\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^5} \, dx\) [453]

Optimal result

Integrand size = 20, antiderivative size = 426 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=-\frac {c \left (35 c^3 d^3-4 b^3 e^3+6 b c e^2 (5 b d-2 a e)-20 c^2 d e (3 b d-a e)\right ) x}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x^2}{2 e^7}-\frac {c^3 (5 c d-4 b e) x^3}{3 e^6}+\frac {c^4 x^4}{4 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{e^9} \] Output:

-c*(35*c^3*d^3-4*b^3*e^3+6*b*c*e^2*(-2*a*e+5*b*d)-20*c^2*d*e*(-a*e+3*b*d)) 
*x/e^8+1/2*c^2*(15*c^2*d^2+6*b^2*e^2-4*c*e*(-a*e+5*b*d))*x^2/e^7-1/3*c^3*( 
-4*b*e+5*c*d)*x^3/e^6+1/4*c^4*x^4/e^5-1/4*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d 
)^4+4/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^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)^2+4*(-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)+( 
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))*ln(e*x+d)/e^9
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\frac {12 c e \left (-35 c^3 d^3+4 b^3 e^3-6 b c e^2 (5 b d-2 a e)+20 c^2 d e (3 b d-a e)\right ) x+6 c^2 e^2 \left (15 c^2 d^2+6 b^2 e^2+4 c e (-5 b d+a e)\right ) x^2+4 c^3 e^3 (-5 c d+4 b e) x^3+3 c^4 e^4 x^4-\frac {3 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^4}+\frac {16 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {12 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}+\frac {48 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right )}{d+e x}+12 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{12 e^9} \] Input:

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^5,x]
 

Output:

(12*c*e*(-35*c^3*d^3 + 4*b^3*e^3 - 6*b*c*e^2*(5*b*d - 2*a*e) + 20*c^2*d*e* 
(3*b*d - a*e))*x + 6*c^2*e^2*(15*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-5*b*d + a*e 
))*x^2 + 4*c^3*e^3*(-5*c*d + 4*b*e)*x^3 + 3*c^4*e^4*x^4 - (3*(c*d^2 + e*(- 
(b*d) + a*e))^4)/(d + e*x)^4 + (16*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e) 
)^3)/(d + e*x)^3 - (12*(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)^2 + (48*(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) + 12*(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))*Log[d + e*x])/(12*e^9)
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 426, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)}+\frac {c \left (20 c^2 d e (3 b d-a e)-6 b c e^2 (5 b d-2 a e)+4 b^3 e^3-35 c^3 d^3\right )}{e^8}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8 (d+e x)^2}+\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (d+e x)^3}+\frac {c^2 x \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{e^7}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^4}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^5}-\frac {c^3 x^2 (5 c d-4 b e)}{e^6}+\frac {c^4 x^3}{e^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}-\frac {c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac {\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}+\frac {c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac {\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac {c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac {c^4 x^4}{4 e^5}\)

Input:

Int[(a + b*x + c*x^2)^4/(d + e*x)^5,x]
 

Output:

-((c*(35*c^3*d^3 - 4*b^3*e^3 + 6*b*c*e^2*(5*b*d - 2*a*e) - 20*c^2*d*e*(3*b 
*d - a*e))*x)/e^8) + (c^2*(15*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(5*b*d - a*e))*x 
^2)/(2*e^7) - (c^3*(5*c*d - 4*b*e)*x^3)/(3*e^6) + (c^4*x^4)/(4*e^5) - (c*d 
^2 - b*d*e + a*e^2)^4/(4*e^9*(d + e*x)^4) + (4*(2*c*d - b*e)*(c*d^2 - b*d* 
e + a*e^2)^3)/(3*e^9*(d + e*x)^3) - ((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)))/(e^9*(d + e*x)^2) + (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)))/(e^9* 
(d + e*x)) + ((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))*Lo 
g[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. \(887\) vs. \(2(416)=832\).

