\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^9} \, dx\) [457]

Optimal result

Integrand size = 20, antiderivative size = 435 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{8 e^9 (d+e x)^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{7 e^9 (d+e x)^7}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{5 e^9 (d+e x)^5}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{4 e^9 (d+e x)^4}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^3}-\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac {c^4 \log (d+e x)}{e^9} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\frac {c^4 d \left (2283 d^7+17424 d^6 e x+57624 d^5 e^2 x^2+107408 d^4 e^3 x^3+122500 d^3 e^4 x^4+86240 d^2 e^5 x^5+35280 d e^6 x^6+6720 e^7 x^7\right )-3 e^4 \left (35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )-4 c e^3 \left (5 a^3 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+9 a b^2 e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b^3 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )-6 c^2 e^2 \left (3 a^2 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+10 a b e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+15 b^2 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )-60 c^3 e \left (a e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 b \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )+840 c^4 (d+e x)^8 \log (d+e x)}{840 e^9 (d+e x)^8} \] Input:

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^9,x]
 

Output:

(c^4*d*(2283*d^7 + 17424*d^6*e*x + 57624*d^5*e^2*x^2 + 107408*d^4*e^3*x^3 
+ 122500*d^3*e^4*x^4 + 86240*d^2*e^5*x^5 + 35280*d*e^6*x^6 + 6720*e^7*x^7) 
 - 3*e^4*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8* 
d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x 
^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) 
- 4*c*e^3*(5*a^3*e^3*(d^2 + 8*d*e*x + 28*e^2*x^2) + 9*a^2*b*e^2*(d^3 + 8*d 
^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 9*a*b^2*e*(d^4 + 8*d^3*e*x + 28*d^2* 
e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b^3*(d^5 + 8*d^4*e*x + 28*d^3*e^2 
*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) - 6*c^2*e^2*(3*a^2*e^2 
*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 10*a*b*e 
*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^ 
5*x^5) + 15*b^2*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^ 
2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)) - 60*c^3*e*(a*e*(d^6 + 8*d^5*e*x + 
 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6* 
x^6) + 7*b*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70*d^3*e^4 
*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)) + 840*c^4*(d + e*x)^8*L 
og[d + e*x])/(840*e^9*(d + e*x)^8)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 435, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx\)

\(\Big \downarrow \) 1140

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x + c*x^2)^4/(d + e*x)^9,x]
 

Output:

-1/8*(c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)^8) + (4*(2*c*d - b*e)*(c*d^2 
 - b*d*e + a*e^2)^3)/(7*e^9*(d + e*x)^7) - ((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)^6) + (4*(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 
)))/(5*e^9*(d + e*x)^5) - (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))/(4*e^9*(d + e*x)^4) + (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - 
c*e*(7*b*d - 3*a*e)))/(3*e^9*(d + e*x)^3) - (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)^2) + (4*c^3*(2*c*d - b*e))/(e^9*(d + 
 e*x)) + (c^4*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. \(888\) vs. \(2(423)=846\).

