\(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx\) [38]

Optimal result

Integrand size = 20, antiderivative size = 163 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{6 e^5 (d+e x)^6}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{5 e^5 (d+e x)^5}+\frac {b^2 (4 b B d-A b e-3 a B e)}{4 e^5 (d+e x)^4}-\frac {b^3 B}{3 e^5 (d+e x)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {10 a^3 e^3 (6 A e+B (d+7 e x))+6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )}{420 e^5 (d+e x)^7} \] Input:

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^8,x]
 

Output:

-1/420*(10*a^3*e^3*(6*A*e + B*(d + 7*e*x)) + 6*a^2*b*e^2*(5*A*e*(d + 7*e*x 
) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + 3*a*b^2*e*(4*A*e*(d^2 + 7*d*e*x + 
21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + b^3*(3* 
A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 
 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)))/(e^5*(d + e*x)^7)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 163, 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 = {86, 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.

\(\displaystyle \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 (-3 a B e-A b e+4 b B d)}{4 e^5 (d+e x)^4}-\frac {3 b (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (d+e x)^5}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac {(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}-\frac {b^3 B}{3 e^5 (d+e x)^3}\)

Input:

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^8,x]
 

Output:

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

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.66

method	result	size
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{3 e}-\frac {b^{2} \left (3 A b e +9 B a e +4 B b d \right ) x^{3}}{12 e^{2}}-\frac {b \left (12 A a b \,e^{2}+3 A \,b^{2} d e +12 B \,a^{2} e^{2}+9 B a b d e +4 b^{2} B \,d^{2}\right ) x^{2}}{20 e^{3}}-\frac {\left (30 A \,a^{2} b \,e^{3}+12 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +10 B \,a^{3} e^{3}+12 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{60 e^{4}}-\frac {60 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}+12 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}+12 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{420 e^{5}}}{\left (e x +d \right )^{7}}\)	\(270\)
default	\(-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{7 e^{5} \left (e x +d \right )^{7}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{4 e^{5} \left (e x +d \right )^{4}}-\frac {b^{3} B}{3 e^{5} \left (e x +d \right )^{3}}\)	\(281\)
gosper	\(-\frac {140 B \,x^{4} b^{3} e^{4}+105 A \,x^{3} b^{3} e^{4}+315 B \,x^{3} a \,b^{2} e^{4}+140 B \,x^{3} b^{3} d \,e^{3}+252 A \,x^{2} a \,b^{2} e^{4}+63 A \,x^{2} b^{3} d \,e^{3}+252 B \,x^{2} a^{2} b \,e^{4}+189 B \,x^{2} a \,b^{2} d \,e^{3}+84 B \,x^{2} b^{3} d^{2} e^{2}+210 A x \,a^{2} b \,e^{4}+84 A x a \,b^{2} d \,e^{3}+21 A x \,b^{3} d^{2} e^{2}+70 B x \,a^{3} e^{4}+84 B x \,a^{2} b d \,e^{3}+63 B x a \,b^{2} d^{2} e^{2}+28 B x \,b^{3} d^{3} e +60 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}+12 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}+12 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{420 e^{5} \left (e x +d \right )^{7}}\)	\(301\)
