1140

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(416)=832\).

Time = 0.88 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.10

method	result	size
norman	\(\frac {\frac {c^{4} x^{8}}{e}-\frac {15 a^{4} e^{8}+10 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+30 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+10 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}-1089 b \,c^{3} d^{7} e +2178 c^{4} d^{8}}{105 e^{9}}-\frac {\left (4 e^{2} a \,c^{3}+6 b^{2} c^{2} e^{2}-28 c^{3} d e b +56 c^{4} d^{2}\right ) x^{6}}{e^{3}}-\frac {\left (6 a b \,c^{2} e^{3}+12 a \,c^{3} d \,e^{2}+2 b^{3} c \,e^{3}+18 d \,e^{2} b^{2} c^{2}-126 b \,c^{3} d^{2} e +252 d^{3} c^{4}\right ) x^{5}}{e^{4}}-\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+30 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+10 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}-770 d^{3} e b \,c^{3}+1540 d^{4} c^{4}\right ) x^{4}}{3 e^{5}}-\frac {\left (9 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+3 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}+60 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}-875 b \,c^{3} d^{4} e +1750 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}-\frac {\left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+30 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+10 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-959 b \,c^{3} d^{5} e +1918 d^{6} c^{4}\right ) x^{2}}{5 e^{7}}-\frac {\left (10 a^{3} b \,e^{7}+4 d \,e^{6} c \,a^{3}+6 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+30 a b \,c^{2} d^{4} e^{3}+60 d^{5} e^{2} a \,c^{3}+b^{4} d^{3} e^{4}+10 b^{3} c \,d^{4} e^{3}+90 b^{2} c^{2} d^{5} e^{2}-1029 b \,c^{3} d^{6} e +2058 d^{7} c^{4}\right ) x}{15 e^{8}}}{\left (e x +d \right )^{7}}+\frac {4 c^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{9}}\)	\(892\)
risch	\(\frac {c^{4} x}{e^{8}}+\frac {\left (-4 a \,c^{3} e^{7}-6 b^{2} c^{2} e^{7}+28 b \,c^{3} d \,e^{6}-28 c^{4} d^{2} e^{5}\right ) x^{6}-2 c \,e^{4} \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 -63 d^{2} e b \,c^{2}+70 d^{3} c^{3}\right ) x^{5}-\frac {e^{3} \left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}+30 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}+10 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}-770 d^{3} e b \,c^{3}+910 d^{4} c^{4}\right ) x^{4}}{3}-\frac {e^{2} \left (9 a^{2} b c \,e^{5}+6 a^{2} c^{2} d \,e^{4}+3 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}+60 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}-875 b \,c^{3} d^{4} e +1078 c^{4} d^{5}\right ) x^{3}}{3}-\frac {e \left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+30 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+b^{4} d^{2} e^{4}+10 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-959 b \,c^{3} d^{5} e +1218 d^{6} c^{4}\right ) x^{2}}{5}+\left (-\frac {2}{3} a^{3} b \,e^{7}-\frac {4}{15} d \,e^{6} c \,a^{3}-\frac {2}{5} a^{2} b^{2} d \,e^{6}-\frac {3}{5} a^{2} b c \,d^{2} e^{5}-\frac {2}{5} d^{3} e^{4} a^{2} c^{2}-\frac {1}{5} a \,b^{3} d^{2} e^{5}-\frac {4}{5} a \,b^{2} c \,d^{3} e^{4}-2 a b \,c^{2} d^{4} e^{3}-4 d^{5} e^{2} a \,c^{3}-\frac {1}{15} b^{4} d^{3} e^{4}-\frac {2}{3} b^{3} c \,d^{4} e^{3}-6 b^{2} c^{2} d^{5} e^{2}+\frac {343}{5} b \,c^{3} d^{6} e -\frac {446}{5} d^{7} c^{4}\right ) x -\frac {15 a^{4} e^{8}+10 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+30 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+10 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}-1089 b \,c^{3} d^{7} e +1443 c^{4} d^{8}}{105 e}}{e^{8} \left (e x +d \right )^{7}}+\frac {4 c^{3} \ln \left (e x +d \right ) b}{e^{8}}-\frac {8 c^{4} \ln \left (e x +d \right ) d}{e^{9}}\)	\(892\)
default	\(\frac {c^{4} x}{e^{8}}-\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}}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{4 e^{9} \left (e x +d \right )^{4}}-\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}}{7 e^{9} \left (e x +d \right )^{7}}+\frac {4 c^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{9}}-\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}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{e^{9} \left (e x +d \right )}-\frac {2 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 )}{e^{9} \left (e x +d \right )^{2}}-\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}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(906\)
parallelrisch	\(\text {Expression too large to display}\)	\(1296\)