Time = 0.89 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.04

method	result	size
risch	\(\frac {-\frac {4 c^{3} \left (b e -2 c d \right ) x^{7}}{e^{2}}-\frac {c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}+14 b c d e -42 c^{2} d^{2}\right ) x^{6}}{e^{3}}-\frac {2 c \left (6 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+2 b^{3} e^{3}+9 d \,e^{2} b^{2} c +42 d^{2} e b \,c^{2}-154 d^{3} c^{3}\right ) x^{5}}{3 e^{4}}-\frac {\left (18 e^{4} a^{2} c^{2}+36 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}+20 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}+420 d^{3} e b \,c^{3}-1750 d^{4} c^{4}\right ) x^{4}}{12 e^{5}}-\frac {\left (36 a^{2} b c \,e^{5}+18 a^{2} c^{2} d \,e^{4}+12 a \,b^{3} e^{5}+36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}+60 a \,c^{3} d^{3} e^{2}+3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}+420 b \,c^{3} d^{4} e -1918 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {\left (20 e^{6} c \,a^{3}+30 a^{2} b^{2} e^{6}+36 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+12 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}+420 b \,c^{3} d^{5} e -2058 d^{6} c^{4}\right ) x^{2}}{30 e^{7}}-\frac {\left (60 a^{3} b \,e^{7}+20 d \,e^{6} c \,a^{3}+30 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}+60 d^{5} e^{2} a \,c^{3}+3 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}+90 b^{2} c^{2} d^{5} e^{2}+420 b \,c^{3} d^{6} e -2178 d^{7} c^{4}\right ) x}{105 e^{8}}-\frac {105 a^{4} e^{8}+60 a^{3} b d \,e^{7}+20 a^{3} c \,d^{2} e^{6}+30 a^{2} b^{2} d^{2} e^{6}+36 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}+420 b \,c^{3} d^{7} e -2283 c^{4} d^{8}}{840 e^{9}}}{\left (e x +d \right )^{8}}+\frac {c^{4} \ln \left (e x +d \right )}{e^{9}}\)	\(889\)
norman	\(\frac {-\frac {105 a^{4} e^{8}+60 a^{3} b d \,e^{7}+20 a^{3} c \,d^{2} e^{6}+30 a^{2} b^{2} d^{2} e^{6}+36 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}+420 b \,c^{3} d^{7} e -2283 c^{4} d^{8}}{840 e^{9}}-\frac {4 \left (b e \,c^{3}-2 c^{4} d \right ) x^{7}}{e^{2}}-\frac {\left (2 e^{2} a \,c^{3}+3 b^{2} c^{2} e^{2}+14 c^{3} d e b -42 c^{4} d^{2}\right ) x^{6}}{e^{3}}-\frac {2 \left (6 a b \,c^{2} e^{3}+6 a \,c^{3} d \,e^{2}+2 b^{3} c \,e^{3}+9 d \,e^{2} b^{2} c^{2}+42 b \,c^{3} d^{2} e -154 d^{3} c^{4}\right ) x^{5}}{3 e^{4}}-\frac {\left (18 e^{4} a^{2} c^{2}+36 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}+60 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}+20 d \,e^{3} b^{3} c +90 d^{2} e^{2} b^{2} c^{2}+420 d^{3} e b \,c^{3}-1750 d^{4} c^{4}\right ) x^{4}}{12 e^{5}}-\frac {\left (36 a^{2} b c \,e^{5}+18 a^{2} c^{2} d \,e^{4}+12 a \,b^{3} e^{5}+36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}+60 a \,c^{3} d^{3} e^{2}+3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}+420 b \,c^{3} d^{4} e -1918 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {\left (20 e^{6} c \,a^{3}+30 a^{2} b^{2} e^{6}+36 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+12 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}+420 b \,c^{3} d^{5} e -2058 d^{6} c^{4}\right ) x^{2}}{30 e^{7}}-\frac {\left (60 a^{3} b \,e^{7}+20 d \,e^{6} c \,a^{3}+30 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}+60 d^{5} e^{2} a \,c^{3}+3 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}+90 b^{2} c^{2} d^{5} e^{2}+420 b \,c^{3} d^{6} e -2178 d^{7} c^{4}\right ) x}{105 e^{8}}}{\left (e x +d \right )^{8}}+\frac {c^{4} \ln \left (e x +d \right )}{e^{9}}\)	\(899\)
default	\(-\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 )}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{4 e^{9} \left (e x +d \right )^{4}}-\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}}{7 e^{9} \left (e x +d \right )^{7}}+\frac {c^{4} \ln \left (e x +d \right )}{e^{9}}-\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}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {4 c^{3} \left (b e -2 c d \right )}{e^{9} \left (e x +d \right )}-\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}}{8 e^{9} \left (e x +d \right )^{8}}-\frac {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 )^{2}}-\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}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(912\)
parallelrisch	\(\text {Expression too large to display}\)	\(1170\)

Input:

int((c*x^2+b*x+a)^4/(e*x+d)^9,x,method=_RETURNVERBOSE)
 

Output:

(-4*c^3*(b*e-2*c*d)/e^2*x^7-c^2*(2*a*c*e^2+3*b^2*e^2+14*b*c*d*e-42*c^2*d^2 
)/e^3*x^6-2/3*c*(6*a*b*c*e^3+6*a*c^2*d*e^2+2*b^3*e^3+9*b^2*c*d*e^2+42*b*c^ 
2*d^2*e-154*c^3*d^3)/e^4*x^5-1/12*(18*a^2*c^2*e^4+36*a*b^2*c*e^4+60*a*b*c^ 
2*d*e^3+60*a*c^3*d^2*e^2+3*b^4*e^4+20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2+420*b 
*c^3*d^3*e-1750*c^4*d^4)/e^5*x^4-1/15*(36*a^2*b*c*e^5+18*a^2*c^2*d*e^4+12* 
a*b^3*e^5+36*a*b^2*c*d*e^4+60*a*b*c^2*d^2*e^3+60*a*c^3*d^3*e^2+3*b^4*d*e^4 
+20*b^3*c*d^2*e^3+90*b^2*c^2*d^3*e^2+420*b*c^3*d^4*e-1918*c^4*d^5)/e^6*x^3 
-1/30*(20*a^3*c*e^6+30*a^2*b^2*e^6+36*a^2*b*c*d*e^5+18*a^2*c^2*d^2*e^4+12* 
a*b^3*d*e^5+36*a*b^2*c*d^2*e^4+60*a*b*c^2*d^3*e^3+60*a*c^3*d^4*e^2+3*b^4*d 
^2*e^4+20*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2+420*b*c^3*d^5*e-2058*c^4*d^6)/e 
^7*x^2-1/105*(60*a^3*b*e^7+20*a^3*c*d*e^6+30*a^2*b^2*d*e^6+36*a^2*b*c*d^2* 
e^5+18*a^2*c^2*d^3*e^4+12*a*b^3*d^2*e^5+36*a*b^2*c*d^3*e^4+60*a*b*c^2*d^4* 
e^3+60*a*c^3*d^5*e^2+3*b^4*d^3*e^4+20*b^3*c*d^4*e^3+90*b^2*c^2*d^5*e^2+420 
*b*c^3*d^6*e-2178*c^4*d^7)/e^8*x-1/840*(105*a^4*e^8+60*a^3*b*d*e^7+20*a^3* 
c*d^2*e^6+30*a^2*b^2*d^2*e^6+36*a^2*b*c*d^3*e^5+18*a^2*c^2*d^4*e^4+12*a*b^ 
3*d^3*e^5+36*a*b^2*c*d^4*e^4+60*a*b*c^2*d^5*e^3+60*a*c^3*d^6*e^2+3*b^4*d^4 
*e^4+20*b^3*c*d^5*e^3+90*b^2*c^2*d^6*e^2+420*b*c^3*d^7*e-2283*c^4*d^8)/e^9 
)/(e*x+d)^8+c^4*ln(e*x+d)/e^9
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (423) = 846\).

Time = 0.08 (sec) , antiderivative size = 998, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^9,x, algorithm="fricas")
 

Output:

1/840*(2283*c^4*d^8 - 420*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 105*a^4*e^8 - 30* 
(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(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 - 12*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 10*(3*a 
^2*b^2 + 2*a^3*c)*d^2*e^6 + 3360*(2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 840*(42*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 + 560*(154*c^ 
4*d^3*e^5 - 42*b*c^3*d^2*e^6 - 3*(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 + 70*(1750*c^4*d^4*e^4 - 420*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 - 3*(b^4 + 12*a*b^2 
*c + 6*a^2*c^2)*e^8)*x^4 + 56*(1918*c^4*d^5*e^3 - 420*b*c^3*d^4*e^4 - 30*( 
3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 3*(b^4 + 1 
2*a*b^2*c + 6*a^2*c^2)*d*e^7 - 12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 28*(2058* 
c^4*d^6*e^2 - 420*b*c^3*d^5*e^3 - 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 20*(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 - 12* 
(a*b^3 + 3*a^2*b*c)*d*e^7 - 10*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(2178*c^ 
4*d^7*e - 420*b*c^3*d^6*e^2 - 60*a^3*b*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^5* 
e^3 - 20*(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 - 12*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - 10*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x 
 + 840*(c^4*e^8*x^8 + 8*c^4*d*e^7*x^7 + 28*c^4*d^2*e^6*x^6 + 56*c^4*d^3*e^ 
5*x^5 + 70*c^4*d^4*e^4*x^4 + 56*c^4*d^5*e^3*x^3 + 28*c^4*d^6*e^2*x^2 + 8*c 
^4*d^7*e*x + c^4*d^8)*log(e*x + d))/(e^17*x^8 + 8*d*e^16*x^7 + 28*d^2*e...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**4/(e*x+d)**9,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (423) = 846\).

