\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^3} \, dx\) [70]

Optimal result

Integrand size = 22, antiderivative size = 300 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {c \left (A c d e \left (10 c d^2+9 a e^2\right )-3 B \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )\right ) x}{e^7}-\frac {c^2 \left (10 B c d^3-6 A c d^2 e+9 a B d e^2-3 a A e^3\right ) x^2}{2 e^6}+\frac {c^2 \left (2 B c d^2-A c d e+a B e^2\right ) x^3}{e^5}-\frac {c^3 (3 B d-A e) x^4}{4 e^4}+\frac {B c^3 x^5}{5 e^3}+\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{2 e^8 (d+e x)^2}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^8 (d+e x)}-\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {5 A e \left (-2 a^3 e^6+6 a^2 c d e^4 (3 d+4 e x)+6 a c^2 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )\right )+B \left (-10 a^3 e^6 (d+2 e x)+30 a^2 c e^4 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+10 a c^2 e^2 \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+c^3 \left (-130 d^7+160 d^6 e x+500 d^5 e^2 x^2+140 d^4 e^3 x^3-35 d^3 e^4 x^4+14 d^2 e^5 x^5-7 d e^6 x^6+4 e^7 x^7\right )\right )-60 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^2 \log (d+e x)}{20 e^8 (d+e x)^2} \] Input:

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

Output:

(5*A*e*(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 4*e*x) + 6*a*c^2*e^2*(7*d^4 + 2* 
d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + c^3*(22*d^6 - 16*d^5*e 
*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x 
^6)) + B*(-10*a^3*e^6*(d + 2*e*x) + 30*a^2*c*e^4*(-5*d^3 - 4*d^2*e*x + 4*d 
*e^2*x^2 + 2*e^3*x^3) + 10*a*c^2*e^2*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 
 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + c^3*(-130*d^7 + 160*d^6*e*x 
 + 500*d^5*e^2*x^2 + 140*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7 
*d*e^6*x^6 + 4*e^7*x^7)) - 60*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 
 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^2*Log[d + e*x])/(20*e^8*(d + e*x)^2)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.

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

\(\Big \downarrow \) 652

\(\displaystyle \int \left (\frac {c \left (3 B \left (a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )-A c d e \left (9 a e^2+10 c d^2\right )\right )}{e^7}-\frac {3 c^2 x^2 \left (-a B e^2+A c d e-2 B c d^2\right )}{e^5}+\frac {c^2 x \left (3 a A e^3-9 a B d e^2+6 A c d^2 e-10 B c d^3\right )}{e^6}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^2}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^3}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)}+\frac {c^3 x^3 (A e-3 B d)}{e^4}+\frac {B c^3 x^4}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((c*(A*c*d*e*(10*c*d^2 + 9*a*e^2) - 3*B*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2* 
e^4))*x)/e^7) - (c^2*(10*B*c*d^3 - 6*A*c*d^2*e + 9*a*B*d*e^2 - 3*a*A*e^3)* 
x^2)/(2*e^6) + (c^2*(2*B*c*d^2 - A*c*d*e + a*B*e^2)*x^3)/e^5 - (c^3*(3*B*d 
 - A*e)*x^4)/(4*e^4) + (B*c^3*x^5)/(5*e^3) + ((B*d - A*e)*(c*d^2 + a*e^2)^ 
3)/(2*e^8*(d + e*x)^2) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e 
^2))/(e^8*(d + e*x)) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a 
*B*d*e^2 - a*A*e^3)*Log[d + e*x])/e^8
 

Defintions of rubi rules used

rule 652

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c 
*x^2)^p, x], x] /; FreeQ[{a, 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 = 0.71 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.42

