\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^4} \, dx\) [452]

Optimal result

Integrand size = 20, antiderivative size = 417 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\frac {\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac {2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac {c^3 (c d-b e) x^4}{e^5}+\frac {c^4 x^5}{5 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^4}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^2}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \] Output:

(35*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+4*b*d)-40*c^3*d^2*e*(-a*e+2*b*d)+6 
*c^2*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2))*x/e^8-2*c*(5*c^3*d^3-b^3*e^3-2*c^ 
2*d*e*(-2*a*e+5*b*d)+3*b*c*e^2*(-a*e+2*b*d))*x^2/e^7+2/3*c^2*(5*c^2*d^2+3* 
b^2*e^2-2*c*e*(-a*e+4*b*d))*x^3/e^6-c^3*(-b*e+c*d)*x^4/e^5+1/5*c^4*x^5/e^4 
-1/3*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^3+2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2 
)^3/e^9/(e*x+d)^2-2*(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)-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))*ln(e*x+d)/e^9
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\frac {15 e \left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (-2 b d+a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x+30 c e^2 \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)+3 b c e^2 (-2 b d+a e)\right ) x^2+10 c^2 e^3 \left (5 c^2 d^2+3 b^2 e^2+2 c e (-4 b d+a e)\right ) x^3+15 c^3 e^4 (-c d+b e) x^4+3 c^4 e^5 x^5-\frac {5 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^3}+\frac {30 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}-\frac {30 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{d+e x}-60 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right ) \log (d+e x)}{15 e^9} \] Input:

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

Output:

(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*( 
-2*b*d + a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2 
*(-5*c^3*d^3 + b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d + a 
*e))*x^2 + 10*c^2*e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 
 15*c^3*e^4*(-(c*d) + b*e)*x^4 + 3*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a 
*e))^4)/(d + e*x)^3 + (30*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + 
 e*x)^2 - (30*(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) - 60*(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))*Log[d + e*x])/(15*e^9)
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 417, 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.

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

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4}{e^8}+\frac {4 c x \left (2 c^2 d e (5 b d-2 a e)-3 b c e^2 (2 b d-a e)+b^3 e^3-5 c^3 d^3\right )}{e^7}+\frac {2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^8 (d+e x)^2}+\frac {4 (2 c d-b e) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^8 (d+e x)}+\frac {2 c^2 x^2 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {\left (a e^2-b d e+c d^2\right )^4}{e^8 (d+e x)^4}+\frac {4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)^3}-\frac {4 c^3 x^3 (c d-b e)}{e^5}+\frac {c^4 x^4}{e^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}-\frac {2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac {c^3 x^4 (c d-b e)}{e^5}+\frac {c^4 x^5}{5 e^4}\)

Input:

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

Output:

((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d 
 - a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c 
^3*d^3 - b^3*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*x^ 
2)/e^7 + (2*c^2*(5*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) 
 - (c^3*(c*d - b*e)*x^4)/e^5 + (c^4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2) 
^4/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9* 
(d + e*x)^2) - (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)))/(e^9*(d + e*x)) - (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))*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. \(886\) vs. \(2(411)=822\).