Input:

int((c*x^2+b*x+a)^4/(e*x+d)^8,x,method=_RETURNVERBOSE)
 

Output:

(c^4/e*x^8-1/105*(15*a^4*e^8+10*a^3*b*d*e^7+4*a^3*c*d^2*e^6+6*a^2*b^2*d^2* 
e^6+9*a^2*b*c*d^3*e^5+6*a^2*c^2*d^4*e^4+3*a*b^3*d^3*e^5+12*a*b^2*c*d^4*e^4 
+30*a*b*c^2*d^5*e^3+60*a*c^3*d^6*e^2+b^4*d^4*e^4+10*b^3*c*d^5*e^3+90*b^2*c 
^2*d^6*e^2-1089*b*c^3*d^7*e+2178*c^4*d^8)/e^9-(4*a*c^3*e^2+6*b^2*c^2*e^2-2 
8*b*c^3*d*e+56*c^4*d^2)/e^3*x^6-(6*a*b*c^2*e^3+12*a*c^3*d*e^2+2*b^3*c*e^3+ 
18*b^2*c^2*d*e^2-126*b*c^3*d^2*e+252*c^4*d^3)/e^4*x^5-1/3*(6*a^2*c^2*e^4+1 
2*a*b^2*c*e^4+30*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4+10*b^3*c*d*e^3+90* 
b^2*c^2*d^2*e^2-770*b*c^3*d^3*e+1540*c^4*d^4)/e^5*x^4-1/3*(9*a^2*b*c*e^5+6 
*a^2*c^2*d*e^4+3*a*b^3*e^5+12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3+60*a*c^3*d^ 
3*e^2+b^4*d*e^4+10*b^3*c*d^2*e^3+90*b^2*c^2*d^3*e^2-875*b*c^3*d^4*e+1750*c 
^4*d^5)/e^6*x^3-1/5*(4*a^3*c*e^6+6*a^2*b^2*e^6+9*a^2*b*c*d*e^5+6*a^2*c^2*d 
^2*e^4+3*a*b^3*d*e^5+12*a*b^2*c*d^2*e^4+30*a*b*c^2*d^3*e^3+60*a*c^3*d^4*e^ 
2+b^4*d^2*e^4+10*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2-959*b*c^3*d^5*e+1918*c^4 
*d^6)/e^7*x^2-1/15*(10*a^3*b*e^7+4*a^3*c*d*e^6+6*a^2*b^2*d*e^6+9*a^2*b*c*d 
^2*e^5+6*a^2*c^2*d^3*e^4+3*a*b^3*d^2*e^5+12*a*b^2*c*d^3*e^4+30*a*b*c^2*d^4 
*e^3+60*a*c^3*d^5*e^2+b^4*d^3*e^4+10*b^3*c*d^4*e^3+90*b^2*c^2*d^5*e^2-1029 
*b*c^3*d^6*e+2058*c^4*d^7)/e^8*x)/(e*x+d)^7+4*c^3/e^9*(b*e-2*c*d)*ln(e*x+d 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1082 vs. \(2 (416) = 832\).