method	result	size
norman	\(\frac {\frac {\left (6 A \,a^{2} c d \,e^{5}+36 A a \,c^{2} d^{3} e^{3}+30 A \,c^{3} d^{5} e -B \,a^{3} e^{6}-18 B \,a^{2} c \,d^{2} e^{4}-60 B a \,c^{2} d^{4} e^{2}-42 B \,c^{3} d^{6}\right ) x}{e^{7}}-\frac {A \,a^{3} e^{7}-9 A \,a^{2} c \,d^{2} e^{5}-54 A a \,c^{2} d^{4} e^{3}-45 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}+27 B \,a^{2} c \,d^{3} e^{4}+90 B a \,c^{2} d^{5} e^{2}+63 B \,c^{3} d^{7}}{2 e^{8}}+\frac {B \,c^{3} x^{7}}{5 e}-\frac {c \left (6 A a c d \,e^{3}+5 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-10 B a c \,d^{2} e^{2}-7 B \,c^{2} d^{4}\right ) x^{3}}{e^{5}}-\frac {c^{2} \left (5 A c d e -10 B a \,e^{2}-7 B c \,d^{2}\right ) x^{5}}{10 e^{3}}+\frac {c^{2} \left (6 A a \,e^{3}+5 A c \,d^{2} e -10 B a d \,e^{2}-7 B c \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {c^{3} \left (5 A e -7 B d \right ) x^{6}}{20 e^{2}}}{\left (e x +d \right )^{2}}+\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(425\)
default	\(-\frac {c \left (-\frac {1}{5} B \,c^{2} x^{5} e^{4}-\frac {1}{4} A \,c^{2} e^{4} x^{4}+\frac {3}{4} B \,c^{2} d \,e^{3} x^{4}+A \,c^{2} d \,e^{3} x^{3}-B a c \,e^{4} x^{3}-2 B \,c^{2} d^{2} e^{2} x^{3}-\frac {3}{2} A a c \,e^{4} x^{2}-3 A \,c^{2} d^{2} e^{2} x^{2}+\frac {9}{2} B a c d \,e^{3} x^{2}+5 B \,c^{2} d^{3} e \,x^{2}+9 A a c d \,e^{3} x +10 A \,c^{2} d^{3} e x -3 B \,a^{2} e^{4} x -18 B a c \,d^{2} e^{2} x -15 B \,c^{2} d^{4} x \right )}{e^{7}}+\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{e^{8} \left (e x +d \right )}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{2 e^{8} \left (e x +d \right )^{2}}\)	\(451\)
risch	\(\frac {B \,c^{3} x^{5}}{5 e^{3}}+\frac {c^{3} A \,x^{4}}{4 e^{3}}-\frac {3 c^{3} B d \,x^{4}}{4 e^{4}}-\frac {c^{3} A d \,x^{3}}{e^{4}}+\frac {c^{2} B a \,x^{3}}{e^{3}}+\frac {2 c^{3} B \,d^{2} x^{3}}{e^{5}}+\frac {3 c^{2} A a \,x^{2}}{2 e^{3}}+\frac {3 c^{3} A \,d^{2} x^{2}}{e^{5}}-\frac {9 c^{2} B a d \,x^{2}}{2 e^{4}}-\frac {5 c^{3} B \,d^{3} x^{2}}{e^{6}}-\frac {9 c^{2} A a d x}{e^{4}}-\frac {10 c^{3} A \,d^{3} x}{e^{6}}+\frac {3 c B \,a^{2} x}{e^{3}}+\frac {18 c^{2} B a \,d^{2} x}{e^{5}}+\frac {15 c^{3} B \,d^{4} x}{e^{7}}+\frac {\left (6 A \,a^{2} c d \,e^{5}+12 A a \,c^{2} d^{3} e^{3}+6 A \,c^{3} d^{5} e -B \,a^{3} e^{6}-9 B \,a^{2} c \,d^{2} e^{4}-15 B a \,c^{2} d^{4} e^{2}-7 B \,c^{3} d^{6}\right ) x -\frac {A \,a^{3} e^{7}-9 A \,a^{2} c \,d^{2} e^{5}-21 A a \,c^{2} d^{4} e^{3}-11 