Time = 0.07 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^9,x, algorithm="maxima")
 

Output:

1/840*(2283*c^4*d^8 - 420*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 105*a^4*e^8 - 30* 
(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(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 - 12*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 10*(3*a 
^2*b^2 + 2*a^3*c)*d^2*e^6 + 3360*(2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 840*(42*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 + 560*(154*c^ 
4*d^3*e^5 - 42*b*c^3*d^2*e^6 - 3*(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 + 70*(1750*c^4*d^4*e^4 - 420*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 - 3*(b^4 + 12*a*b^2 
*c + 6*a^2*c^2)*e^8)*x^4 + 56*(1918*c^4*d^5*e^3 - 420*b*c^3*d^4*e^4 - 30*( 
3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 3*(b^4 + 1 
2*a*b^2*c + 6*a^2*c^2)*d*e^7 - 12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 28*(2058* 
c^4*d^6*e^2 - 420*b*c^3*d^5*e^3 - 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 20*(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 - 12* 
(a*b^3 + 3*a^2*b*c)*d*e^7 - 10*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(2178*c^ 
4*d^7*e - 420*b*c^3*d^6*e^2 - 60*a^3*b*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^5* 
e^3 - 20*(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 - 12*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - 10*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x 
)/(e^17*x^8 + 8*d*e^16*x^7 + 28*d^2*e^15*x^6 + 56*d^3*e^14*x^5 + 70*d^4*e^ 
13*x^4 + 56*d^5*e^12*x^3 + 28*d^6*e^11*x^2 + 8*d^7*e^10*x + d^8*e^9) + c^4 
*log(e*x + d)/e^9
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (423) = 846\).

Time = 0.35 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^9,x, algorithm="giac")
 

Output:

c^4*log(abs(e*x + d))/e^9 + 1/840*(3360*(2*c^4*d*e^6 - b*c^3*e^7)*x^7 + 84 
0*(42*c^4*d^2*e^5 - 14*b*c^3*d*e^6 - 3*b^2*c^2*e^7 - 2*a*c^3*e^7)*x^6 + 56 
0*(154*c^4*d^3*e^4 - 42*b*c^3*d^2*e^5 - 9*b^2*c^2*d*e^6 - 6*a*c^3*d*e^6 - 
2*b^3*c*e^7 - 6*a*b*c^2*e^7)*x^5 + 70*(1750*c^4*d^4*e^3 - 420*b*c^3*d^3*e^ 
4 - 90*b^2*c^2*d^2*e^5 - 60*a*c^3*d^2*e^5 - 20*b^3*c*d*e^6 - 60*a*b*c^2*d* 
e^6 - 3*b^4*e^7 - 36*a*b^2*c*e^7 - 18*a^2*c^2*e^7)*x^4 + 56*(1918*c^4*d^5* 
e^2 - 420*b*c^3*d^4*e^3 - 90*b^2*c^2*d^3*e^4 - 60*a*c^3*d^3*e^4 - 20*b^3*c 
*d^2*e^5 - 60*a*b*c^2*d^2*e^5 - 3*b^4*d*e^6 - 36*a*b^2*c*d*e^6 - 18*a^2*c^ 
2*d*e^6 - 12*a*b^3*e^7 - 36*a^2*b*c*e^7)*x^3 + 28*(2058*c^4*d^6*e - 420*b* 
c^3*d^5*e^2 - 90*b^2*c^2*d^4*e^3 - 60*a*c^3*d^4*e^3 - 20*b^3*c*d^3*e^4 - 6 
0*a*b*c^2*d^3*e^4 - 3*b^4*d^2*e^5 - 36*a*b^2*c*d^2*e^5 - 18*a^2*c^2*d^2*e^ 
5 - 12*a*b^3*d*e^6 - 36*a^2*b*c*d*e^6 - 30*a^2*b^2*e^7 - 20*a^3*c*e^7)*x^2 
 + 8*(2178*c^4*d^7 - 420*b*c^3*d^6*e - 90*b^2*c^2*d^5*e^2 - 60*a*c^3*d^5*e 
^2 - 20*b^3*c*d^4*e^3 - 60*a*b*c^2*d^4*e^3 - 3*b^4*d^3*e^4 - 36*a*b^2*c*d^ 
3*e^4 - 18*a^2*c^2*d^3*e^4 - 12*a*b^3*d^2*e^5 - 36*a^2*b*c*d^2*e^5 - 30*a^ 
2*b^2*d*e^6 - 20*a^3*c*d*e^6 - 60*a^3*b*e^7)*x + (2283*c^4*d^8 - 420*b*c^3 
*d^7*e - 90*b^2*c^2*d^6*e^2 - 60*a*c^3*d^6*e^2 - 20*b^3*c*d^5*e^3 - 60*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 - 1 
2*a*b^3*d^3*e^5 - 36*a^2*b*c*d^3*e^5 - 30*a^2*b^2*d^2*e^6 - 20*a^3*c*d^2*e 
^6 - 60*a^3*b*d*e^7 - 105*a^4*e^8)/e)/((e*x + d)^8*e^8)
 