Time = 0.92 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.13

method	result	size
norman	\(\frac {\frac {\left (6 e^{4} a^{2} c^{2}+12 a \,b^{2} c \,e^{4}-30 a b \,c^{2} d \,e^{3}+20 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-10 d \,e^{3} b^{3} c +30 d^{2} e^{2} b^{2} c^{2}-35 d^{3} e b \,c^{3}+14 d^{4} c^{4}\right ) x^{4}}{e^{5}}-\frac {a^{4} e^{8}+2 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-66 a^{2} b c \,d^{3} e^{5}+132 a^{2} c^{2} d^{4} e^{4}-22 a \,b^{3} d^{3} e^{5}+264 a \,b^{2} c \,d^{4} e^{4}-660 a b \,c^{2} d^{5} e^{3}+440 a \,c^{3} d^{6} e^{2}+22 b^{4} d^{4} e^{4}-220 b^{3} c \,d^{5} e^{3}+660 b^{2} c^{2} d^{6} e^{2}-770 b \,c^{3} d^{7} e +308 c^{4} d^{8}}{3 e^{9}}+\frac {c^{4} x^{8}}{5 e}-\frac {\left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+72 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+144 a \,b^{2} c \,d^{2} e^{4}-360 a b \,c^{2} d^{3} e^{3}+240 d^{4} e^{2} a \,c^{3}+12 b^{4} d^{2} e^{4}-120 b^{3} c \,d^{3} e^{3}+360 b^{2} c^{2} d^{4} e^{2}-420 b \,c^{3} d^{5} e +168 d^{6} c^{4}\right ) x^{2}}{e^{7}}-\frac {\left (2 a^{3} b \,e^{7}+4 d \,e^{6} c \,a^{3}+6 a^{2} b^{2} d \,e^{6}-54 a^{2} b c \,d^{2} e^{5}+108 d^{3} e^{4} a^{2} c^{2}-18 a \,b^{3} d^{2} e^{5}+216 a \,b^{2} c \,d^{3} e^{4}-540 a b \,c^{2} d^{4} e^{3}+360 d^{5} e^{2} a \,c^{3}+18 b^{4} d^{3} e^{4}-180 b^{3} c \,d^{4} e^{3}+540 b^{2} c^{2} d^{5} e^{2}-630 b \,c^{3} d^{6} e +252 d^{7} c^{4}\right ) x}{e^{8}}+\frac {c \left (30 a b c \,e^{3}-20 d \,e^{2} a \,c^{2}+10 b^{3} e^{3}-30 d \,e^{2} b^{2} c +35 d^{2} e b \,c^{2}-14 d^{3} c^{3}\right ) x^{5}}{5 e^{4}}+\frac {c^{2} \left (20 a c \,e^{2}+30 b^{2} e^{2}-35 b c d e +14 c^{2} d^{2}\right ) x^{6}}{15 e^{3}}+\frac {c^{3} \left (5 b e -2 c d \right ) x^{7}}{5 e^{2}}}{\left (e x +d \right )^{3}}+\frac {4 \left (3 a^{2} b c \,e^{5}-6 a^{2} c^{2} d \,e^{4}+a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}-20 a \,c^{3} d^{3} e^{2}-b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}-30 b^{2} c^{2} d^{3} e^{2}+35 b \,c^{3} d^{4} e -14 c^{4} d^{5}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(887\)
default	\(\frac {\frac {1}{5} c^{4} e^{4} x^{5}+b \,c^{3} e^{4} x^{4}-c^{4} d \,e^{3} x^{4}+\frac {4}{3} a \,c^{3} e^{4} x^{3}+2 b^{2} c^{2} e^{4} x^{3}-\frac {16}{3} b \,c^{3} d \,e^{3} x^{3}+\frac {10}{3} c^{4} d^{2} e^{2} x^{3}+6 a b \,c^{2} e^{4} x^{2}-8 a \,c^{3} d \,e^{3} x^{2}+2 x^{2} b^{3} c \,e^{4}-12 b^{2} c^{2} d \,e^{3} x^{2}+20 b \,c^{3} d^{2} e^{2} x^{2}-10 c^{4} d^{3} e \,x^{2}+6 e^{4} a^{2} c^{2} x +12 a \,b^{2} c \,e^{4} x -48 a b \,c^{2} d \,e^{3} x +40 d^{2} e^{2} a \,c^{3} x +b^{4} e^{4} x -16 d \,e^{3} b^{3} c x +60 d^{2} e^{2} b^{2} c^{2} x -80 d^{3} e b \,c^{3} x +35 c^{4} d^{4} x}{e^{8}}-\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}}{3 e^{9} \left (e x +d \right )^{3}}+\frac {\left (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}\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}}{e^{9} \left (e x +d \right )}-\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}}{2 e^{9} \left (e x +d \right )^{2}}\)	\(930\)