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}+15 B \,a^{2} c \,d^{3} e^{4}+27 B a \,c^{2} d^{5} e^{2}+13 B \,c^{3} d^{7}}{2 e}}{e^{7} \left (e x +d \right )^{2}}+\frac {3 c \ln \left (e x +d \right ) A \,a^{2}}{e^{3}}+\frac {18 c^{2} \ln \left (e x +d \right ) A a \,d^{2}}{e^{5}}+\frac {15 c^{3} \ln \left (e x +d \right ) A \,d^{4}}{e^{7}}-\frac {9 c \ln \left (e x +d \right ) B \,a^{2} d}{e^{4}}-\frac {30 c^{2} \ln \left (e x +d \right ) B a \,d^{3}}{e^{6}}-\frac {21 c^{3} \ln \left (e x +d \right ) B \,d^{5}}{e^{8}}\)	\(493\)
parallelrisch	\(\frac {720 A \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{4}-270 B \,a^{2} c \,d^{3} e^{4}+90 A \,a^{2} c \,d^{2} e^{5}+540 A a \,c^{2} d^{4} e^{3}+5 A \,x^{6} c^{3} e^{7}+4 B \,x^{7} c^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}-20 B x \,a^{3} e^{7}-1200 B x a \,c^{2} d^{4} e^{3}-360 B \ln \left (e x +d \right ) x \,a^{2} c \,d^{2} e^{5}-900 B a \,c^{2} d^{5} e^{2}+60 A \ln \left (e x +d \right ) x^{2} a^{2} c \,e^{7}+300 A \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{3}-420 B \ln \left (e x +d \right ) x^{2} c^{3} d^{5} e^{2}+25 A \,x^{4} c^{3} d^{2} e^{5}-35 B \,x^{4} c^{3} d^{3} e^{4}-100 A \,x^{3} c^{3} d^{3} e^{4}-630 B \,c^{3} d^{7}-1200 B \ln \left (e x +d \right ) x a \,c^{2} d^{4} e^{3}-180 B \ln \left (e x +d \right ) x^{2} a^{2} c d \,e^{6}+120 A \ln \left (e x +d \right ) x \,a^{2} c d \,e^{6}-600 B \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{3} e^{4}+360 A \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{5}-10 A \,x^{5} c^{3} d \,e^{6}-7 B \,x^{6} c^{3} d \,e^{6}+300 A \ln \left (e x +d \right ) c^{3} d^{6} e +600 A \ln \left (e x +d \right ) x \,c^{3} d^{5} e^{2}-840 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e +60 B \,x^{3} a^{2} c \,e^{7}+140 B \,x^{3} c^{3} d^{4} e^{3}+600 A x \,c^{3} d^{5} e^{2}-840 B x \,c^{3} d^{6} e +20 B \,x^{5} a \,c^{2} e^{7}+14 B \,x^{5} c^{3} d^{2} e^{5}+30 A \,x^{4} a \,c^{2} e^{7}+450 A \,c^{3} d^{6} e -10 B \,a^{3} d \,e^{6}+60 A \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{5}+360 A \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{3}-180 B \ln \left (e x +d \right ) a^{2} c \,d^{3} e^{4}-600 B \ln \left (e x +d \right ) a \,c^{2} d^{5} e^{2}+200 B \,x^{3} a \,c^{2} d^{2} e^{5}+120 A x \,a^{2} c d \,e^{6}+720 A x a \,c^{2} d^{3} e^{4}-360 B x \,a^{2} c \,d^{2} e^{5}-50 B \,x^{4} a \,c^{2} d \,e^{6}-120 A \,x^{3} a \,c^{2} d \,e^{6}-10 A \,a^{3} e^{7}}{20 e^{8} \left (e x +d \right )^{2}}\)	\(746\)