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1168, normalized size of antiderivative = 2.69 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^4/(d + e*x)^9,x)
 

Output:

-((a^4*e^8)/8 - (761*c^4*d^8)/280 - c^4*d^8*log(d + e*x) + (b^4*d^4*e^4)/2 
80 + (b^4*e^8*x^4)/4 + (a*b^3*d^3*e^5)/70 + (a*c^3*d^6*e^2)/14 + (a^3*c*d^ 
2*e^6)/42 + (b^3*c*d^5*e^3)/42 + (4*a*b^3*e^8*x^3)/5 + (2*a^3*c*e^8*x^2)/3 
 + 2*a*c^3*e^8*x^6 + (4*b^3*c*e^8*x^5)/3 + 4*b*c^3*e^8*x^7 + (b^4*d^3*e^5* 
x)/35 + (b^4*d*e^7*x^3)/5 - 8*c^4*d*e^7*x^7 - c^4*e^8*x^8*log(d + e*x) + ( 
a^2*b^2*d^2*e^6)/28 + (3*a^2*c^2*d^4*e^4)/140 + (3*b^2*c^2*d^6*e^2)/28 + a 
^2*b^2*e^8*x^2 + (3*a^2*c^2*e^8*x^4)/2 + 3*b^2*c^2*e^8*x^6 + (b^4*d^2*e^6* 
x^2)/10 - (343*c^4*d^6*e^2*x^2)/5 - (1918*c^4*d^5*e^3*x^3)/15 - (875*c^4*d 
^4*e^4*x^4)/6 - (308*c^4*d^3*e^5*x^5)/3 - 42*c^4*d^2*e^6*x^6 + (a^3*b*d*e^ 
7)/14 + (b*c^3*d^7*e)/2 + (4*a^3*b*e^8*x)/7 - (726*c^4*d^7*e*x)/35 + (4*a^ 
3*c*d*e^7*x)/21 - 8*c^4*d^7*e*x*log(d + e*x) + (3*a^2*c^2*d^2*e^6*x^2)/5 + 
 3*b^2*c^2*d^4*e^4*x^2 + 6*b^2*c^2*d^3*e^5*x^3 + (15*b^2*c^2*d^2*e^6*x^4)/ 
2 + (a*b*c^2*d^5*e^3)/14 + (3*a*b^2*c*d^4*e^4)/70 + (3*a^2*b*c*d^3*e^5)/70 
 + (12*a^2*b*c*e^8*x^3)/5 + 3*a*b^2*c*e^8*x^4 + 4*a*b*c^2*e^8*x^5 + (4*a*b 
^3*d^2*e^6*x)/35 + (2*a^2*b^2*d*e^7*x)/7 + (2*a*b^3*d*e^7*x^2)/5 + (4*a*c^ 
3*d^5*e^3*x)/7 + 4*a*c^3*d*e^7*x^5 + 4*b*c^3*d^6*e^2*x + (4*b^3*c*d^4*e^4* 
x)/21 + (5*b^3*c*d*e^7*x^4)/3 + 14*b*c^3*d*e^7*x^6 - 8*c^4*d*e^7*x^7*log(d 
 + e*x) + (6*a^2*c^2*d^3*e^5*x)/35 + 2*a*c^3*d^4*e^4*x^2 + 4*a*c^3*d^3*e^5 
*x^3 + (6*a^2*c^2*d*e^7*x^3)/5 + 5*a*c^3*d^2*e^6*x^4 + (6*b^2*c^2*d^5*e^3* 
x)/7 + 14*b*c^3*d^5*e^3*x^2 + (2*b^3*c*d^3*e^5*x^2)/3 + 28*b*c^3*d^4*e^...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1193, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^4/(e*x+d)^9,x)
                                                                                    