orering	\(-\frac {140 B \,x^{4} b^{3} e^{4}+105 A \,x^{3} b^{3} e^{4}+315 B \,x^{3} a \,b^{2} e^{4}+140 B \,x^{3} b^{3} d \,e^{3}+252 A \,x^{2} a \,b^{2} e^{4}+63 A \,x^{2} b^{3} d \,e^{3}+252 B \,x^{2} a^{2} b \,e^{4}+189 B \,x^{2} a \,b^{2} d \,e^{3}+84 B \,x^{2} b^{3} d^{2} e^{2}+210 A x \,a^{2} b \,e^{4}+84 A x a \,b^{2} d \,e^{3}+21 A x \,b^{3} d^{2} e^{2}+70 B x \,a^{3} e^{4}+84 B x \,a^{2} b d \,e^{3}+63 B x a \,b^{2} d^{2} e^{2}+28 B x \,b^{3} d^{3} e +60 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}+12 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}+12 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{420 e^{5} \left (e x +d \right )^{7}}\)	\(301\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{3 e}-\frac {\left (3 A \,b^{3} e^{3}+9 B a \,b^{2} e^{3}+4 b^{3} B d \,e^{2}\right ) x^{3}}{12 e^{4}}-\frac {\left (12 A a \,b^{2} e^{4}+3 A \,b^{3} d \,e^{3}+12 B \,a^{2} b \,e^{4}+9 B a \,b^{2} d \,e^{3}+4 b^{3} B \,d^{2} e^{2}\right ) x^{2}}{20 e^{5}}-\frac {\left (30 A \,a^{2} b \,e^{5}+12 A a \,b^{2} d \,e^{4}+3 A \,b^{3} d^{2} e^{3}+10 B \,a^{3} e^{5}+12 B \,a^{2} b d \,e^{4}+9 B a \,b^{2} d^{2} e^{3}+4 b^{3} B \,d^{3} e^{2}\right ) x}{60 e^{6}}-\frac {60 a^{3} A \,e^{6}+30 A \,a^{2} b d \,e^{5}+12 A a \,b^{2} d^{2} e^{4}+3 A \,b^{3} d^{3} e^{3}+10 B \,a^{3} d \,e^{5}+12 B \,a^{2} b \,d^{2} e^{4}+9 B a \,b^{2} d^{3} e^{3}+4 B \,b^{3} d^{4} e^{2}}{420 e^{7}}}{\left (e x +d \right )^{7}}\)	\(306\)
parallelrisch	\(-\frac {140 B \,b^{3} x^{4} e^{6}+105 A \,b^{3} e^{6} x^{3}+315 B a \,b^{2} e^{6} x^{3}+140 B \,b^{3} d \,e^{5} x^{3}+252 A a \,b^{2} e^{6} x^{2}+63 A \,b^{3} d \,e^{5} x^{2}+252 B \,a^{2} b \,e^{6} x^{2}+189 B a \,b^{2} d \,e^{5} x^{2}+84 B \,b^{3} d^{2} e^{4} x^{2}+210 A \,a^{2} b \,e^{6} x +84 A a \,b^{2} d \,e^{5} x +21 A \,b^{3} d^{2} e^{4} x +70 B \,a^{3} e^{6} x +84 B \,a^{2} b d \,e^{5} x +63 B a \,b^{2} d^{2} e^{4} x +28 B \,b^{3} d^{3} e^{3} x +60 a^{3} A \,e^{6}+30 A \,a^{2} b d \,e^{5}+12 A a \,b^{2} d^{2} e^{4}+3 A \,b^{3} d^{3} e^{3}+10 B \,a^{3} d \,e^{5}+12 B \,a^{2} b \,d^{2} e^{4}+9 B a \,b^{2} d^{3} e^{3}+4 B \,b^{3} d^{4} e^{2}}{420 e^{7} \left (e x +d \right )^{7}}\)	\(310\)

Input:

int((b*x+a)^3*(B*x+A)/(e*x+d)^8,x,method=_RETURNVERBOSE)
 

Output:

(-1/3*b^3*B/e*x^4-1/12*b^2/e^2*(3*A*b*e+9*B*a*e+4*B*b*d)*x^3-1/20*b/e^3*(1 
2*A*a*b*e^2+3*A*b^2*d*e+12*B*a^2*e^2+9*B*a*b*d*e+4*B*b^2*d^2)*x^2-1/60/e^4 
*(30*A*a^2*b*e^3+12*A*a*b^2*d*e^2+3*A*b^3*d^2*e+10*B*a^3*e^3+12*B*a^2*b*d* 
e^2+9*B*a*b^2*d^2*e+4*B*b^3*d^3)*x-1/420/e^5*(60*A*a^3*e^4+30*A*a^2*b*d*e^ 
3+12*A*a*b^2*d^2*e^2+3*A*b^3*d^3*e+10*B*a^3*d*e^3+12*B*a^2*b*d^2*e^2+9*B*a 
*b^2*d^3*e+4*B*b^3*d^4))/(e*x+d)^7
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (153) = 306\).