Input:

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

Output:

((6*A*a^2*c*d*e^5+36*A*a*c^2*d^3*e^3+30*A*c^3*d^5*e-B*a^3*e^6-18*B*a^2*c*d 
^2*e^4-60*B*a*c^2*d^4*e^2-42*B*c^3*d^6)/e^7*x-1/2*(A*a^3*e^7-9*A*a^2*c*d^2 
*e^5-54*A*a*c^2*d^4*e^3-45*A*c^3*d^6*e+B*a^3*d*e^6+27*B*a^2*c*d^3*e^4+90*B 
*a*c^2*d^5*e^2+63*B*c^3*d^7)/e^8+1/5*B*c^3*x^7/e-c*(6*A*a*c*d*e^3+5*A*c^2* 
d^3*e-3*B*a^2*e^4-10*B*a*c*d^2*e^2-7*B*c^2*d^4)/e^5*x^3-1/10*c^2*(5*A*c*d* 
e-10*B*a*e^2-7*B*c*d^2)/e^3*x^5+1/4*c^2*(6*A*a*e^3+5*A*c*d^2*e-10*B*a*d*e^ 
2-7*B*c*d^3)/e^4*x^4+1/20*c^3*(5*A*e-7*B*d)/e^2*x^6)/(e*x+d)^2+3*c/e^8*(A* 
a^2*e^5+6*A*a*c*d^2*e^3+5*A*c^2*d^4*e-3*B*a^2*d*e^4-10*B*a*c*d^3*e^2-7*B*c 
^2*d^5)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 691 vs. \(2 (292) = 584\).

