\(\int \frac {(A+B x) (b x+c x^2)^3}{(d+e x)^3} \, dx\) [28]

Optimal result

Integrand size = 24, antiderivative size = 359 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {(c d-b e) \left (A e \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )-3 B d \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )\right ) x}{e^7}+\frac {(c d-b e) \left (3 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) x^2}{2 e^6}+\frac {c (c d-b e) (2 B c d-b B e-A c e) x^3}{e^5}-\frac {c^2 (3 B c d-3 b B e-A c e) x^4}{4 e^4}+\frac {B c^3 x^5}{5 e^3}+\frac {d^3 (B d-A e) (c d-b e)^3}{2 e^8 (d+e x)^2}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (d+e x)}+\frac {3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) \log (d+e x)}{e^8} \] Output:

-(-b*e+c*d)*(A*e*(b^2*e^2-8*b*c*d*e+10*c^2*d^2)-3*B*d*(b^2*e^2-5*b*c*d*e+5 
*c^2*d^2))*x/e^7+1/2*(-b*e+c*d)*(3*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e 
+10*c^2*d^2))*x^2/e^6+c*(-b*e+c*d)*(-A*c*e-B*b*e+2*B*c*d)*x^3/e^5-1/4*c^2* 
(-A*c*e-3*B*b*e+3*B*c*d)*x^4/e^4+1/5*B*c^3*x^5/e^3+1/2*d^3*(-A*e+B*d)*(-b* 
e+c*d)^3/e^8/(e*x+d)^2-d^2*(-b*e+c*d)^2*(B*d*(-4*b*e+7*c*d)-3*A*e*(-b*e+2* 
c*d))/e^8/(e*x+d)+3*d*(-b*e+c*d)*(A*e*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)-B*d*(2 
*b^2*e^2-8*b*c*d*e+7*c^2*d^2))*ln(e*x+d)/e^8
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {20 e (-c d+b e) \left (A e \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )-3 B d \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )\right ) x+10 e^2 (-c d+b e) \left (3 A c e (-2 c d+b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) x^2-20 c e^3 (c d-b e) (-2 B c d+b B e+A c e) x^3+5 c^2 e^4 (-3 B c d+3 b B e+A c e) x^4+4 B c^3 e^5 x^5+\frac {10 d^3 (B d-A e) (c d-b e)^3}{(d+e x)^2}-\frac {20 d^2 (c d-b e)^2 (B d (7 c d-4 b e)+3 A e (-2 c d+b e))}{d+e x}-60 d (c d-b e) \left (-A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )+B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) \log (d+e x)}{20 e^8} \] Input:

Integrate[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^3,x]
 

Output:

(20*e*(-(c*d) + b*e)*(A*e*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - 3*B*d*(5*c^ 
2*d^2 - 5*b*c*d*e + b^2*e^2))*x + 10*e^2*(-(c*d) + b*e)*(3*A*c*e*(-2*c*d + 
 b*e) + B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*x^2 - 20*c*e^3*(c*d - b*e)*( 
-2*B*c*d + b*B*e + A*c*e)*x^3 + 5*c^2*e^4*(-3*B*c*d + 3*b*B*e + A*c*e)*x^4 
 + 4*B*c^3*e^5*x^5 + (10*d^3*(B*d - A*e)*(c*d - b*e)^3)/(d + e*x)^2 - (20* 
d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) + 3*A*e*(-2*c*d + b*e)))/(d + e*x) 
- 60*d*(c*d - b*e)*(-(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)) + B*d*(7*c^2* 
d^2 - 8*b*c*d*e + 2*b^2*e^2))*Log[d + e*x])/(20*e^8)
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.

\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 d (c d-b e) \log (d+e x) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8}-\frac {x (c d-b e) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{e^7}+\frac {x^2 (c d-b e) \left (3 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{2 e^6}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac {d^3 (B d-A e) (c d-b e)^3}{2 e^8 (d+e x)^2}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (d+e x)}+\frac {c x^3 (c d-b e) (-A c e-b B e+2 B c d)}{e^5}+\frac {B c^3 x^5}{5 e^3}\)

Input:

Int[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^3,x]
 

Output:

-(((c*d - b*e)*(A*e*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - 3*B*d*(5*c^2*d^2 
- 5*b*c*d*e + b^2*e^2))*x)/e^7) + ((c*d - b*e)*(3*A*c*e*(2*c*d - b*e) - B* 
(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*x^2)/(2*e^6) + (c*(c*d - b*e)*(2*B*c*d 
 - b*B*e - A*c*e)*x^3)/e^5 - (c^2*(3*B*c*d - 3*b*B*e - A*c*e)*x^4)/(4*e^4) 
 + (B*c^3*x^5)/(5*e^3) + (d^3*(B*d - A*e)*(c*d - b*e)^3)/(2*e^8*(d + e*x)^ 
2) - (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) - 3*A*e*(2*c*d - b*e)))/(e^8* 
(d + e*x)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*d 
*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*Log[d + e*x])/e^8
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.59

method	result	size
norman	\(\frac {\frac {\left (A \,b^{3} e^{4}-6 A \,b^{2} c d \,e^{3}+10 A b \,c^{2} d^{2} e^{2}-5 A \,c^{3} d^{3} e -2 B \,b^{3} d \,e^{3}+10 B \,b^{2} c \,d^{2} e^{2}-15 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) x^{3}}{e^{5}}-\frac {d^{2} \left (9 A d \,b^{3} e^{4}-54 A \,b^{2} c \,d^{2} e^{3}+90 A b \,c^{2} d^{3} e^{2}-45 A \,c^{3} d^{4} e -18 B \,b^{3} d^{2} e^{3}+90 B \,b^{2} c \,d^{3} e^{2}-135 B b \,c^{2} d^{4} e +63 B \,c^{3} d^{5}\right )}{2 e^{8}}+\frac {\left (6 A \,b^{2} c \,e^{3}-10 A b \,c^{2} d \,e^{2}+5 A \,c^{3} d^{2} e +2 B \,e^{3} b^{3}-10 B \,b^{2} c d \,e^{2}+15 B b \,c^{2} d^{2} e -7 B \,c^{3} d^{3}\right ) x^{4}}{4 e^{4}}+\frac {B \,c^{3} x^{7}}{5 e}+\frac {c \left (10 A b c \,e^{2}-5 A \,c^{2} d e +10 B \,e^{2} b^{2}-15 B b c d e +7 B \,c^{2} d^{2}\right ) x^{5}}{10 e^{3}}+\frac {c^{2} \left (5 A c e +15 B b e -7 B c d \right ) x^{6}}{20 e^{2}}-\frac {2 d \left (3 A d \,b^{3} e^{4}-18 A \,b^{2} c \,d^{2} e^{3}+30 A b \,c^{2} d^{3} e^{2}-15 A \,c^{3} d^{4} e -6 B \,b^{3} d^{2} e^{3}+30 B \,b^{2} c \,d^{3} e^{2}-45 B b \,c^{2} d^{4} e +21 B \,c^{3} d^{5}\right ) x}{e^{7}}}{\left (e x +d \right )^{2}}-\frac {3 d \left (A \,b^{3} e^{4}-6 A \,b^{2} c d \,e^{3}+10 A b \,c^{2} d^{2} e^{2}-5 A \,c^{3} d^{3} e -2 B \,b^{3} d \,e^{3}+10 B \,b^{2} c \,d^{2} e^{2}-15 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(571\)
default	\(\frac {\frac {1}{5} B \,c^{3} x^{5} e^{4}+\frac {1}{4} A \,c^{3} e^{4} x^{4}+\frac {3}{4} B b \,c^{2} e^{4} x^{4}-\frac {3}{4} B \,c^{3} d \,e^{3} x^{4}+A b \,c^{2} e^{4} x^{3}-A \,c^{3} d \,e^{3} x^{3}+B \,b^{2} c \,e^{4} x^{3}-3 B b \,c^{2} d \,e^{3} x^{3}+2 B \,c^{3} d^{2} e^{2} x^{3}+\frac {3}{2} A \,b^{2} c \,e^{4} x^{2}-\frac {9}{2} A b \,c^{2} d \,e^{3} x^{2}+3 A \,c^{3} d^{2} e^{2} x^{2}+\frac {1}{2} B \,b^{3} e^{4} x^{2}-\frac {9}{2} B \,b^{2} c d \,e^{3} x^{2}+9 B b \,c^{2} d^{2} e^{2} x^{2}-5 B \,c^{3} d^{3} e \,x^{2}+A \,b^{3} e^{4} x -9 A \,b^{2} c