                                                                                    
 

Output:

(840*log(d + e*x)*c**4*d**9 + 6720*log(d + e*x)*c**4*d**8*e*x + 23520*log( 
d + e*x)*c**4*d**7*e**2*x**2 + 47040*log(d + e*x)*c**4*d**6*e**3*x**3 + 58 
800*log(d + e*x)*c**4*d**5*e**4*x**4 + 47040*log(d + e*x)*c**4*d**4*e**5*x 
**5 + 23520*log(d + e*x)*c**4*d**3*e**6*x**6 + 6720*log(d + e*x)*c**4*d**2 
*e**7*x**7 + 840*log(d + e*x)*c**4*d*e**8*x**8 - 105*a**4*d*e**8 - 60*a**3 
*b*d**2*e**7 - 480*a**3*b*d*e**8*x - 20*a**3*c*d**3*e**6 - 160*a**3*c*d**2 
*e**7*x - 560*a**3*c*d*e**8*x**2 - 30*a**2*b**2*d**3*e**6 - 240*a**2*b**2* 
d**2*e**7*x - 840*a**2*b**2*d*e**8*x**2 - 36*a**2*b*c*d**4*e**5 - 288*a**2 
*b*c*d**3*e**6*x - 1008*a**2*b*c*d**2*e**7*x**2 - 2016*a**2*b*c*d*e**8*x** 
3 - 18*a**2*c**2*d**5*e**4 - 144*a**2*c**2*d**4*e**5*x - 504*a**2*c**2*d** 
3*e**6*x**2 - 1008*a**2*c**2*d**2*e**7*x**3 - 1260*a**2*c**2*d*e**8*x**4 - 
 12*a*b**3*d**4*e**5 - 96*a*b**3*d**3*e**6*x - 336*a*b**3*d**2*e**7*x**2 - 
 672*a*b**3*d*e**8*x**3 - 36*a*b**2*c*d**5*e**4 - 288*a*b**2*c*d**4*e**5*x 
 - 1008*a*b**2*c*d**3*e**6*x**2 - 2016*a*b**2*c*d**2*e**7*x**3 - 2520*a*b* 
*2*c*d*e**8*x**4 - 60*a*b*c**2*d**6*e**3 - 480*a*b*c**2*d**5*e**4*x - 1680 
*a*b*c**2*d**4*e**5*x**2 - 3360*a*b*c**2*d**3*e**6*x**3 - 4200*a*b*c**2*d* 
*2*e**7*x**4 - 3360*a*b*c**2*d*e**8*x**5 - 60*a*c**3*d**7*e**2 - 480*a*c** 
3*d**6*e**3*x - 1680*a*c**3*d**5*e**4*x**2 - 3360*a*c**3*d**4*e**5*x**3 - 
4200*a*c**3*d**3*e**6*x**4 - 3360*a*c**3*d**2*e**7*x**5 - 1680*a*c**3*d*e* 
*8*x**6 - 3*b**4*d**5*e**4 - 24*b**4*d**4*e**5*x - 84*b**4*d**3*e**6*x*...