Time = 0.08 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {4 \, B c^{3} e^{7} x^{7} - 130 \, B c^{3} d^{7} + 110 \, A c^{3} d^{6} e - 270 \, B a c^{2} d^{5} e^{2} + 210 \, A a c^{2} d^{4} e^{3} - 150 \, B a^{2} c d^{3} e^{4} + 90 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 10 \, A a^{3} e^{7} - {\left (7 \, B c^{3} d e^{6} - 5 \, A c^{3} e^{7}\right )} x^{6} + 2 \, {\left (7 \, B c^{3} d^{2} e^{5} - 5 \, A c^{3} d e^{6} + 10 \, B a c^{2} e^{7}\right )} x^{5} - 5 \, {\left (7 \, B c^{3} d^{3} e^{4} - 5 \, A c^{3} d^{2} e^{5} + 10 \, B a c^{2} d e^{6} - 6 \, A a c^{2} e^{7}\right )} x^{4} + 20 \, {\left (7 \, B c^{3} d^{4} e^{3} - 5 \, A c^{3} d^{3} e^{4} + 10 \, B a c^{2} d^{2} e^{5} - 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \, {\left (50 \, B c^{3} d^{5} e^{2} - 34 \, A c^{3} d^{4} e^{3} + 63 \, B a c^{2} d^{3} e^{4} - 33 \, A a c^{2} d^{2} e^{5} + 12 \, B a^{2} c d e^{6}\right )} x^{2} + 20 \, {\left (8 \, B c^{3} d^{6} e - 4 \, A c^{3} d^{5} e^{2} + 3 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} - 6 \, B a^{2} c d^{2} e^{5} + 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x - 60 \, {\left (7 \, B c^{3} d^{7} - 5 \, A c^{3} d^{6} e + 10 \, B a c^{2} d^{5} e^{2} - 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - A a^{2} c d^{2} e^{5} + {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B a c^{2} d^{3} e^{4} - 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - A a^{2} c e^{7}\right )} x^{2} + 2 \, {\left (7 \, B c^{3} d^{6} e - 5 \, A c^{3} d^{5} e^{2} + 10 \, B a c^{2} d^{4} e^{3} - 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} - A a^{2} c d e^{6}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^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 + 110*A*c^3*d^6*e - 270*B*a*c^2*d^5* 
e^2 + 210*A*a*c^2*d^4*e^3 - 150*B*a^2*c*d^3*e^4 + 90*A*a^2*c*d^2*e^5 - 10* 
B*a^3*d*e^6 - 10*A*a^3*e^7 - (7*B*c^3*d*e^6 - 5*A*c^3*e^7)*x^6 + 2*(7*B*c^ 
3*d^2*e^5 - 5*A*c^3*d*e^6 + 10*B*a*c^2*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 5*A 
*c^3*d^2*e^5 + 10*B*a*c^2*d*e^6 - 6*A*a*c^2*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 
 - 5*A*c^3*d^3*e^4 + 10*B*a*c^2*d^2*e^5 - 6*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7) 
*x^3 + 10*(50*B*c^3*d^5*e^2 - 34*A*c^3*d^4*e^3 + 63*B*a*c^2*d^3*e^4 - 33*A 
*a*c^2*d^2*e^5 + 12*B*a^2*c*d*e^6)*x^2 + 20*(8*B*c^3*d^6*e - 4*A*c^3*d^5*e 
^2 + 3*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4 - 6*B*a^2*c*d^2*e^5 + 6*A*a^2*c 
*d*e^6 - B*a^3*e^7)*x - 60*(7*B*c^3*d^7 - 5*A*c^3*d^6*e + 10*B*a*c^2*d^5*e 
^2 - 6*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - A*a^2*c*d^2*e^5 + (7*B*c^3*d^ 
5*e^2 - 5*A*c^3*d^4*e^3 + 10*B*a*c^2*d^3*e^4 - 6*A*a*c^2*d^2*e^5 + 3*B*a^2 
*c*d*e^6 - A*a^2*c*e^7)*x^2 + 2*(7*B*c^3*d^6*e - 5*A*c^3*d^5*e^2 + 10*B*a* 
c^2*d^4*e^3 - 6*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 - A*a^2*c*d*e^6)*x)*lo 
g(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)
 