Time = 0.07 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \, {\left (4 \, B b^{3} d e^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \, {\left (4 \, B b^{3} d^{2} e^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \, {\left (4 \, B b^{3} d^{3} e + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^8,x, algorithm="fricas")
 

Output:

-1/420*(140*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 60*A*a^3*e^4 + 3*(3*B*a*b^2 + A* 
b^3)*d^3*e + 12*(B*a^2*b + A*a*b^2)*d^2*e^2 + 10*(B*a^3 + 3*A*a^2*b)*d*e^3 
 + 35*(4*B*b^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 21*(4*B*b^3*d^2*e^ 
2 + 3*(3*B*a*b^2 + A*b^3)*d*e^3 + 12*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 7*(4*B 
*b^3*d^3*e + 3*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 
+ 10*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^ 
5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e 
^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (153) = 306\).

Time = 0.06 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \, {\left (4 \, B b^{3} d e^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \, {\left (4 \, B b^{3} d^{2} e^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \, {\left (4 \, B b^{3} d^{3} e + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^8,x, algorithm="maxima")
 

Output:

-1/420*(140*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 60*A*a^3*e^4 + 3*(3*B*a*b^2 + A* 
b^3)*d^3*e + 12*(B*a^2*b + A*a*b^2)*d^2*e^2 + 10*(B*a^3 + 3*A*a^2*b)*d*e^3 
 + 35*(4*B*b^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 21*(4*B*b^3*d^2*e^ 
2 + 3*(3*B*a*b^2 + A*b^3)*d*e^3 + 12*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 7*(4*B 
*b^3*d^3*e + 3*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 
+ 10*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^ 
5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e 
^5)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {140 \, B b^{3} e^{4} x^{4} + 140 \, B b^{3} d e^{3} x^{3} + 315 \, B a b^{2} e^{4} x^{3} + 105 \, A b^{3} e^{4} x^{3} + 84 \, B b^{3} d^{2} e^{2} x^{2} + 189 \, B a b^{2} d e^{3} x^{2} + 63 \, A b^{3} d e^{3} x^{2} + 252 \, B a^{2} b e^{4} x^{2} + 252 \, A a b^{2} e^{4} x^{2} + 28 \, B b^{3} d^{3} e x + 63 \, B a b^{2} d^{2} e^{2} x + 21 \, A b^{3} d^{2} e^{2} x + 84 \, B a^{2} b d e^{3} x + 84 \, A a b^{2} d e^{3} x + 70 \, B a^{3} e^{4} x + 210 \, A a^{2} b e^{4} x + 4 \, B b^{3} d^{4} + 9 \, B a b^{2} d^{3} e + 3 \, A b^{3} d^{3} e + 12 \, B a^{2} b d^{2} e^{2} + 12 \, A a b^{2} d^{2} e^{2} + 10 \, B a^{3} d e^{3} + 30 \, A a^{2} b d e^{3} + 60 \, A a^{3} e^{4}}{420 \, {\left (e x + d\right )}^{7} e^{5}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^8,x, algorithm="giac")
 

Output:

-1/420*(140*B*b^3*e^4*x^4 + 140*B*b^3*d*e^3*x^3 + 315*B*a*b^2*e^4*x^3 + 10 
5*A*b^3*e^4*x^3 + 84*B*b^3*d^2*e^2*x^2 + 189*B*a*b^2*d*e^3*x^2 + 63*A*b^3* 
d*e^3*x^2 + 252*B*a^2*b*e^4*x^2 + 252*A*a*b^2*e^4*x^2 + 28*B*b^3*d^3*e*x + 
 63*B*a*b^2*d^2*e^2*x + 21*A*b^3*d^2*e^2*x + 84*B*a^2*b*d*e^3*x + 84*A*a*b 
^2*d*e^3*x + 70*B*a^3*e^4*x + 210*A*a^2*b*e^4*x + 4*B*b^3*d^4 + 9*B*a*b^2* 
d^3*e + 3*A*b^3*d^3*e + 12*B*a^2*b*d^2*e^2 + 12*A*a*b^2*d^2*e^2 + 10*B*a^3 
*d*e^3 + 30*A*a^2*b*d*e^3 + 60*A*a^3*e^4)/((e*x + d)^7*e^5)
 