d \,e^{3} x +18 A b \,c^{2} d^{2} e^{2} x -10 A \,c^{3} d^{3} e x -3 B \,b^{3} d \,e^{3} x +18 B \,b^{2} c \,d^{2} e^{2} x -30 B b \,c^{2} d^{3} e x +15 B \,c^{3} d^{4} x}{e^{7}}-\frac {3 d \left (A \,b^{3} e^{4}-6 A \,b^{2} c d \,e^{3}+10 A b \,c^{2} d^{2} e^{2}-5 A \,c^{3} d^{3} e -2 B \,b^{3} d \,e^{3}+10 B \,b^{2} c \,d^{2} e^{2}-15 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {d^{2} \left (3 A \,b^{3} e^{4}-12 A \,b^{2} c d \,e^{3}+15 A b \,c^{2} d^{2} e^{2}-6 A \,c^{3} d^{3} e -4 B \,b^{3} d \,e^{3}+15 B \,b^{2} c \,d^{2} e^{2}-18 B b \,c^{2} d^{3} e +7 B \,c^{3} d^{4}\right )}{e^{8} \left (e x +d \right )}+\frac {d^{3} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right )}{2 e^{8} \left (e x +d \right )^{2}}\)	\(613\)
risch	\(\frac {A \,c^{3} x^{4}}{4 e^{3}}+\frac {B \,b^{3} x^{2}}{2 e^{3}}+\frac {A \,b^{3} x}{e^{3}}+\frac {3 B b \,c^{2} x^{4}}{4 e^{3}}-\frac {3 B \,c^{3} d \,x^{4}}{4 e^{4}}+\frac {A b \,c^{2} x^{3}}{e^{3}}-\frac {A \,c^{3} d \,x^{3}}{e^{4}}+\frac {B \,b^{2} c \,x^{3}}{e^{3}}+\frac {2 B \,c^{3} d^{2} x^{3}}{e^{5}}+\frac {3 A \,b^{2} c \,x^{2}}{2 e^{3}}-\frac {3 d \ln \left (e x +d \right ) A \,b^{3}}{e^{4}}+\frac {15 d^{4} \ln \left (e x +d \right ) A \,c^{3}}{e^{7}}+\frac {6 d^{2} \ln \left (e x +d \right ) B \,b^{3}}{e^{5}}-\frac {21 d^{5} \ln \left (e x +d \right ) B \,c^{3}}{e^{8}}+\frac {3 A \,c^{3} d^{2} x^{2}}{e^{5}}-\frac {5 B \,c^{3} d^{3} x^{2}}{e^{6}}-\frac {10 A \,c^{3} d^{3} x}{e^{6}}-\frac {3 B \,b^{3} d x}{e^{4}}+\frac {15 B \,c^{3} d^{4} x}{e^{7}}+\frac {18 B \,b^{2} c \,d^{2} x}{e^{5}}+\frac {9 B b \,c^{2} d^{2} x^{2}}{e^{5}}-\frac {9 A \,b^{2} c d x}{e^{4}}+\frac {18 A b \,c^{2} d^{2} x}{e^{5}}+\frac {B \,c^{3} x^{5}}{5 e^{3}}+\frac {\left (-3 A \,b^{3} d^{2} e^{4}+12 A \,b^{2} c \,d^{3} e^{3}-15 A b \,c^{2} d^{4} e^{2}+6 A \,c^{3} d^{5} e +4 B \,b^{3} d^{3} e^{3}-15 B \,b^{2} c \,d^{4} e^{2}+18 B b \,c^{2} d^{5} e -7 B \,c^{3} d^{6}\right ) x -\frac {d^{3} \left (5 A \,b^{3} e^{4}-21 A \,b^{2} c d \,e^{3}+27 A b \,c^{2} d^{2} e^{2}-11 A \,c^{3} d^{3} e -7 B \,b^{3} d \,e^{3}+27 B \,b^{2} c \,d^{2} e^{2}-33 B b \,c^{2} d^{3} e +13 B \,c^{3} d^{4}\right )}{2 e}}{e^{7} \left (e x +d \right )^{2}}+\frac {45 d^{4} \ln \left (e x +d \right ) B b \,c^{2}}{e^{7}}-\frac {3 B b \,c^{2} d \,x^{3}}{e^{4}}-\frac {9 A b \,c^{2} d \,x^{2}}{2 e^{4}}-\frac {9 B \,b^{2} c d \,x^{2}}{2 e^{4}}-\frac {30 B b \,c^{2} d^{3} x}{e^{6}}+\frac {18 d^{2} \ln \left (e x +d \right ) A \,b^{2} c}{e^{5}}-\frac {30 d^{3} \ln \left (e x +d \right ) A b \,c^{2}}{e^{6}}-\frac {30 d^{3} \ln \left (e x +d \right ) B \,b^{2} c}{e^{6}}\)	\(666\)
parallelrisch	\(\text {Expression too large to display}\)	\(1012\)