Sympy [A] (verification not implemented)

Time = 2.72 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {B c^{3} x^{5}}{5 e^{3}} - \frac {3 c \left (a e^{2} + c d^{2}\right ) \left (- A a e^{3} - 5 A c d^{2} e + 3 B a d e^{2} + 7 B c d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A c^{3}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}}\right ) + x^{3} \left (- \frac {A c^{3} d}{e^{4}} + \frac {B a c^{2}}{e^{3}} + \frac {2 B c^{3} d^{2}}{e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A a c^{2}}{2 e^{3}} + \frac {3 A c^{3} d^{2}}{e^{5}} - \frac {9 B a c^{2} d}{2 e^{4}} - \frac {5 B c^{3} d^{3}}{e^{6}}\right ) + x \left (- \frac {9 A a c^{2} d}{e^{4}} - \frac {10 A c^{3} d^{3}}{e^{6}} + \frac {3 B a^{2} c}{e^{3}} + \frac {18 B a c^{2} d^{2}}{e^{5}} + \frac {15 B c^{3} d^{4}}{e^{7}}\right ) + \frac {- A a^{3} e^{7} + 9 A a^{2} c d^{2} e^{5} + 21 A a c^{2} d^{4} e^{3} + 11 A c^{3} d^{6} e - B a^{3} d e^{6} - 15 B a^{2} c d^{3} e^{4} - 27 B a c^{2} d^{5} e^{2} - 13 B c^{3} d^{7} + x \left (12 A a^{2} c d e^{6} + 24 A a c^{2} d^{3} e^{4} + 12 A c^{3} d^{5} e^{2} - 2 B a^{3} e^{7} - 18 B a^{2} c d^{2} e^{5} - 30 B a c^{2} d^{4} e^{3} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \] Input:

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

Output:

B*c**3*x**5/(5*e**3) - 3*c*(a*e**2 + c*d**2)*(-A*a*e**3 - 5*A*c*d**2*e + 3 
*B*a*d*e**2 + 7*B*c*d**3)*log(d + e*x)/e**8 + x**4*(A*c**3/(4*e**3) - 3*B* 
c**3*d/(4*e**4)) + x**3*(-A*c**3*d/e**4 + B*a*c**2/e**3 + 2*B*c**3*d**2/e* 
*5) + x**2*(3*A*a*c**2/(2*e**3) + 3*A*c**3*d**2/e**5 - 9*B*a*c**2*d/(2*e** 
4) - 5*B*c**3*d**3/e**6) + x*(-9*A*a*c**2*d/e**4 - 10*A*c**3*d**3/e**6 + 3 
*B*a**2*c/e**3 + 18*B*a*c**2*d**2/e**5 + 15*B*c**3*d**4/e**7) + (-A*a**3*e 
**7 + 9*A*a**2*c*d**2*e**5 + 21*A*a*c**2*d**4*e**3 + 11*A*c**3*d**6*e - B* 
a**3*d*e**6 - 15*B*a**2*c*d**3*e**4 - 27*B*a*c**2*d**5*e**2 - 13*B*c**3*d* 
*7 + x*(12*A*a**2*c*d*e**6 + 24*A*a*c**2*d**3*e**4 + 12*A*c**3*d**5*e**2 - 
 2*B*a**3*e**7 - 18*B*a**2*c*d**2*e**5 - 30*B*a*c**2*d**4*e**3 - 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 = 464, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {13 \, B c^{3} d^{7} - 11 \, A c^{3} d^{6} e + 27 \, B a c^{2} d^{5} e^{2} - 21 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + A a^{3} e^{7} + 2 \, {\left (7 \, B c^{3} d^{6} e - 6 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B c^{3} e^{4} x^{5} - 5 \, {\left (3 \, B c^{3} d e^{3} - A c^{3} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B c^{3} d^{2} e^{2} - A c^{3} d e^{3} + B a c^{2} e^{4}\right )} x^{3} - 10 \, {\left (10 \, B c^{3} d^{3} e - 6 \, A c^{3} d^{2} e^{2} + 9 \, B a c^{2} d e^{3} - 3 \, A a c^{2} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B c^{3} d^{4} - 10 \, A c^{3} d^{3} e + 18 \, B a c^{2} d^{2} e^{2} - 9 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:

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

Output:

-1/2*(13*B*c^3*d^7 - 11*A*c^3*d^6*e + 27*B*a*c^2*d^5*e^2 - 21*A*a*c^2*d^4* 
e^3 + 15*B*a^2*c*d^3*e^4 - 9*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 + A*a^3*e^7 + 2 
*(7*B*c^3*d^6*e - 6*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^ 
4 + 9*B*a^2*c*d^2*e^5 - 6*A*a^2*c*d*e^6 + B*a^3*e^7)*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 - A*c^3*e^4)*x^4 
 + 20*(2*B*c^3*d^2*e^2 - A*c^3*d*e^3 + B*a*c^2*e^4)*x^3 - 10*(10*B*c^3*d^3 
*e - 6*A*c^3*d^2*e^2 + 9*B*a*c^2*d*e^3 - 3*A*a*c^2*e^4)*x^2 + 20*(15*B*c^3 
*d^4 - 10*A*c^3*d^3*e + 18*B*a*c^2*d^2*e^2 - 9*A*a*c^2*d*e^3 + 3*B*a^2*c*e 
^4)*x)/e^7 - 3*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c 
^2*d^2*e^3 + 3*B*a^2*c*d*e^4 - A*a^2*c*e^5)*log(e*x + d)/e^8
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.56 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {3 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} - \frac {13 \, B c^{3} d^{7} - 11 \, A c^{3} d^{6} e + 27 \, B a c^{2} d^{5} e^{2} - 21 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + A a^{3} e^{7} + 2 \, {\left (7 \, B c^{3} d^{6} e - 6 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{8}} + \frac {4 \, B c^{3} e^{12} x^{5} - 15 \, B c^{3} d e^{11} x^{4} + 5 \, A c^{3} e^{12} x^{4} + 40 \, B c^{3} d^{2} e^{10} x^{3} - 20 \, A c^{3} d e^{11} x^{3} + 20 \, B a c^{2} e^{12} x^{3} - 100 \, B c^{3} d^{3} e^{9} x^{2} + 60 \, A c^{3} d^{2} e^{10} x^{2} - 90 \, B a c^{2} d e^{11} x^{2} + 30 \, A a c^{2} e^{12} x^{2} + 300 \, B c^{3} d^{4} e^{8} x - 200 \, A c^{3} d^{3} e^{9} x + 360 \, B a c^{2} d^{2} e^{10} x - 180 \, A a c^{2} d e^{11} x + 60 \, B a^{2} c e^{12} x}{20 \, e^{15}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx=x^4\,\left (\frac {A\,c^3}{4\,e^3}-\frac {3\,B\,c^3\,d}{4\,e^4}\right )-\frac {\frac {B\,a^3\,d\,e^6+A\,a^3\,e^7+15\,B\,a^2\,c\,d^3\,e^4-9\,A\,a^2\,c\,d^2\,e^5+27\,B\,a\,c^2\,d^5\,e^2-21\,A\,a\,c^2\,d^4\,e^3+13\,B\,c^3\,d^7-11\,A\,c^3\,d^6\,e}{2\,e}+x\,\left (B\,a^3\,e^6+9\,B\,a^2\,c\,d^2\,e^4-6\,A\,a^2\,c\,d\,e^5+15\,B\,a\,c^2\,d^4\,e^2-12\,A\,a\,c^2\,d^3\,e^3+7\,B\,c^3\,d^6-6\,A\,c^3\,d^5\,e\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}-x\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^3}-\frac {B\,c^3\,d^3}{e^6}\right )}{e}-\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e^2}+\frac {d^3\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^3}-\frac {3\,B\,a^2\,c}{e^3}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {B\,a\,c^2}{e^3}+\frac {B\,c^3\,d^2}{e^5}\right )+x^2\,\left (\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{2\,e}-\frac {3\,d^2\,\left (\frac {A\,c^3}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{2\,e^2}+\frac {3\,A\,a\,c^2}{2\,e^3}-\frac {B\,c^3\,d^3}{2\,e^6}\right )-\frac {\ln \left (d+e\,x\right )\,\left (9\,B\,a^2\,c\,d\,e^4-3\,A\,a^2\,c\,e^5+30\,B\,a\,c^2\,d^3\,e^2-18\,A\,a\,c^2\,d^2\,e^3+21\,B\,c^3\,d^5-15\,A\,c^3\,d^4\,e\right )}{e^8}+\frac {B\,c^3\,x^5}{5\,e^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.62 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(60*log(d + e*x)*a**3*c*d**3*e**5 + 120*log(d + e*x)*a**3*c*d**2*e**6*x + 
60*log(d + e*x)*a**3*c*d*e**7*x**2 - 180*log(d + e*x)*a**2*b*c*d**4*e**4 - 
 360*log(d + e*x)*a**2*b*c*d**3*e**5*x - 180*log(d + e*x)*a**2*b*c*d**2*e* 
*6*x**2 + 360*log(d + e*x)*a**2*c**2*d**5*e**3 + 720*log(d + e*x)*a**2*c** 
2*d**4*e**4*x + 360*log(d + e*x)*a**2*c**2*d**3*e**5*x**2 - 600*log(d + e* 
x)*a*b*c**2*d**6*e**2 - 1200*log(d + e*x)*a*b*c**2*d**5*e**3*x - 600*log(d 
 + e*x)*a*b*c**2*d**4*e**4*x**2 + 300*log(d + e*x)*a*c**3*d**7*e + 600*log 
(d + e*x)*a*c**3*d**6*e**2*x + 300*log(d + e*x)*a*c**3*d**5*e**3*x**2 - 42 
0*log(d + e*x)*b*c**3*d**8 - 840*log(d + e*x)*b*c**3*d**7*e*x - 420*log(d 
+ e*x)*b*c**3*d**6*e**2*x**2 - 10*a**4*d*e**7 + 10*a**3*b*e**8*x**2 + 30*a 
**3*c*d**3*e**5 - 60*a**3*c*d*e**7*x**2 - 90*a**2*b*c*d**4*e**4 + 180*a**2 
*b*c*d**2*e**6*x**2 + 60*a**2*b*c*d*e**7*x**3 + 180*a**2*c**2*d**5*e**3 - 
360*a**2*c**2*d**3*e**5*x**2 - 120*a**2*c**2*d**2*e**6*x**3 + 30*a**2*c**2 
*d*e**7*x**4 - 300*a*b*c**2*d**6*e**2 + 600*a*b*c**2*d**4*e**4*x**2 + 200* 
a*b*c**2*d**3*e**5*x**3 - 50*a*b*c**2*d**2*e**6*x**4 + 20*a*b*c**2*d*e**7* 
x**5 + 150*a*c**3*d**7*e - 300*a*c**3*d**5*e**3*x**2 - 100*a*c**3*d**4*e** 
4*x**3 + 25*a*c**3*d**3*e**5*x**4 - 10*a*c**3*d**2*e**6*x**5 + 5*a*c**3*d* 
e**7*x**6 - 210*b*c**3*d**8 + 420*b*c**3*d**6*e**2*x**2 + 140*b*c**3*d**5* 
e**3*x**3 - 35*b*c**3*d**4*e**4*x**4 + 14*b*c**3*d**3*e**5*x**5 - 7*b*c**3 
*d**2*e**6*x**6 + 4*b*c**3*d*e**7*x**7)/(20*d*e**8*(d**2 + 2*d*e*x + e*...