risch	\(\frac {40 \ln \left (e x +d \right ) b^{3} c \,d^{2}}{e^{6}}-\frac {120 \ln \left (e x +d \right ) b^{2} c^{2} d^{3}}{e^{7}}+\frac {140 \ln \left (e x +d \right ) b \,c^{3} d^{4}}{e^{8}}+\frac {4 \ln \left (e x +d \right ) a \,b^{3}}{e^{4}}-\frac {4 \ln \left (e x +d \right ) b^{4} d}{e^{5}}-\frac {56 \ln \left (e x +d \right ) c^{4} d^{5}}{e^{9}}+\frac {12 a \,b^{2} c x}{e^{4}}+\frac {40 d^{2} a \,c^{3} x}{e^{6}}-\frac {16 d \,b^{3} c x}{e^{5}}+\frac {60 d^{2} b^{2} c^{2} x}{e^{6}}-\frac {80 d^{3} b \,c^{3} x}{e^{7}}+\frac {12 \ln \left (e x +d \right ) a^{2} b c}{e^{4}}-\frac {24 \ln \left (e x +d \right ) a^{2} c^{2} d}{e^{5}}-\frac {80 \ln \left (e x +d \right ) a \,c^{3} d^{3}}{e^{7}}-\frac {16 b \,c^{3} d \,x^{3}}{3 e^{5}}+\frac {6 a b \,c^{2} x^{2}}{e^{4}}+\frac {b^{4} x}{e^{4}}+\frac {b \,c^{3} x^{4}}{e^{4}}-\frac {c^{4} d \,x^{4}}{e^{5}}+\frac {4 a \,c^{3} x^{3}}{3 e^{4}}+\frac {2 b^{2} c^{2} x^{3}}{e^{4}}+\frac {10 c^{4} d^{2} x^{3}}{3 e^{6}}+\frac {2 x^{2} b^{3} c}{e^{4}}-\frac {10 c^{4} d^{3} x^{2}}{e^{7}}+\frac {6 a^{2} c^{2} x}{e^{4}}+\frac {35 c^{4} d^{4} x}{e^{8}}-\frac {8 a \,c^{3} d \,x^{2}}{e^{5}}-\frac {12 b^{2} c^{2} d \,x^{2}}{e^{5}}+\frac {20 b \,c^{3} d^{2} x^{2}}{e^{6}}-\frac {48 \ln \left (e x +d \right ) a \,b^{2} c d}{e^{5}}+\frac {120 \ln \left (e x +d \right ) a b \,c^{2} d^{2}}{e^{6}}+\frac {\left (-4 e^{7} c \,a^{3}-6 a^{2} b^{2} e^{7}+36 a^{2} b c d \,e^{6}-36 d^{2} e^{5} a^{2} c^{2}+12 a \,b^{3} d \,e^{6}-72 a \,b^{2} c \,d^{2} e^{5}+120 a b \,c^{2} d^{3} e^{4}-60 d^{4} e^{3} a \,c^{3}-6 b^{4} d^{2} e^{5}+40 b^{3} c \,d^{3} e^{4}-90 b^{2} c^{2} d^{4} e^{3}+84 b \,c^{3} d^{5} e^{2}-28 c^{4} d^{6} e \right ) x^{2}+\left (-2 a^{3} b \,e^{7}-4 d \,e^{6} c \,a^{3}-6 a^{2} b^{2} d \,e^{6}+54 a^{2} b c \,d^{2} e^{5}-60 d^{3} e^{4} a^{2} c^{2}+18 a \,b^{3} d^{2} e^{5}-120 a \,b^{2} c \,d^{3} e^{4}+210 a b \,c^{2} d^{4} e^{3}-108 d^{5} e^{2} a \,c^{3}-10 b^{4} d^{3} e^{4}+70 b^{3} c \,d^{4} e^{3}-162 b^{2} c^{2} d^{5} e^{2}+154 b \,c^{3} d^{6} e -52 d^{7} c^{4}\right ) x -\frac {a^{4} e^{8}+2 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-66 a^{2} b c \,d^{3} e^{5}+78 a^{2} c^{2} d^{4} e^{4}-22 a \,b^{3} d^{3} e^{5}+156 a \,b^{2} c \,d^{4} e^{4}-282 a b \,c^{2} d^{5} e^{3}+148 a \,c^{3} d^{6} e^{2}+13 b^{4} d^{4} e^{4}-94 b^{3} c \,d^{5} e^{3}+222 b^{2} c^{2} d^{6} e^{2}-214 b \,c^{3} d^{7} e +73 c^{4} d^{8}}{3 e}}{e^{8} \left (e x +d \right )^{3}}+\frac {c^{4} x^{5}}{5 e^{4}}-\frac {48 a b \,c^{2} d x}{e^{5}}\)	\(984\)
parallelrisch	\(\text {Expression too large to display}\)	\(1724\)