Time = 0.90 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.08

method	result	size
norman	\(\frac {-\frac {3 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}+36 a^{2} b c \,d^{3} e^{5}-150 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}-300 a \,b^{2} c \,d^{4} e^{4}+1500 a b \,c^{2} d^{5} e^{3}-1500 a \,c^{3} d^{6} e^{2}-25 b^{4} d^{4} e^{4}+500 b^{3} c \,d^{5} e^{3}-2250 b^{2} c^{2} d^{6} e^{2}+3500 b \,c^{3} d^{7} e -1750 c^{4} d^{8}}{12 e^{9}}+\frac {c^{4} x^{8}}{4 e}-\frac {4 \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}+60 a b \,c^{2} d^{2} e^{3}-60 a \,c^{3} d^{3} e^{2}-b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-90 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -70 c^{4} d^{5}\right ) x^{3}}{e^{6}}-\frac {\left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}-54 d^{2} e^{4} a^{2} c^{2}+6 a \,b^{3} d \,e^{5}-108 a \,b^{2} c \,d^{2} e^{4}+540 a b \,c^{2} d^{3} e^{3}-540 d^{4} e^{2} a \,c^{3}-9 b^{4} d^{2} e^{4}+180 b^{3} c \,d^{3} e^{3}-810 b^{2} c^{2} d^{4} e^{2}+1260 b \,c^{3} d^{5} e -630 d^{6} c^{4}\right ) x^{2}}{e^{7}}-\frac {2 \left (2 a^{3} b \,e^{7}+2 d \,e^{6} c \,a^{3}+3 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}-66 d^{3} e^{4} a^{2} c^{2}+6 a \,b^{3} d^{2} e^{5}-132 a \,b^{2} c \,d^{3} e^{4}+660 a b \,c^{2} d^{4} e^{3}-660 d^{5} e^{2} a \,c^{3}-11 b^{4} d^{3} e^{4}+220 b^{3} c \,d^{4} e^{3}-990 b^{2} c^{2} d^{5} e^{2}+1540 b \,c^{3} d^{6} e -770 d^{7} c^{4}\right ) x}{3 e^{8}}+\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 +14 d^{2} e b \,c^{2}-7 d^{3} c^{3}\right ) x^{5}}{e^{4}}+\frac {c^{2} \left (6 a c \,e^{2}+9 b^{2} e^{2}-14 b c d e +7 c^{2} d^{2}\right ) x^{6}}{3 e^{3}}+\frac {2 c^{3} \left (2 b e -c d \right ) x^{7}}{3 e^{2}}}{\left (e x +d \right )^{4}}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(888\)
default	\(\frac {c \left (\frac {1}{4} c^{3} x^{4} e^{3}+\frac {4}{3} b \,c^{2} e^{3} x^{3}-\frac {5}{3} c^{3} d \,e^{2} x^{3}+2 a \,c^{2} e^{3} x^{2}+3 b^{2} c \,e^{3} x^{2}-10 b \,c^{2} d \,e^{2} x^{2}+\frac {15}{2} c^{3} d^{2} e \,x^{2}+12 a b c \,e^{3} x -20 d \,e^{2} a \,c^{2} x +4 b^{3} e^{3} x -30 d \,e^{2} b^{2} c x +60 d^{2} e b \,c^{2} x -35 d^{3} c^{3} x \right )}{e^{8}}-\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}}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{4 e^{9} \left (e x +d \right )^{4}}+\frac {\left (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}\right ) \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}}{e^{9} \left (e x +d \right )}-\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}}{2 e^{9} \left (e x +d \right )^{2}}\)	\(912\)