Mupad [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {\frac {10\,B\,a^3\,d\,e^3+60\,A\,a^3\,e^4+12\,B\,a^2\,b\,d^2\,e^2+30\,A\,a^2\,b\,d\,e^3+9\,B\,a\,b^2\,d^3\,e+12\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+3\,A\,b^3\,d^3\,e}{420\,e^5}+\frac {x\,\left (10\,B\,a^3\,e^3+12\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+9\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{60\,e^4}+\frac {b^2\,x^3\,\left (3\,A\,b\,e+9\,B\,a\,e+4\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (12\,B\,a^2\,e^2+9\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{20\,e^3}+\frac {B\,b^3\,x^4}{3\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \] Input:

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^8,x)
 

Output:

-((60*A*a^3*e^4 + 4*B*b^3*d^4 + 3*A*b^3*d^3*e + 10*B*a^3*d*e^3 + 12*A*a*b^ 
2*d^2*e^2 + 12*B*a^2*b*d^2*e^2 + 30*A*a^2*b*d*e^3 + 9*B*a*b^2*d^3*e)/(420* 
e^5) + (x*(10*B*a^3*e^3 + 4*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e + 1 
2*A*a*b^2*d*e^2 + 9*B*a*b^2*d^2*e + 12*B*a^2*b*d*e^2))/(60*e^4) + (b^2*x^3 
*(3*A*b*e + 9*B*a*e + 4*B*b*d))/(12*e^2) + (b*x^2*(12*B*a^2*e^2 + 4*B*b^2* 
d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 9*B*a*b*d*e))/(20*e^3) + (B*b^3*x^4)/(3 
*e))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d 
^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=\frac {-35 b^{4} e^{4} x^{4}-105 a \,b^{3} e^{4} x^{3}-35 b^{4} d \,e^{3} x^{3}-126 a^{2} b^{2} e^{4} x^{2}-63 a \,b^{3} d \,e^{3} x^{2}-21 b^{4} d^{2} e^{2} x^{2}-70 a^{3} b \,e^{4} x -42 a^{2} b^{2} d \,e^{3} x -21 a \,b^{3} d^{2} e^{2} x -7 b^{4} d^{3} e x -15 a^{4} e^{4}-10 a^{3} b d \,e^{3}-6 a^{2} b^{2} d^{2} e^{2}-3 a \,b^{3} d^{3} e -b^{4} d^{4}}{105 e^{5} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:

int((b*x+a)^3*(B*x+A)/(e*x+d)^8,x)
 

Output:

( - 15*a**4*e**4 - 10*a**3*b*d*e**3 - 70*a**3*b*e**4*x - 6*a**2*b**2*d**2* 
e**2 - 42*a**2*b**2*d*e**3*x - 126*a**2*b**2*e**4*x**2 - 3*a*b**3*d**3*e - 
 21*a*b**3*d**2*e**2*x - 63*a*b**3*d*e**3*x**2 - 105*a*b**3*e**4*x**3 - b* 
*4*d**4 - 7*b**4*d**3*e*x - 21*b**4*d**2*e**2*x**2 - 35*b**4*d*e**3*x**3 - 
 35*b**4*e**4*x**4)/(105*e**5*(d**7 + 7*d**6*e*x + 21*d**5*e**2*x**2 + 35* 
d**4*e**3*x**3 + 35*d**3*e**4*x**4 + 21*d**2*e**5*x**5 + 7*d*e**6*x**6 + e 
**7*x**7))