Input:

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

Output:

((6*a^2*c^2*e^4+12*a*b^2*c*e^4-30*a*b*c^2*d*e^3+20*a*c^3*d^2*e^2+b^4*e^4-1 
0*b^3*c*d*e^3+30*b^2*c^2*d^2*e^2-35*b*c^3*d^3*e+14*c^4*d^4)/e^5*x^4-1/3*(a 
^4*e^8+2*a^3*b*d*e^7+4*a^3*c*d^2*e^6+6*a^2*b^2*d^2*e^6-66*a^2*b*c*d^3*e^5+ 
132*a^2*c^2*d^4*e^4-22*a*b^3*d^3*e^5+264*a*b^2*c*d^4*e^4-660*a*b*c^2*d^5*e 
^3+440*a*c^3*d^6*e^2+22*b^4*d^4*e^4-220*b^3*c*d^5*e^3+660*b^2*c^2*d^6*e^2- 
770*b*c^3*d^7*e+308*c^4*d^8)/e^9+1/5*c^4/e*x^8-(4*a^3*c*e^6+6*a^2*b^2*e^6- 
36*a^2*b*c*d*e^5+72*a^2*c^2*d^2*e^4-12*a*b^3*d*e^5+144*a*b^2*c*d^2*e^4-360 
*a*b*c^2*d^3*e^3+240*a*c^3*d^4*e^2+12*b^4*d^2*e^4-120*b^3*c*d^3*e^3+360*b^ 
2*c^2*d^4*e^2-420*b*c^3*d^5*e+168*c^4*d^6)/e^7*x^2-(2*a^3*b*e^7+4*a^3*c*d* 
e^6+6*a^2*b^2*d*e^6-54*a^2*b*c*d^2*e^5+108*a^2*c^2*d^3*e^4-18*a*b^3*d^2*e^ 
5+216*a*b^2*c*d^3*e^4-540*a*b*c^2*d^4*e^3+360*a*c^3*d^5*e^2+18*b^4*d^3*e^4 
-180*b^3*c*d^4*e^3+540*b^2*c^2*d^5*e^2-630*b*c^3*d^6*e+252*c^4*d^7)/e^8*x+ 
1/5*c*(30*a*b*c*e^3-20*a*c^2*d*e^2+10*b^3*e^3-30*b^2*c*d*e^2+35*b*c^2*d^2* 
e-14*c^3*d^3)/e^4*x^5+1/15*c^2*(20*a*c*e^2+30*b^2*e^2-35*b*c*d*e+14*c^2*d^ 
2)/e^3*x^6+1/5*c^3*(5*b*e-2*c*d)/e^2*x^7)/(e*x+d)^3+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+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2- 
b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/e 
^9*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (411) = 822\).

Time = 0.10 (sec) , antiderivative size = 1282, normalized size of antiderivative = 3.07 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/15*(3*c^4*e^8*x^8 - 365*c^4*d^8 + 1070*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 5* 
a^4*e^8 - 370*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 470*(b^3*c + 3*a*b*c^2)*d^5* 
e^3 - 65*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 110*(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 - 3*(2*c^4*d*e^7 - 5*b*c^3*e^8) 
*x^7 + (14*c^4*d^2*e^6 - 35*b*c^3*d*e^7 + 10*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^ 
6 - 3*(14*c^4*d^3*e^5 - 35*b*c^3*d^2*e^6 + 10*(3*b^2*c^2 + 2*a*c^3)*d*e^7 
- 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 15*(14*c^4*d^4*e^4 - 35*b*c^3*d^3*e^5 
+ 10*(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 + 5*(235*c^4*d^5*e^3 - 556*b*c^3*d^4*e^4 
 + 146*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 126*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 9 
*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7)*x^3 + 15*(67*c^4*d^6*e^2 - 136*b*c^ 
3*d^5*e^3 + 26*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 6*(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 - 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 - 15*(17*c^4*d^7*e - 74*b*c^3*d^6*e^ 
2 + 2*a^3*b*e^8 + 34*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 54*(b^3*c + 3*a*b*c^2 
)*d^4*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 18*(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 - 60*(14*c^4*d^8 - 35*b*c^3 
*d^7*e + 10*(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 - (a*b^3 + 3*a^2*b*c)*d^3*e^5 + 
(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 -...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (413) = 826\).