risch	\(\frac {c^{4} x^{4}}{4 e^{5}}+\frac {4 c^{3} b \,x^{3}}{3 e^{5}}-\frac {5 c^{4} d \,x^{3}}{3 e^{6}}+\frac {2 c^{3} a \,x^{2}}{e^{5}}+\frac {3 c^{2} b^{2} x^{2}}{e^{5}}-\frac {10 c^{3} b d \,x^{2}}{e^{6}}+\frac {15 c^{4} d^{2} x^{2}}{2 e^{7}}+\frac {12 c^{2} a b x}{e^{5}}-\frac {20 c^{3} d a x}{e^{6}}+\frac {4 c \,b^{3} x}{e^{5}}-\frac {30 c^{2} d \,b^{2} x}{e^{6}}+\frac {60 c^{3} d^{2} b x}{e^{7}}-\frac {35 c^{4} d^{3} x}{e^{8}}+\frac {\left (-12 a^{2} b c \,e^{7}+24 a^{2} c^{2} d \,e^{6}-4 a \,b^{3} e^{7}+48 a \,b^{2} c d \,e^{6}-120 a b \,c^{2} d^{2} e^{5}+80 a \,c^{3} d^{3} e^{4}+4 b^{4} d \,e^{6}-40 b^{3} c \,d^{2} e^{5}+120 b^{2} c^{2} d^{3} e^{4}-140 d^{4} b \,c^{3} e^{3}+56 c^{4} d^{5} e^{2}\right ) x^{3}-e \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}+18 a^{2} b c d \,e^{5}-54 d^{2} e^{4} a^{2} c^{2}+6 a \,b^{3} d \,e^{5}-108 a \,b^{2} c \,d^{2} e^{4}+300 a b \,c^{2} d^{3} e^{3}-210 d^{4} e^{2} a \,c^{3}-9 b^{4} d^{2} e^{4}+100 b^{3} c \,d^{3} e^{3}-315 b^{2} c^{2} d^{4} e^{2}+378 b \,c^{3} d^{5} e -154 d^{6} c^{4}\right ) x^{2}+\left (-\frac {4}{3} a^{3} b \,e^{7}-\frac {4}{3} d \,e^{6} c \,a^{3}-2 a^{2} b^{2} d \,e^{6}-12 a^{2} b c \,d^{2} e^{5}+44 d^{3} e^{4} a^{2} c^{2}-4 a \,b^{3} d^{2} e^{5}+88 a \,b^{2} c \,d^{3} e^{4}-260 a b \,c^{2} d^{4} e^{3}+188 d^{5} e^{2} a \,c^{3}+\frac {22}{3} b^{4} d^{3} e^{4}-\frac {260}{3} b^{3} c \,d^{4} e^{3}+282 b^{2} c^{2} d^{5} e^{2}-\frac {1036}{3} b \,c^{3} d^{6} e +\frac {428}{3} d^{7} c^{4}\right ) x -\frac {3 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}+36 a^{2} b c \,d^{3} e^{5}-150 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}-300 a \,b^{2} c \,d^{4} e^{4}+924 a b \,c^{2} d^{5} e^{3}-684 a \,c^{3} d^{6} e^{2}-25 b^{4} d^{4} e^{4}+308 b^{3} c \,d^{5} e^{3}-1026 b^{2} c^{2} d^{6} e^{2}+1276 b \,c^{3} d^{7} e -533 c^{4} d^{8}}{12 e}}{e^{8} \left (e x +d \right )^{4}}+\frac {6 \ln \left (e x +d \right ) a^{2} c^{2}}{e^{5}}+\frac {12 \ln \left (e x +d \right ) a \,b^{2} c}{e^{5}}-\frac {60 \ln \left (e x +d \right ) a b \,c^{2} d}{e^{6}}+\frac {60 \ln \left (e x +d \right ) d^{2} a \,c^{3}}{e^{7}}+\frac {b^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {20 \ln \left (e x +d \right ) d \,b^{3} c}{e^{6}}+\frac {90 \ln \left (e x +d \right ) d^{2} b^{2} c^{2}}{e^{7}}-\frac {140 \ln \left (e x +d \right ) d^{3} b \,c^{3}}{e^{8}}+\frac {70 \ln \left (e x +d \right ) d^{4} c^{4}}{e^{9}}\)	\(957\)
parallelrisch	\(\text {Expression too large to display}\)	\(1784\)