Input:

int((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

(1/e^5*(A*b^3*e^4-6*A*b^2*c*d*e^3+10*A*b*c^2*d^2*e^2-5*A*c^3*d^3*e-2*B*b^3 
*d*e^3+10*B*b^2*c*d^2*e^2-15*B*b*c^2*d^3*e+7*B*c^3*d^4)*x^3-1/2*d^2*(9*A*b 
^3*d*e^4-54*A*b^2*c*d^2*e^3+90*A*b*c^2*d^3*e^2-45*A*c^3*d^4*e-18*B*b^3*d^2 
*e^3+90*B*b^2*c*d^3*e^2-135*B*b*c^2*d^4*e+63*B*c^3*d^5)/e^8+1/4*(6*A*b^2*c 
*e^3-10*A*b*c^2*d*e^2+5*A*c^3*d^2*e+2*B*b^3*e^3-10*B*b^2*c*d*e^2+15*B*b*c^ 
2*d^2*e-7*B*c^3*d^3)/e^4*x^4+1/5*B*c^3*x^7/e+1/10*c*(10*A*b*c*e^2-5*A*c^2* 
d*e+10*B*b^2*e^2-15*B*b*c*d*e+7*B*c^2*d^2)/e^3*x^5+1/20*c^2*(5*A*c*e+15*B* 
b*e-7*B*c*d)/e^2*x^6-2*d*(3*A*b^3*d*e^4-18*A*b^2*c*d^2*e^3+30*A*b*c^2*d^3* 
e^2-15*A*c^3*d^4*e-6*B*b^3*d^2*e^3+30*B*b^2*c*d^3*e^2-45*B*b*c^2*d^4*e+21* 
B*c^3*d^5)/e^7*x)/(e*x+d)^2-3*d/e^8*(A*b^3*e^4-6*A*b^2*c*d*e^3+10*A*b*c^2* 
d^2*e^2-5*A*c^3*d^3*e-2*B*b^3*d*e^3+10*B*b^2*c*d^2*e^2-15*B*b*c^2*d^3*e+7* 
B*c^3*d^4)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (351) = 702\).

Time = 0.09 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="fricas")
 

Output:

1/20*(4*B*c^3*e^7*x^7 - 130*B*c^3*d^7 - 50*A*b^3*d^3*e^4 + 110*(3*B*b*c^2 
+ A*c^3)*d^6*e - 270*(B*b^2*c + A*b*c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*b^2*c)* 
d^4*e^3 - (7*B*c^3*d*e^6 - 5*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 2*(7*B*c^3*d^2 
*e^5 - 5*(3*B*b*c^2 + A*c^3)*d*e^6 + 10*(B*b^2*c + A*b*c^2)*e^7)*x^5 - 5*( 
7*B*c^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 10*(B*b^2*c + A*b*c^2)*d 
*e^6 - 2*(B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 + A*b^3*e^7 - 
5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 10*(B*b^2*c + A*b*c^2)*d^2*e^5 - 2*(B*b^3 
+ 3*A*b^2*c)*d*e^6)*x^3 + 10*(50*B*c^3*d^5*e^2 + 4*A*b^3*d*e^6 - 34*(3*B*b 
*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c + A*b*c^2)*d^3*e^4 - 11*(B*b^3 + 3*A*b 
^2*c)*d^2*e^5)*x^2 + 20*(8*B*c^3*d^6*e - 2*A*b^3*d^2*e^5 - 4*(3*B*b*c^2 + 
A*c^3)*d^5*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4*e^3 + (B*b^3 + 3*A*b^2*c)*d^3*e 
^4)*x - 60*(7*B*c^3*d^7 + A*b^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^6*e + 10 
*(B*b^2*c + A*b*c^2)*d^5*e^2 - 2*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + (7*B*c^3*d^ 
5*e^2 + A*b^3*d*e^6 - 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 10*(B*b^2*c + A*b*c^ 
2)*d^3*e^4 - 2*(B*b^3 + 3*A*b^2*c)*d^2*e^5)*x^2 + 2*(7*B*c^3*d^6*e + A*b^3 
*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 10*(B*b^2*c + A*b*c^2)*d^4*e^3 
- 2*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + 
d^2*e^8)
 

Sympy [A] (verification not implemented)