Time = 56.46 (sec) , antiderivative size = 944, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

c**4*x**5/(5*e**4) + x**4*(b*c**3/e**4 - c**4*d/e**5) + x**3*(4*a*c**3/(3* 
e**4) + 2*b**2*c**2/e**4 - 16*b*c**3*d/(3*e**5) + 10*c**4*d**2/(3*e**6)) + 
 x**2*(6*a*b*c**2/e**4 - 8*a*c**3*d/e**5 + 2*b**3*c/e**4 - 12*b**2*c**2*d/ 
e**5 + 20*b*c**3*d**2/e**6 - 10*c**4*d**3/e**7) + x*(6*a**2*c**2/e**4 + 12 
*a*b**2*c/e**4 - 48*a*b*c**2*d/e**5 + 40*a*c**3*d**2/e**6 + b**4/e**4 - 16 
*b**3*c*d/e**5 + 60*b**2*c**2*d**2/e**6 - 80*b*c**3*d**3/e**7 + 35*c**4*d* 
*4/e**8) + (-a**4*e**8 - 2*a**3*b*d*e**7 - 4*a**3*c*d**2*e**6 - 6*a**2*b** 
2*d**2*e**6 + 66*a**2*b*c*d**3*e**5 - 78*a**2*c**2*d**4*e**4 + 22*a*b**3*d 
**3*e**5 - 156*a*b**2*c*d**4*e**4 + 282*a*b*c**2*d**5*e**3 - 148*a*c**3*d* 
*6*e**2 - 13*b**4*d**4*e**4 + 94*b**3*c*d**5*e**3 - 222*b**2*c**2*d**6*e** 
2 + 214*b*c**3*d**7*e - 73*c**4*d**8 + x**2*(-12*a**3*c*e**8 - 18*a**2*b** 
2*e**8 + 108*a**2*b*c*d*e**7 - 108*a**2*c**2*d**2*e**6 + 36*a*b**3*d*e**7 
- 216*a*b**2*c*d**2*e**6 + 360*a*b*c**2*d**3*e**5 - 180*a*c**3*d**4*e**4 - 
 18*b**4*d**2*e**6 + 120*b**3*c*d**3*e**5 - 270*b**2*c**2*d**4*e**4 + 252* 
b*c**3*d**5*e**3 - 84*c**4*d**6*e**2) + x*(-6*a**3*b*e**8 - 12*a**3*c*d*e* 
*7 - 18*a**2*b**2*d*e**7 + 162*a**2*b*c*d**2*e**6 - 180*a**2*c**2*d**3*e** 
5 + 54*a*b**3*d**2*e**6 - 360*a*b**2*c*d**3*e**5 + 630*a*b*c**2*d**4*e**4 
- 324*a*c**3*d**5*e**3 - 30*b**4*d**3*e**5 + 210*b**3*c*d**4*e**4 - 486*b* 
*2*c**2*d**5*e**3 + 462*b*c**3*d**6*e**2 - 156*c**4*d**7*e))/(3*d**3*e**9 
+ 9*d**2*e**10*x + 9*d*e**11*x**2 + 3*e**12*x**3) + 4*(b*e - 2*c*d)*(a*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (411) = 822\).

Time = 0.07 (sec) , antiderivative size = 827, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 2*a^3*b*d*e^7 + a^4*e^8 + 74*(3*b^2*c 
^2 + 2*a*c^3)*d^6*e^2 - 94*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 13*(b^4 + 12*a*b^ 
2*c + 6*a^2*c^2)*d^4*e^4 - 22*(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 + 6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 15*(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 - 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 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + a^3*b*e^8 + 27*(3*b^2*c^2 
+ 2*a*c^3)*d^5*e^3 - 35*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 5*(b^4 + 12*a*b^2*c 
+ 6*a^2*c^2)*d^3*e^5 - 9*(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^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*e^9) + 1/15*(3* 
c^4*e^4*x^5 - 15*(c^4*d*e^3 - b*c^3*e^4)*x^4 + 10*(5*c^4*d^2*e^2 - 8*b*c^3 
*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 30*(5*c^4*d^3*e - 10*b*c^3*d^2*e 
^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 - (b^3*c + 3*a*b*c^2)*e^4)*x^2 + 15*(35 
*c^4*d^4 - 80*b*c^3*d^3*e + 20*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 16*(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)*x)/e^8 - 4*(14*c^4 
*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a 
*b*c^2)*d^2*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b* 
c)*e^5)*log(e*x + d)/e^9
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (411) = 822\).