Input:

int((c*x^2+b*x+a)^4/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

(-1/12*(3*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+36*a^2*b 
*c*d^3*e^5-150*a^2*c^2*d^4*e^4+12*a*b^3*d^3*e^5-300*a*b^2*c*d^4*e^4+1500*a 
*b*c^2*d^5*e^3-1500*a*c^3*d^6*e^2-25*b^4*d^4*e^4+500*b^3*c*d^5*e^3-2250*b^ 
2*c^2*d^6*e^2+3500*b*c^3*d^7*e-1750*c^4*d^8)/e^9+1/4*c^4/e*x^8-4*(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+60*a*b*c^2*d^2*e^3-60*a*c 
^3*d^3*e^2-b^4*d*e^4+20*b^3*c*d^2*e^3-90*b^2*c^2*d^3*e^2+140*b*c^3*d^4*e-7 
0*c^4*d^5)/e^6*x^3-(2*a^3*c*e^6+3*a^2*b^2*e^6+18*a^2*b*c*d*e^5-54*a^2*c^2* 
d^2*e^4+6*a*b^3*d*e^5-108*a*b^2*c*d^2*e^4+540*a*b*c^2*d^3*e^3-540*a*c^3*d^ 
4*e^2-9*b^4*d^2*e^4+180*b^3*c*d^3*e^3-810*b^2*c^2*d^4*e^2+1260*b*c^3*d^5*e 
-630*c^4*d^6)/e^7*x^2-2/3*(2*a^3*b*e^7+2*a^3*c*d*e^6+3*a^2*b^2*d*e^6+18*a^ 
2*b*c*d^2*e^5-66*a^2*c^2*d^3*e^4+6*a*b^3*d^2*e^5-132*a*b^2*c*d^3*e^4+660*a 
*b*c^2*d^4*e^3-660*a*c^3*d^5*e^2-11*b^4*d^3*e^4+220*b^3*c*d^4*e^3-990*b^2* 
c^2*d^5*e^2+1540*b*c^3*d^6*e-770*c^4*d^7)/e^8*x+2*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+14*b*c^2*d^2*e-7*c^3*d^3)/e^4*x^5+1/3*c^2*(6* 
a*c*e^2+9*b^2*e^2-14*b*c*d*e+7*c^2*d^2)/e^3*x^6+2/3*c^3*(2*b*e-c*d)/e^2*x^ 
7)/(e*x+d)^4+1/e^9*(6*a^2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3 
*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d^3*e+70*c^4* 
d^4)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1303 vs. \(2 (416) = 832\).

Time = 0.09 (sec) , antiderivative size = 1303, normalized size of antiderivative = 3.06 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^5,x, algorithm="fricas")
 

Output:

1/12*(3*c^4*e^8*x^8 + 533*c^4*d^8 - 1276*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 3*a 
^4*e^8 + 342*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 308*(b^3*c + 3*a*b*c^2)*d^5*e 
^3 + 25*(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 - 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 8*(c^4*d*e^7 - 2*b*c^3*e^8)*x^7 
+ 4*(7*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 2 
4*(7*c^4*d^3*e^5 - 14*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 - (1217*c^4*d^4*e^4 - 2224*b*c^3*d^3*e^5 + 408* 
(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 192*(b^3*c + 3*a*b*c^2)*d*e^7)*x^4 - 4*(37 
7*c^4*d^5*e^3 - 544*b*c^3*d^4*e^4 + 48*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 48* 
(b^3*c + 3*a*b*c^2)*d^2*e^6 - 12*(b^4 + 12*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 + 6*(43*c^4*d^6*e^2 - 296*b*c^3*d^5*e^3 + 13 
2*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 168*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 18*(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 - 2*(3* 
a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 4*(323*c^4*d^7*e - 856*b*c^3*d^6*e^2 - 4*a^3 
*b*e^8 + 252*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 248*(b^3*c + 3*a*b*c^2)*d^4*e 
^4 + 22*(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 - 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 12*(70*c^4*d^8 - 140*b*c^3*d^7* 
e + 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 + (b 
^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + (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...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**4/(e*x+d)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (416) = 832\).