Time = 3.75 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.84 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {B c^{3} x^{5}}{5 e^{3}} + \frac {3 d \left (b e - c d\right ) \left (- A b^{2} e^{3} + 5 A b c d e^{2} - 5 A c^{2} d^{2} e + 2 B b^{2} d e^{2} - 8 B b c d^{2} e + 7 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A c^{3}}{4 e^{3}} + \frac {3 B b c^{2}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}}\right ) + x^{3} \left (\frac {A b c^{2}}{e^{3}} - \frac {A c^{3} d}{e^{4}} + \frac {B b^{2} c}{e^{3}} - \frac {3 B b c^{2} d}{e^{4}} + \frac {2 B c^{3} d^{2}}{e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A b^{2} c}{2 e^{3}} - \frac {9 A b c^{2} d}{2 e^{4}} + \frac {3 A c^{3} d^{2}}{e^{5}} + \frac {B b^{3}}{2 e^{3}} - \frac {9 B b^{2} c d}{2 e^{4}} + \frac {9 B b c^{2} d^{2}}{e^{5}} - \frac {5 B c^{3} d^{3}}{e^{6}}\right ) + x \left (\frac {A b^{3}}{e^{3}} - \frac {9 A b^{2} c d}{e^{4}} + \frac {18 A b c^{2} d^{2}}{e^{5}} - \frac {10 A c^{3} d^{3}}{e^{6}} - \frac {3 B b^{3} d}{e^{4}} + \frac {18 B b^{2} c d^{2}}{e^{5}} - \frac {30 B b c^{2} d^{3}}{e^{6}} + \frac {15 B c^{3} d^{4}}{e^{7}}\right ) + \frac {- 5 A b^{3} d^{3} e^{4} + 21 A b^{2} c d^{4} e^{3} - 27 A b c^{2} d^{5} e^{2} + 11 A c^{3} d^{6} e + 7 B b^{3} d^{4} e^{3} - 27 B b^{2} c d^{5} e^{2} + 33 B b c^{2} d^{6} e - 13 B c^{3} d^{7} + x \left (- 6 A b^{3} d^{2} e^{5} + 24 A b^{2} c d^{3} e^{4} - 30 A b c^{2} d^{4} e^{3} + 12 A c^{3} d^{5} e^{2} + 8 B b^{3} d^{3} e^{4} - 30 B b^{2} c d^{4} e^{3} + 36 B b c^{2} d^{5} e^{2} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \] Input:

integrate((B*x+A)*(c*x**2+b*x)**3/(e*x+d)**3,x)
 

Output:

B*c**3*x**5/(5*e**3) + 3*d*(b*e - c*d)*(-A*b**2*e**3 + 5*A*b*c*d*e**2 - 5* 
A*c**2*d**2*e + 2*B*b**2*d*e**2 - 8*B*b*c*d**2*e + 7*B*c**2*d**3)*log(d + 
e*x)/e**8 + x**4*(A*c**3/(4*e**3) + 3*B*b*c**2/(4*e**3) - 3*B*c**3*d/(4*e* 
*4)) + x**3*(A*b*c**2/e**3 - A*c**3*d/e**4 + B*b**2*c/e**3 - 3*B*b*c**2*d/ 
e**4 + 2*B*c**3*d**2/e**5) + x**2*(3*A*b**2*c/(2*e**3) - 9*A*b*c**2*d/(2*e 
**4) + 3*A*c**3*d**2/e**5 + B*b**3/(2*e**3) - 9*B*b**2*c*d/(2*e**4) + 9*B* 
b*c**2*d**2/e**5 - 5*B*c**3*d**3/e**6) + x*(A*b**3/e**3 - 9*A*b**2*c*d/e** 
4 + 18*A*b*c**2*d**2/e**5 - 10*A*c**3*d**3/e**6 - 3*B*b**3*d/e**4 + 18*B*b 
**2*c*d**2/e**5 - 30*B*b*c**2*d**3/e**6 + 15*B*c**3*d**4/e**7) + (-5*A*b** 
3*d**3*e**4 + 21*A*b**2*c*d**4*e**3 - 27*A*b*c**2*d**5*e**2 + 11*A*c**3*d* 
*6*e + 7*B*b**3*d**4*e**3 - 27*B*b**2*c*d**5*e**2 + 33*B*b*c**2*d**6*e - 1 
3*B*c**3*d**7 + x*(-6*A*b**3*d**2*e**5 + 24*A*b**2*c*d**3*e**4 - 30*A*b*c* 
*2*d**4*e**3 + 12*A*c**3*d**5*e**2 + 8*B*b**3*d**3*e**4 - 30*B*b**2*c*d**4 
*e**3 + 36*B*b*c**2*d**5*e**2 - 14*B*c**3*d**6*e))/(2*d**2*e**8 + 4*d*e**9 
*x + 2*e**10*x**2)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {13 \, B c^{3} d^{7} + 5 \, A b^{3} d^{3} e^{4} - 11 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 27 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 7 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + 2 \, {\left (7 \, B c^{3} d^{6} e + 3 \, A b^{3} d^{2} e^{5} - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B c^{3} e^{4} x^{5} - 5 \, {\left (3 \, B c^{3} d e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B c^{3} d^{2} e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + {\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{3} - 10 \, {\left (10 \, B c^{3} d^{3} e - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{3} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B c^{3} d^{4} + A b^{3} e^{4} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} - 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B c^{3} d^{5} + A b^{3} d e^{4} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="maxima")
 

Output:

-1/2*(13*B*c^3*d^7 + 5*A*b^3*d^3*e^4 - 11*(3*B*b*c^2 + A*c^3)*d^6*e + 27*( 
B*b^2*c + A*b*c^2)*d^5*e^2 - 7*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + 2*(7*B*c^3*d^ 
6*e + 3*A*b^3*d^2*e^5 - 6*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + A*b* 
c^2)*d^4*e^3 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)/(e^10*x^2 + 2*d*e^9*x + d 
^2*e^8) + 1/20*(4*B*c^3*e^4*x^5 - 5*(3*B*c^3*d*e^3 - (3*B*b*c^2 + A*c^3)*e 
^4)*x^4 + 20*(2*B*c^3*d^2*e^2 - (3*B*b*c^2 + A*c^3)*d*e^3 + (B*b^2*c + A*b 
*c^2)*e^4)*x^3 - 10*(10*B*c^3*d^3*e - 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 9*(B 
*b^2*c + A*b*c^2)*d*e^3 - (B*b^3 + 3*A*b^2*c)*e^4)*x^2 + 20*(15*B*c^3*d^4 
+ A*b^3*e^4 - 10*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + A*b*c^2)*d^2*e^ 
2 - 3*(B*b^3 + 3*A*b^2*c)*d*e^3)*x)/e^7 - 3*(7*B*c^3*d^5 + A*b^3*d*e^4 - 5 
*(3*B*b*c^2 + A*c^3)*d^4*e + 10*(B*b^2*c + A*b*c^2)*d^3*e^2 - 2*(B*b^3 + 3 
*A*b^2*c)*d^2*e^3)*log(e*x + d)/e^8
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {3 \, {\left (7 \, B c^{3} d^{5} - 15 \, B b c^{2} d^{4} e - 5 \, A c^{3} d^{4} e + 10 \, B b^{2} c d^{3} e^{2} + 10 \, A b c^{2} d^{3} e^{2} - 2 \, B b^{3} d^{2} e^{3} - 6 \, A b^{2} c d^{2} e^{3} + A b^{3} d e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} - \frac {13 \, B c^{3} d^{7} - 33 \, B b c^{2} d^{6} e - 11 \, A c^{3} d^{6} e + 27 \, B b^{2} c d^{5} e^{2} + 27 \, A b c^{2} d^{5} e^{2} - 7 \, B b^{3} d^{4} e^{3} - 21 \, A b^{2} c d^{4} e^{3} + 5 \, A b^{3} d^{3} e^{4} + 2 \, {\left (7 \, B c^{3} d^{6} e - 18 \, B b c^{2} d^{5} e^{2} - 6 \, A c^{3} d^{5} e^{2} + 15 \, B b^{2} c d^{4} e^{3} + 15 \, A b c^{2} d^{4} e^{3} - 4 \, B b^{3} d^{3} e^{4} - 12 \, A b^{2} c d^{3} e^{4} + 3 \, A b^{3} d^{2} e^{5}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{8}} + \frac {4 \, B c^{3} e^{12} x^{5} - 15 \, B c^{3} d e^{11} x^{4} + 15 \, B b c^{2} e^{12} x^{4} + 5 \, A c^{3} e^{12} x^{4} + 40 \, B c^{3} d^{2} e^{10} x^{3} - 60 \, B b c^{2} d e^{11} x^{3} - 20 \, A c^{3} d e^{11} x^{3} + 20 \, B b^{2} c e^{12} x^{3} + 20 \, A b c^{2} e^{12} x^{3} - 100 \, B c^{3} d^{3} e^{9} x^{2} + 180 \, B b c^{2} d^{2} e^{10} x^{2} + 60 \, A c^{3} d^{2} e^{10} x^{2} - 90 \, B b^{2} c d e^{11} x^{2} - 90 \, A b c^{2} d e^{11} x^{2} + 10 \, B b^{3} e^{12} x^{2} + 30 \, A b^{2} c e^{12} x^{2} + 300 \, B c^{3} d^{4} e^{8} x - 600 \, B b c^{2} d^{3} e^{9} x - 200 \, A c^{3} d^{3} e^{9} x + 360 \, B b^{2} c d^{2} e^{10} x + 360 \, A b c^{2} d^{2} e^{10} x - 60 \, B b^{3} d e^{11} x - 180 \, A b^{2} c d e^{11} x + 20 \, A b^{3} e^{12} x}{20 \, e^{15}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="giac")
 

Output:

-3*(7*B*c^3*d^5 - 15*B*b*c^2*d^4*e - 5*A*c^3*d^4*e + 10*B*b^2*c*d^3*e^2 + 
10*A*b*c^2*d^3*e^2 - 2*B*b^3*d^2*e^3 - 6*A*b^2*c*d^2*e^3 + A*b^3*d*e^4)*lo 
g(abs(e*x + d))/e^8 - 1/2*(13*B*c^3*d^7 - 33*B*b*c^2*d^6*e - 11*A*c^3*d^6* 
e + 27*B*b^2*c*d^5*e^2 + 27*A*b*c^2*d^5*e^2 - 7*B*b^3*d^4*e^3 - 21*A*b^2*c 
*d^4*e^3 + 5*A*b^3*d^3*e^4 + 2*(7*B*c^3*d^6*e - 18*B*b*c^2*d^5*e^2 - 6*A*c 
^3*d^5*e^2 + 15*B*b^2*c*d^4*e^3 + 15*A*b*c^2*d^4*e^3 - 4*B*b^3*d^3*e^4 - 1 
2*A*b^2*c*d^3*e^4 + 3*A*b^3*d^2*e^5)*x)/((e*x + d)^2*e^8) + 1/20*(4*B*c^3* 
e^12*x^5 - 15*B*c^3*d*e^11*x^4 + 15*B*b*c^2*e^12*x^4 + 5*A*c^3*e^12*x^4 + 
40*B*c^3*d^2*e^10*x^3 - 60*B*b*c^2*d*e^11*x^3 - 20*A*c^3*d*e^11*x^3 + 20*B 
*b^2*c*e^12*x^3 + 20*A*b*c^2*e^12*x^3 - 100*B*c^3*d^3*e^9*x^2 + 180*B*b*c^ 
2*d^2*e^10*x^2 + 60*A*c^3*d^2*e^10*x^2 - 90*B*b^2*c*d*e^11*x^2 - 90*A*b*c^ 
2*d*e^11*x^2 + 10*B*b^3*e^12*x^2 + 30*A*b^2*c*e^12*x^2 + 300*B*c^3*d^4*e^8 
*x - 600*B*b*c^2*d^3*e^9*x - 200*A*c^3*d^3*e^9*x + 360*B*b^2*c*d^2*e^10*x 
+ 360*A*b*c^2*d^2*e^10*x - 60*B*b^3*d*e^11*x - 180*A*b^2*c*d*e^11*x + 20*A 
*b^3*e^12*x)/e^15
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 828, normalized size of antiderivative = 2.31 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=x\,\left (\frac {A\,b^3}{e^3}-\frac {3\,d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e^3}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^2}-\frac {B\,c^3\,d^3}{e^6}\right )}{e}-\frac {d^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^3}+\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e^2}\right )+x^4\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{4\,e^3}-\frac {3\,B\,c^3\,d}{4\,e^4}\right )-\frac {x\,\left (-4\,B\,b^3\,d^3\,e^3+3\,A\,b^3\,d^2\,e^4+15\,B\,b^2\,c\,d^4\,e^2-12\,A\,b^2\,c\,d^3\,e^3-18\,B\,b\,c^2\,d^5\,e+15\,A\,b\,c^2\,d^4\,e^2+7\,B\,c^3\,d^6-6\,A\,c^3\,d^5\,e\right )+\frac {-7\,B\,b^3\,d^4\,e^3+5\,A\,b^3\,d^3\,e^4+27\,B\,b^2\,c\,d^5\,e^2-21\,A\,b^2\,c\,d^4\,e^3-33\,B\,b\,c^2\,d^6\,e+27\,A\,b\,c^2\,d^5\,e^2+13\,B\,c^3\,d^7-11\,A\,c^3\,d^6\,e}{2\,e}}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}+x^2\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{2\,e^3}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{2\,e}-\frac {3\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{2\,e^2}-\frac {B\,c^3\,d^3}{2\,e^6}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {b\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {B\,c^3\,d^2}{e^5}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-6\,B\,b^3\,d^2\,e^3+3\,A\,b^3\,d\,e^4+30\,B\,b^2\,c\,d^3\,e^2-18\,A\,b^2\,c\,d^2\,e^3-45\,B\,b\,c^2\,d^4\,e+30\,A\,b\,c^2\,d^3\,e^2+21\,B\,c^3\,d^5-15\,A\,c^3\,d^4\,e\right )}{e^8}+\frac {B\,c^3\,x^5}{5\,e^3} \] Input:

int(((b*x + c*x^2)^3*(A + B*x))/(d + e*x)^3,x)
 

Output:

x*((A*b^3)/e^3 - (3*d*((B*b^3 + 3*A*b^2*c)/e^3 + (3*d*((3*d*((A*c^3 + 3*B* 
b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*b*c*(A*c + B*b))/e^3 + (3*B*c^3*d^2) 
/e^5))/e - (3*d^2*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^2 - (B*c^ 
3*d^3)/e^6))/e - (d^3*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^3 + ( 
3*d^2*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*b*c*(A*c + 
 B*b))/e^3 + (3*B*c^3*d^2)/e^5))/e^2) + x^4*((A*c^3 + 3*B*b*c^2)/(4*e^3) - 
 (3*B*c^3*d)/(4*e^4)) - (x*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 3*A*b^3*d^2*e^4 
- 4*B*b^3*d^3*e^3 + 15*A*b*c^2*d^4*e^2 - 12*A*b^2*c*d^3*e^3 + 15*B*b^2*c*d 
^4*e^2 - 18*B*b*c^2*d^5*e) + (13*B*c^3*d^7 - 11*A*c^3*d^6*e + 5*A*b^3*d^3* 
e^4 - 7*B*b^3*d^4*e^3 + 27*A*b*c^2*d^5*e^2 - 21*A*b^2*c*d^4*e^3 + 27*B*b^2 
*c*d^5*e^2 - 33*B*b*c^2*d^6*e)/(2*e))/(d^2*e^7 + e^9*x^2 + 2*d*e^8*x) + x^ 
2*((B*b^3 + 3*A*b^2*c)/(2*e^3) + (3*d*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3* 
B*c^3*d)/e^4))/e - (3*b*c*(A*c + B*b))/e^3 + (3*B*c^3*d^2)/e^5))/(2*e) - ( 
3*d^2*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/(2*e^2) - (B*c^3*d^3)/( 
2*e^6)) - x^3*((d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (b*c*(A 
*c + B*b))/e^3 + (B*c^3*d^2)/e^5) - (log(d + e*x)*(21*B*c^3*d^5 + 3*A*b^3* 
d*e^4 - 15*A*c^3*d^4*e - 6*B*b^3*d^2*e^3 + 30*A*b*c^2*d^3*e^2 - 18*A*b^2*c 
*d^2*e^3 + 30*B*b^2*c*d^3*e^2 - 45*B*b*c^2*d^4*e))/e^8 + (B*c^3*x^5)/(5*e^ 
3)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1032, normalized size of antiderivative = 2.87 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x)
 

Output:

( - 60*log(d + e*x)*a*b**3*d**3*e**4 - 120*log(d + e*x)*a*b**3*d**2*e**5*x 
 - 60*log(d + e*x)*a*b**3*d*e**6*x**2 + 360*log(d + e*x)*a*b**2*c*d**4*e** 
3 + 720*log(d + e*x)*a*b**2*c*d**3*e**4*x + 360*log(d + e*x)*a*b**2*c*d**2 
*e**5*x**2 - 600*log(d + e*x)*a*b*c**2*d**5*e**2 - 1200*log(d + e*x)*a*b*c 
**2*d**4*e**3*x - 600*log(d + e*x)*a*b*c**2*d**3*e**4*x**2 + 300*log(d + e 
*x)*a*c**3*d**6*e + 600*log(d + e*x)*a*c**3*d**5*e**2*x + 300*log(d + e*x) 
*a*c**3*d**4*e**3*x**2 + 120*log(d + e*x)*b**4*d**4*e**3 + 240*log(d + e*x 
)*b**4*d**3*e**4*x + 120*log(d + e*x)*b**4*d**2*e**5*x**2 - 600*log(d + e* 
x)*b**3*c*d**5*e**2 - 1200*log(d + e*x)*b**3*c*d**4*e**3*x - 600*log(d + e 
*x)*b**3*c*d**3*e**4*x**2 + 900*log(d + e*x)*b**2*c**2*d**6*e + 1800*log(d 
 + e*x)*b**2*c**2*d**5*e**2*x + 900*log(d + e*x)*b**2*c**2*d**4*e**3*x**2 
- 420*log(d + e*x)*b*c**3*d**7 - 840*log(d + e*x)*b*c**3*d**6*e*x - 420*lo 
g(d + e*x)*b*c**3*d**5*e**2*x**2 - 30*a*b**3*d**3*e**4 + 60*a*b**3*d*e**6* 
x**2 + 20*a*b**3*e**7*x**3 + 180*a*b**2*c*d**4*e**3 - 360*a*b**2*c*d**2*e* 
*5*x**2 - 120*a*b**2*c*d*e**6*x**3 + 30*a*b**2*c*e**7*x**4 - 300*a*b*c**2* 
d**5*e**2 + 600*a*b*c**2*d**3*e**4*x**2 + 200*a*b*c**2*d**2*e**5*x**3 - 50 
*a*b*c**2*d*e**6*x**4 + 20*a*b*c**2*e**7*x**5 + 150*a*c**3*d**6*e - 300*a* 
c**3*d**4*e**3*x**2 - 100*a*c**3*d**3*e**4*x**3 + 25*a*c**3*d**2*e**5*x**4 
 - 10*a*c**3*d*e**6*x**5 + 5*a*c**3*e**7*x**6 + 60*b**4*d**4*e**3 - 120*b* 
*4*d**2*e**5*x**2 - 40*b**4*d*e**6*x**3 + 10*b**4*e**7*x**4 - 300*b**3*...