Time = 0.38 (sec) , antiderivative size = 933, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-4*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 
10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a 
^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*log(abs(e*x + d))/e^9 - 1/3*(73* 
c^4*d^8 - 214*b*c^3*d^7*e + 222*b^2*c^2*d^6*e^2 + 148*a*c^3*d^6*e^2 - 94*b 
^3*c*d^5*e^3 - 282*a*b*c^2*d^5*e^3 + 13*b^4*d^4*e^4 + 156*a*b^2*c*d^4*e^4 
+ 78*a^2*c^2*d^4*e^4 - 22*a*b^3*d^3*e^5 - 66*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 + 2*a^3*b*d*e^7 + a^4*e^8 + 6*(14*c^4*d^6*e^2 - 4 
2*b*c^3*d^5*e^3 + 45*b^2*c^2*d^4*e^4 + 30*a*c^3*d^4*e^4 - 20*b^3*c*d^3*e^5 
 - 60*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 - 6*a*b^3*d*e^7 - 18*a^2*b*c*d*e^7 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8)*x^ 
2 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + 81*b^2*c^2*d^5*e^3 + 54*a*c^3*d^5 
*e^3 - 35*b^3*c*d^4*e^4 - 105*a*b*c^2*d^4*e^4 + 5*b^4*d^3*e^5 + 60*a*b^2*c 
*d^3*e^5 + 30*a^2*c^2*d^3*e^5 - 9*a*b^3*d^2*e^6 - 27*a^2*b*c*d^2*e^6 + 3*a 
^2*b^2*d*e^7 + 2*a^3*c*d*e^7 + a^3*b*e^8)*x)/((e*x + d)^3*e^9) + 1/15*(3*c 
^4*e^16*x^5 - 15*c^4*d*e^15*x^4 + 15*b*c^3*e^16*x^4 + 50*c^4*d^2*e^14*x^3 
- 80*b*c^3*d*e^15*x^3 + 30*b^2*c^2*e^16*x^3 + 20*a*c^3*e^16*x^3 - 150*c^4* 
d^3*e^13*x^2 + 300*b*c^3*d^2*e^14*x^2 - 180*b^2*c^2*d*e^15*x^2 - 120*a*c^3 
*d*e^15*x^2 + 30*b^3*c*e^16*x^2 + 90*a*b*c^2*e^16*x^2 + 525*c^4*d^4*e^12*x 
 - 1200*b*c^3*d^3*e^13*x + 900*b^2*c^2*d^2*e^14*x + 600*a*c^3*d^2*e^14*x - 
 240*b^3*c*d*e^15*x - 720*a*b*c^2*d*e^15*x + 15*b^4*e^16*x + 180*a*b^2*...
 

Mupad [B] (verification not implemented)

Time = 5.48 (sec) , antiderivative size = 1143, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