Time = 0.07 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^5,x, algorithm="maxima")
 

Output:

1/12*(533*c^4*d^8 - 1276*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 3*a^4*e^8 + 342*(3* 
b^2*c^2 + 2*a*c^3)*d^6*e^2 - 308*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 25*(b^4 + 1 
2*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 - 2*(3*a^2 
*b^2 + 2*a^3*c)*d^2*e^6 + 48*(14*c^4*d^5*e^3 - 35*b*c^3*d^4*e^4 + 10*(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 - (a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 12*(154*c^4*d^6*e^ 
2 - 378*b*c^3*d^5*e^3 + 105*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 100*(b^3*c + 3 
*a*b*c^2)*d^3*e^5 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 6*(a*b^3 + 
3*a^2*b*c)*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(214*c^4*d^7*e - 518 
*b*c^3*d^6*e^2 - 2*a^3*b*e^8 + 141*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 130*(b^ 
3*c + 3*a*b*c^2)*d^4*e^4 + 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 6*( 
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^13*x^4 + 4 
*d*e^12*x^3 + 6*d^2*e^11*x^2 + 4*d^3*e^10*x + d^4*e^9) + 1/12*(3*c^4*e^3*x 
^4 - 4*(5*c^4*d*e^2 - 4*b*c^3*e^3)*x^3 + 6*(15*c^4*d^2*e - 20*b*c^3*d*e^2 
+ 2*(3*b^2*c^2 + 2*a*c^3)*e^3)*x^2 - 12*(35*c^4*d^3 - 60*b*c^3*d^2*e + 10* 
(3*b^2*c^2 + 2*a*c^3)*d*e^2 - 4*(b^3*c + 3*a*b*c^2)*e^3)*x)/e^8 + (70*c^4* 
d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a 
*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^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. 1301 vs. \(2 (416) = 832\).

Time = 0.42 (sec) , antiderivative size = 1301, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^4/(e*x+d)^5,x, algorithm="giac")
 

Output:

1/12*(3*c^4 - 16*(2*c^4*d*e - b*c^3*e^2)/((e*x + d)*e) + 12*(14*c^4*d^2*e^ 
2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*e^4 + 2*a*c^3*e^4)/((e*x + d)^2*e^2) - 48*( 
14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e^5 - b^3* 
c*e^6 - 3*a*b*c^2*e^6)/((e*x + d)^3*e^3))*(e*x + d)^4/e^9 - (70*c^4*d^4 - 
140*b*c^3*d^3*e + 90*b^2*c^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20*b^3*c*d*e^3 - 
 60*a*b*c^2*d*e^3 + b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*log(abs(e*x 
+ d)/((e*x + d)^2*abs(e)))/e^9 + 1/12*(672*c^4*d^5*e^43/(e*x + d) - 168*c^ 
4*d^6*e^43/(e*x + d)^2 + 32*c^4*d^7*e^43/(e*x + d)^3 - 3*c^4*d^8*e^43/(e*x 
 + d)^4 - 1680*b*c^3*d^4*e^44/(e*x + d) + 504*b*c^3*d^5*e^44/(e*x + d)^2 - 
 112*b*c^3*d^6*e^44/(e*x + d)^3 + 12*b*c^3*d^7*e^44/(e*x + d)^4 + 1440*b^2 
*c^2*d^3*e^45/(e*x + d) + 960*a*c^3*d^3*e^45/(e*x + d) - 540*b^2*c^2*d^4*e 
^45/(e*x + d)^2 - 360*a*c^3*d^4*e^45/(e*x + d)^2 + 144*b^2*c^2*d^5*e^45/(e 
*x + d)^3 + 96*a*c^3*d^5*e^45/(e*x + d)^3 - 18*b^2*c^2*d^6*e^45/(e*x + d)^ 
4 - 12*a*c^3*d^6*e^45/(e*x + d)^4 - 480*b^3*c*d^2*e^46/(e*x + d) - 1440*a* 
b*c^2*d^2*e^46/(e*x + d) + 240*b^3*c*d^3*e^46/(e*x + d)^2 + 720*a*b*c^2*d^ 
3*e^46/(e*x + d)^2 - 80*b^3*c*d^4*e^46/(e*x + d)^3 - 240*a*b*c^2*d^4*e^46/ 
(e*x + d)^3 + 12*b^3*c*d^5*e^46/(e*x + d)^4 + 36*a*b*c^2*d^5*e^46/(e*x + d 
)^4 + 48*b^4*d*e^47/(e*x + d) + 576*a*b^2*c*d*e^47/(e*x + d) + 288*a^2*c^2 
*d*e^47/(e*x + d) - 36*b^4*d^2*e^47/(e*x + d)^2 - 432*a*b^2*c*d^2*e^47/(e* 
x + d)^2 - 216*a^2*c^2*d^2*e^47/(e*x + d)^2 + 16*b^4*d^3*e^47/(e*x + d)...
 