Time = 0.09 (sec) , antiderivative size = 1082, normalized size of antiderivative = 2.55 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="fricas")
 

Output:

1/105*(105*c^4*e^8*x^8 + 735*c^4*d*e^7*x^7 - 1443*c^4*d^8 + 1089*b*c^3*d^7 
*e - 10*a^3*b*d*e^7 - 15*a^4*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 10*( 
b^3*c + 3*a*b*c^2)*d^5*e^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 - 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 105*(7*c^4*d 
^2*e^6 - 28*b*c^3*d*e^7 + 2*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 105*(105*c^4* 
d^3*e^5 - 126*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 2*(b^3*c + 3 
*a*b*c^2)*e^8)*x^5 - 35*(805*c^4*d^4*e^4 - 770*b*c^3*d^3*e^5 + 30*(3*b^2*c 
^2 + 2*a*c^3)*d^2*e^6 + 10*(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 - 35*(1015*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30*(3*b^ 
2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(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 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 21*(1183*c^4*d^6 
*e^2 - 959*b*c^3*d^5*e^3 + 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 10*(b^3*c + 
3*a*b*c^2)*d^3*e^5 + (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 + 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 - 7*(1323*c^4*d^7*e - 1 
029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 10*( 
b^3*c + 3*a*b*c^2)*d^4*e^4 + (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 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x - 420*(2*c^4* 
d^8 - b*c^3*d^7*e + (2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 7*(2*c^4*d^2*e^6 - b*c 
^3*d*e^7)*x^6 + 21*(2*c^4*d^3*e^5 - b*c^3*d^2*e^6)*x^5 + 35*(2*c^4*d^4*e^4 
 - b*c^3*d^3*e^5)*x^4 + 35*(2*c^4*d^5*e^3 - b*c^3*d^4*e^4)*x^3 + 21*(2*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**4/(e*x+d)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (416) = 832\).

Time = 0.08 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="maxima")
 

Output:

-1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 10*a^3*b*d*e^7 + 15*a^4*e^8 + 30 
*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^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 + 2*(3*a^2* 
b^2 + 2*a^3*c)*d^2*e^6 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + (3*b^2*c^2 
 + 2*a*c^3)*e^8)*x^6 + 210*(70*c^4*d^3*e^5 - 63*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 + 35*(910*c^4*d^4*e^4 - 
 770*b*c^3*d^3*e^5 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 10*(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 + 35*(1078*c^4*d^5*e^ 
3 - 875*b*c^3*d^4*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(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 + 3*(a*b^3 + 3*a^2* 
b*c)*e^8)*x^3 + 21*(1218*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(3*b^2*c^2 + 
 2*a*c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + (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 + 2*(3*a^2*b^2 + 2*a^3*c)* 
e^8)*x^2 + 7*(1338*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(3*b 
^2*c^2 + 2*a*c^3)*d^5*e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (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 + 2*(3*a^2*b^2 + 
 2*a^3*c)*d*e^7)*x)/(e^16*x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^ 
13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e^11*x^2 + 7*d^6*e^10*x + d^7*e^9) + c^4 
*x/e^8 - 4*(2*c^4*d - b*c^3*e)*log(e*x + d)/e^9
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (416) = 832\).

Time = 0.36 (sec) , antiderivative size = 900, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="giac")
 

Output:

c^4*x/e^8 - 4*(2*c^4*d - b*c^3*e)*log(abs(e*x + d))/e^9 - 1/105*(1443*c^4* 
d^8 - 1089*b*c^3*d^7*e + 90*b^2*c^2*d^6*e^2 + 60*a*c^3*d^6*e^2 + 10*b^3*c* 
d^5*e^3 + 30*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*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 + 6*a^2*b^2*d^2*e^6 + 4*a^ 
3*c*d^2*e^6 + 10*a^3*b*d*e^7 + 15*a^4*e^8 + 210*(14*c^4*d^2*e^6 - 14*b*c^3 
*d*e^7 + 3*b^2*c^2*e^8 + 2*a*c^3*e^8)*x^6 + 210*(70*c^4*d^3*e^5 - 63*b*c^3 
*d^2*e^6 + 9*b^2*c^2*d*e^7 + 6*a*c^3*d*e^7 + b^3*c*e^8 + 3*a*b*c^2*e^8)*x^ 
5 + 35*(910*c^4*d^4*e^4 - 770*b*c^3*d^3*e^5 + 90*b^2*c^2*d^2*e^6 + 60*a*c^ 
3*d^2*e^6 + 10*b^3*c*d*e^7 + 30*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 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 90*b^2*c^ 
2*d^3*e^5 + 60*a*c^3*d^3*e^5 + 10*b^3*c*d^2*e^6 + 30*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 + 3*a*b^3*e^8 + 9*a^2*b*c*e^8) 
*x^3 + 21*(1218*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 90*b^2*c^2*d^4*e^4 + 60* 
a*c^3*d^4*e^4 + 10*b^3*c*d^3*e^5 + 30*a*b*c^2*d^3*e^5 + b^4*d^2*e^6 + 12*a 
*b^2*c*d^2*e^6 + 6*a^2*c^2*d^2*e^6 + 3*a*b^3*d*e^7 + 9*a^2*b*c*d*e^7 + 6*a 
^2*b^2*e^8 + 4*a^3*c*e^8)*x^2 + 7*(1338*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 9 
0*b^2*c^2*d^5*e^3 + 60*a*c^3*d^5*e^3 + 10*b^3*c*d^4*e^4 + 30*a*b*c^2*d^4*e 
^4 + b^4*d^3*e^5 + 12*a*b^2*c*d^3*e^5 + 6*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 + 6*a^2*b^2*d*e^7 + 4*a^3*c*d*e^7 + 10*a^3*b*e^8)*x) 
/((e*x + d)^7*e^9)
 

Mupad [B] (verification not implemented)

Time = 5.53 (sec) , antiderivative size = 1306, normalized size of antiderivative = 3.08 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^4/(d + e*x)^8,x)
 

Output:

-((a^4*e^8)/7 + (481*c^4*d^8)/35 + 8*c^4*d^8*log(d + e*x) + (b^4*d^4*e^4)/ 
105 + (b^4*e^8*x^4)/3 - c^4*e^8*x^8 + (a*b^3*d^3*e^5)/35 + (4*a*c^3*d^6*e^ 
2)/7 + (4*a^3*c*d^2*e^6)/105 + (2*b^3*c*d^5*e^3)/21 + a*b^3*e^8*x^3 + (4*a 
^3*c*e^8*x^2)/5 + 4*a*c^3*e^8*x^6 + 2*b^3*c*e^8*x^5 + (b^4*d^3*e^5*x)/15 + 
 (b^4*d*e^7*x^3)/3 - 7*c^4*d*e^7*x^7 + (2*a^2*b^2*d^2*e^6)/35 + (2*a^2*c^2 
*d^4*e^4)/35 + (6*b^2*c^2*d^6*e^2)/7 + (6*a^2*b^2*e^8*x^2)/5 + 2*a^2*c^2*e 
^8*x^4 + 6*b^2*c^2*e^8*x^6 + (b^4*d^2*e^6*x^2)/5 + (1183*c^4*d^6*e^2*x^2)/ 
5 + (1015*c^4*d^5*e^3*x^3)/3 + (805*c^4*d^4*e^4*x^4)/3 + 105*c^4*d^3*e^5*x 
^5 + 7*c^4*d^2*e^6*x^6 + (2*a^3*b*d*e^7)/21 - (363*b*c^3*d^7*e)/35 + (2*a^ 
3*b*e^8*x)/3 + (441*c^4*d^7*e*x)/5 - 4*b*c^3*d^7*e*log(d + e*x) + (4*a^3*c 
*d*e^7*x)/15 + 56*c^4*d^7*e*x*log(d + e*x) + (6*a^2*c^2*d^2*e^6*x^2)/5 + 1 
8*b^2*c^2*d^4*e^4*x^2 + 30*b^2*c^2*d^3*e^5*x^3 + 30*b^2*c^2*d^2*e^6*x^4 + 
(2*a*b*c^2*d^5*e^3)/7 + (4*a*b^2*c*d^4*e^4)/35 + (3*a^2*b*c*d^3*e^5)/35 + 
3*a^2*b*c*e^8*x^3 + 4*a*b^2*c*e^8*x^4 + 6*a*b*c^2*e^8*x^5 + (a*b^3*d^2*e^6 
*x)/5 + (2*a^2*b^2*d*e^7*x)/5 + (3*a*b^3*d*e^7*x^2)/5 + 4*a*c^3*d^5*e^3*x 
+ 12*a*c^3*d*e^7*x^5 - (343*b*c^3*d^6*e^2*x)/5 + (2*b^3*c*d^4*e^4*x)/3 + ( 
10*b^3*c*d*e^7*x^4)/3 - 28*b*c^3*d*e^7*x^6 - 4*b*c^3*e^8*x^7*log(d + e*x) 
+ 8*c^4*d*e^7*x^7*log(d + e*x) + (2*a^2*c^2*d^3*e^5*x)/5 + 12*a*c^3*d^4*e^ 
4*x^2 + 20*a*c^3*d^3*e^5*x^3 + 2*a^2*c^2*d*e^7*x^3 + 20*a*c^3*d^2*e^6*x^4 
+ 6*b^2*c^2*d^5*e^3*x - (959*b*c^3*d^5*e^3*x^2)/5 + 2*b^3*c*d^3*e^5*x^2...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1223, normalized size of antiderivative = 2.88 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^4/(e*x+d)^8,x)
                                                                                    