x^4*((b*c^3)/e^4 - (c^4*d)/e^5) - x^2*((2*c^4*d^3)/e^7 + (3*d^2*((4*b*c^3) 
/e^4 - (4*c^4*d)/e^5))/e^2 - (2*d*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e 
 - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (2*b*c*(3*a*c + b^2)) 
/e^4) - x^3*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/(3*e) - (4*a*c^3 + 6*b^ 
2*c^2)/(3*e^4) + (2*c^4*d^2)/e^6) + x*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^4 
+ (6*d^2*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/ 
e^4 + (6*c^4*d^2)/e^6))/e^2 + (4*d*((4*c^4*d^3)/e^7 + (6*d^2*((4*b*c^3)/e^ 
4 - (4*c^4*d)/e^5))/e^2 - (4*d*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - 
(4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (4*b*c*(3*a*c + b^2))/e^ 
4))/e - (c^4*d^4)/e^8 - (4*d^3*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^3) - (x* 
(52*c^4*d^7 + 2*a^3*b*e^7 + 10*b^4*d^3*e^4 - 18*a*b^3*d^2*e^5 + 6*a^2*b^2* 
d*e^6 + 108*a*c^3*d^5*e^2 - 70*b^3*c*d^4*e^3 + 60*a^2*c^2*d^3*e^4 + 162*b^ 
2*c^2*d^5*e^2 + 4*a^3*c*d*e^6 - 154*b*c^3*d^6*e - 210*a*b*c^2*d^4*e^3 + 12 
0*a*b^2*c*d^3*e^4 - 54*a^2*b*c*d^2*e^5) + (a^4*e^8 + 73*c^4*d^8 + 13*b^4*d 
^4*e^4 - 22*a*b^3*d^3*e^5 + 148*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 94*b^3*c 
*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 78*a^2*c^2*d^4*e^4 + 222*b^2*c^2*d^6*e^2 + 
2*a^3*b*d*e^7 - 214*b*c^3*d^7*e - 282*a*b*c^2*d^5*e^3 + 156*a*b^2*c*d^4*e^ 
4 - 66*a^2*b*c*d^3*e^5)/(3*e) + x^2*(4*a^3*c*e^7 + 28*c^4*d^6*e + 6*a^2*b^ 
2*e^7 + 6*b^4*d^2*e^5 + 60*a*c^3*d^4*e^3 - 84*b*c^3*d^5*e^2 - 40*b^3*c*d^3 
*e^4 + 36*a^2*c^2*d^2*e^5 + 90*b^2*c^2*d^4*e^3 - 12*a*b^3*d*e^6 - 36*a^...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1735, normalized size of antiderivative = 4.16 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(180*log(d + e*x)*a**2*b*c*d**4*e**5 + 540*log(d + e*x)*a**2*b*c*d**3*e**6 
*x + 540*log(d + e*x)*a**2*b*c*d**2*e**7*x**2 + 180*log(d + e*x)*a**2*b*c* 
d*e**8*x**3 - 360*log(d + e*x)*a**2*c**2*d**5*e**4 - 1080*log(d + e*x)*a** 
2*c**2*d**4*e**5*x - 1080*log(d + e*x)*a**2*c**2*d**3*e**6*x**2 - 360*log( 
d + e*x)*a**2*c**2*d**2*e**7*x**3 + 60*log(d + e*x)*a*b**3*d**4*e**5 + 180 
*log(d + e*x)*a*b**3*d**3*e**6*x + 180*log(d + e*x)*a*b**3*d**2*e**7*x**2 
+ 60*log(d + e*x)*a*b**3*d*e**8*x**3 - 720*log(d + e*x)*a*b**2*c*d**5*e**4 
 - 2160*log(d + e*x)*a*b**2*c*d**4*e**5*x - 2160*log(d + e*x)*a*b**2*c*d** 
3*e**6*x**2 - 720*log(d + e*x)*a*b**2*c*d**2*e**7*x**3 + 1800*log(d + e*x) 
*a*b*c**2*d**6*e**3 + 5400*log(d + e*x)*a*b*c**2*d**5*e**4*x + 5400*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 
 - 1200*log(d + e*x)*a*c**3*d**7*e**2 - 3600*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 - 1200*log(d + e*x)*a*c**3*d* 
*4*e**5*x**3 - 60*log(d + e*x)*b**4*d**5*e**4 - 180*log(d + e*x)*b**4*d**4 
*e**5*x - 180*log(d + e*x)*b**4*d**3*e**6*x**2 - 60*log(d + e*x)*b**4*d**2 
*e**7*x**3 + 600*log(d + e*x)*b**3*c*d**6*e**3 + 1800*log(d + e*x)*b**3*c* 
d**5*e**4*x + 1800*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 - 1800*log(d + e*x)*b**2*c**2*d**7*e**2 - 5400*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 
- 1800*log(d + e*x)*b**2*c**2*d**4*e**5*x**3 + 2100*log(d + e*x)*b*c**3...