Mupad [B] (verification not implemented)

Time = 5.34 (sec) , antiderivative size = 1005, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((a + b*x + c*x^2)^4/(d + e*x)^5,x)
 

Output:

x^3*((4*b*c^3)/(3*e^5) - (5*c^4*d)/(3*e^6)) - (x*((4*a^3*b*e^7)/3 - (428*c 
^4*d^7)/3 - (22*b^4*d^3*e^4)/3 + 4*a*b^3*d^2*e^5 + 2*a^2*b^2*d*e^6 - 188*a 
*c^3*d^5*e^2 + (260*b^3*c*d^4*e^3)/3 - 44*a^2*c^2*d^3*e^4 - 282*b^2*c^2*d^ 
5*e^2 + (4*a^3*c*d*e^6)/3 + (1036*b*c^3*d^6*e)/3 + 260*a*b*c^2*d^4*e^3 - 8 
8*a*b^2*c*d^3*e^4 + 12*a^2*b*c*d^2*e^5) - x^3*(4*b^4*d*e^6 - 4*a*b^3*e^7 + 
 56*c^4*d^5*e^2 + 80*a*c^3*d^3*e^4 + 24*a^2*c^2*d*e^6 - 140*b*c^3*d^4*e^3 
- 40*b^3*c*d^2*e^5 + 120*b^2*c^2*d^3*e^4 - 12*a^2*b*c*e^7 + 48*a*b^2*c*d*e 
^6 - 120*a*b*c^2*d^2*e^5) + (3*a^4*e^8 - 533*c^4*d^8 - 25*b^4*d^4*e^4 + 12 
*a*b^3*d^3*e^5 - 684*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 308*b^3*c*d^5*e^3 + 
 6*a^2*b^2*d^2*e^6 - 150*a^2*c^2*d^4*e^4 - 1026*b^2*c^2*d^6*e^2 + 4*a^3*b* 
d*e^7 + 1276*b*c^3*d^7*e + 924*a*b*c^2*d^5*e^3 - 300*a*b^2*c*d^4*e^4 + 36* 
a^2*b*c*d^3*e^5)/(12*e) + x^2*(2*a^3*c*e^7 - 154*c^4*d^6*e + 3*a^2*b^2*e^7 
 - 9*b^4*d^2*e^5 - 210*a*c^3*d^4*e^3 + 378*b*c^3*d^5*e^2 + 100*b^3*c*d^3*e 
^4 - 54*a^2*c^2*d^2*e^5 - 315*b^2*c^2*d^4*e^3 + 6*a*b^3*d*e^6 + 18*a^2*b*c 
*d*e^6 + 300*a*b*c^2*d^3*e^4 - 108*a*b^2*c*d^2*e^5))/(d^4*e^8 + e^12*x^4 + 
 4*d^3*e^9*x + 4*d*e^11*x^3 + 6*d^2*e^10*x^2) - x^2*((5*d*((4*b*c^3)/e^5 - 
 (5*c^4*d)/e^6))/(2*e) - (4*a*c^3 + 6*b^2*c^2)/(2*e^5) + (5*c^4*d^2)/e^7) 
- x*((10*c^4*d^3)/e^8 + (10*d^2*((4*b*c^3)/e^5 - (5*c^4*d)/e^6))/e^2 - (5* 
d*((5*d*((4*b*c^3)/e^5 - (5*c^4*d)/e^6))/e - (4*a*c^3 + 6*b^2*c^2)/e^5 + ( 
10*c^4*d^2)/e^7))/e - (4*b*c*(3*a*c + b^2))/e^5) + (c^4*x^4)/(4*e^5) + ...
 