                                                                                    
 

Output:

(420*log(d + e*x)*b*c**3*d**8*e + 2940*log(d + e*x)*b*c**3*d**7*e**2*x + 8 
820*log(d + e*x)*b*c**3*d**6*e**3*x**2 + 14700*log(d + e*x)*b*c**3*d**5*e* 
*4*x**3 + 14700*log(d + e*x)*b*c**3*d**4*e**5*x**4 + 8820*log(d + e*x)*b*c 
**3*d**3*e**6*x**5 + 2940*log(d + e*x)*b*c**3*d**2*e**7*x**6 + 420*log(d + 
 e*x)*b*c**3*d*e**8*x**7 - 840*log(d + e*x)*c**4*d**9 - 5880*log(d + e*x)* 
c**4*d**8*e*x - 17640*log(d + e*x)*c**4*d**7*e**2*x**2 - 29400*log(d + e*x 
)*c**4*d**6*e**3*x**3 - 29400*log(d + e*x)*c**4*d**5*e**4*x**4 - 17640*log 
(d + e*x)*c**4*d**4*e**5*x**5 - 5880*log(d + e*x)*c**4*d**3*e**6*x**6 - 84 
0*log(d + e*x)*c**4*d**2*e**7*x**7 - 15*a**4*d*e**8 - 10*a**3*b*d**2*e**7 
- 70*a**3*b*d*e**8*x - 4*a**3*c*d**3*e**6 - 28*a**3*c*d**2*e**7*x - 84*a** 
3*c*d*e**8*x**2 - 6*a**2*b**2*d**3*e**6 - 42*a**2*b**2*d**2*e**7*x - 126*a 
**2*b**2*d*e**8*x**2 - 9*a**2*b*c*d**4*e**5 - 63*a**2*b*c*d**3*e**6*x - 18 
9*a**2*b*c*d**2*e**7*x**2 - 315*a**2*b*c*d*e**8*x**3 - 6*a**2*c**2*d**5*e* 
*4 - 42*a**2*c**2*d**4*e**5*x - 126*a**2*c**2*d**3*e**6*x**2 - 210*a**2*c* 
*2*d**2*e**7*x**3 - 210*a**2*c**2*d*e**8*x**4 - 3*a*b**3*d**4*e**5 - 21*a* 
b**3*d**3*e**6*x - 63*a*b**3*d**2*e**7*x**2 - 105*a*b**3*d*e**8*x**3 - 12* 
a*b**2*c*d**5*e**4 - 84*a*b**2*c*d**4*e**5*x - 252*a*b**2*c*d**3*e**6*x**2 
 - 420*a*b**2*c*d**2*e**7*x**3 - 420*a*b**2*c*d*e**8*x**4 - 30*a*b*c**2*d* 
*6*e**3 - 210*a*b*c**2*d**5*e**4*x - 630*a*b*c**2*d**4*e**5*x**2 - 1050*a* 
b*c**2*d**3*e**6*x**3 - 1050*a*b*c**2*d**2*e**7*x**4 - 630*a*b*c**2*d*e...