Reduce [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 1777, normalized size of antiderivative = 4.17 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^4/(e*x+d)^5,x)
                                                                                    
                                                                                    
 

Output:

(72*log(d + e*x)*a**2*c**2*d**5*e**4 + 288*log(d + e*x)*a**2*c**2*d**4*e** 
5*x + 432*log(d + e*x)*a**2*c**2*d**3*e**6*x**2 + 288*log(d + e*x)*a**2*c* 
*2*d**2*e**7*x**3 + 72*log(d + e*x)*a**2*c**2*d*e**8*x**4 + 144*log(d + e* 
x)*a*b**2*c*d**5*e**4 + 576*log(d + e*x)*a*b**2*c*d**4*e**5*x + 864*log(d 
+ e*x)*a*b**2*c*d**3*e**6*x**2 + 576*log(d + e*x)*a*b**2*c*d**2*e**7*x**3 
+ 144*log(d + e*x)*a*b**2*c*d*e**8*x**4 - 720*log(d + e*x)*a*b*c**2*d**6*e 
**3 - 2880*log(d + e*x)*a*b*c**2*d**5*e**4*x - 4320*log(d + e*x)*a*b*c**2* 
d**4*e**5*x**2 - 2880*log(d + e*x)*a*b*c**2*d**3*e**6*x**3 - 720*log(d + e 
*x)*a*b*c**2*d**2*e**7*x**4 + 720*log(d + e*x)*a*c**3*d**7*e**2 + 2880*log 
(d + e*x)*a*c**3*d**6*e**3*x + 4320*log(d + e*x)*a*c**3*d**5*e**4*x**2 + 2 
880*log(d + e*x)*a*c**3*d**4*e**5*x**3 + 720*log(d + e*x)*a*c**3*d**3*e**6 
*x**4 + 12*log(d + e*x)*b**4*d**5*e**4 + 48*log(d + e*x)*b**4*d**4*e**5*x 
+ 72*log(d + e*x)*b**4*d**3*e**6*x**2 + 48*log(d + e*x)*b**4*d**2*e**7*x** 
3 + 12*log(d + e*x)*b**4*d*e**8*x**4 - 240*log(d + e*x)*b**3*c*d**6*e**3 - 
 960*log(d + e*x)*b**3*c*d**5*e**4*x - 1440*log(d + e*x)*b**3*c*d**4*e**5* 
x**2 - 960*log(d + e*x)*b**3*c*d**3*e**6*x**3 - 240*log(d + e*x)*b**3*c*d* 
*2*e**7*x**4 + 1080*log(d + e*x)*b**2*c**2*d**7*e**2 + 4320*log(d + e*x)*b 
**2*c**2*d**6*e**3*x + 6480*log(d + e*x)*b**2*c**2*d**5*e**4*x**2 + 4320*l 
og(d + e*x)*b**2*c**2*d**4*e**5*x**3 + 1080*log(d + e*x)*b**2*c**2*d**3*e* 
*6*x**4 - 1680*log(d + e*x)*b*c**3*d**8*e - 6720*log(d + e*x)*b*c**3*d*...