\(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx\) [37]

Optimal result

Integrand size = 20, antiderivative size = 133 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4} \] Output:

-1/6*(-A*e+B*d)*(b*x+a)^4/e/(-a*e+b*d)/(e*x+d)^6+1/15*(A*b*e-3*B*a*e+2*B*b 
*d)*(b*x+a)^4/e/(-a*e+b*d)^2/(e*x+d)^5+1/60*b*(A*b*e-3*B*a*e+2*B*b*d)*(b*x 
+a)^4/e/(-a*e+b*d)^3/(e*x+d)^4
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {2 a^3 e^3 (5 A e+B (d+6 e x))+3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{60 e^5 (d+e x)^6} \] Input:

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^7,x]
 

Output:

-1/60*(2*a^3*e^3*(5*A*e + B*(d + 6*e*x)) + 3*a^2*b*e^2*(2*A*e*(d + 6*e*x) 
+ B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 3*a*b^2*e*(A*e*(d^2 + 6*d*e*x + 15*e^2 
*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + b^3*(A*e*(d^3 + 
 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 15*d^2*e^ 
2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)))/(e^5*(d + e*x)^6)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {87, 55, 48}

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.

\(\displaystyle \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {b (a+b x)^4}{20 (d+e x)^4 (b d-a e)^2}+\frac {(a+b x)^4}{5 (d+e x)^5 (b d-a e)}\right ) (-3 a B e+A b e+2 b B d)}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\)

Input:

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^7,x]
 

Output:

-1/6*((B*d - A*e)*(a + b*x)^4)/(e*(b*d - a*e)*(d + e*x)^6) + ((2*b*B*d + A 
*b*e - 3*a*B*e)*((a + b*x)^4/(5*(b*d - a*e)*(d + e*x)^5) + (b*(a + b*x)^4) 
/(20*(b*d - a*e)^2*(d + e*x)^4)))/(3*e*(b*d - a*e))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(127)=254\).

Time = 0.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.00

method	result	size
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {b^{2} \left (A b e +3 B a e +2 B b d \right ) x^{3}}{3 e^{2}}-\frac {b \left (3 A a b \,e^{2}+A \,b^{2} d e +3 B \,a^{2} e^{2}+3 B a b d e +2 b^{2} B \,d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (6 A \,a^{2} b \,e^{3}+3 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +2 B \,a^{3} e^{3}+3 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\)	\(266\)
default	\(-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {b^{3} B}{2 e^{5} \left (e x +d \right )^{2}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{4 e^{5} \left (e x +d \right )^{4}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{3 e^{5} \left (e x +d \right )^{3}}\)	\(281\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {\left (A \,b^{3} e^{2}+3 B a \,b^{2} e^{2}+2 b^{3} B d e \right ) x^{3}}{3 e^{3}}-\frac {\left (3 A a \,b^{2} e^{3}+A \,b^{3} d \,e^{2}+3 B \,a^{2} b \,e^{3}+3 B a \,b^{2} d \,e^{2}+2 b^{3} B \,d^{2} e \right ) x^{2}}{4 e^{4}}-\frac {\left (6 A \,a^{2} b \,e^{4}+3 A a \,b^{2} d \,e^{3}+A \,b^{3} d^{2} e^{2}+2 B \,a^{3} e^{4}+3 B \,a^{2} b d \,e^{3}+3 B a \,b^{2} d^{2} e^{2}+2 b^{3} B \,d^{3} e \right ) x}{10 e^{5}}-\frac {10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(294\)
gosper	\(-\frac {30 B \,x^{4} b^{3} e^{4}+20 A \,x^{3} b^{3} e^{4}+60 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}+15 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}+45 B \,x^{2} a \,b^{2} d \,e^{3}+30 B \,x^{2} b^{3} d^{2} e^{2}+36 A x \,a^{2} b \,e^{4}+18 A x a \,b^{2} d \,e^{3}+6 A x \,b^{3} d^{2} e^{2}+12 B x \,a^{3} e^{4}+18 B x \,a^{2} b d \,e^{3}+18 B x a \,b^{2} d^{2} e^{2}+12 B x \,b^{3} d^{3} e +10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\)	\(300\)
orering	\(-\frac {30 B \,x^{4} b^{3} e^{4}+20 A \,x^{3} b^{3} e^{4}+60 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}+15 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}+45 B \,x^{2} a \,b^{2} d \,e^{3}+30 B \,x^{2} b^{3} d^{2} e^{2}+36 A x \,a^{2} b \,e^{4}+18 A x a \,b^{2} d \,e^{3}+6 A x \,b^{3} d^{2} e^{2}+12 B x \,a^{3} e^{4}+18 B x \,a^{2} b d \,e^{3}+18 B x a \,b^{2} d^{2} e^{2}+12 B x \,b^{3} d^{3} e +10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\)	\(300\)
parallelrisch	\(-\frac {30 B \,b^{3} x^{4} e^{5}+20 A \,b^{3} e^{5} x^{3}+60 B a \,b^{2} e^{5} x^{3}+40 B \,b^{3} d \,e^{4} x^{3}+45 A a \,b^{2} e^{5} x^{2}+15 A \,b^{3} d \,e^{4} x^{2}+45 B \,a^{2} b \,e^{5} x^{2}+45 B a \,b^{2} d \,e^{4} x^{2}+30 B \,b^{3} d^{2} e^{3} x^{2}+36 A \,a^{2} b \,e^{5} x +18 A a \,b^{2} d \,e^{4} x +6 A \,b^{3} d^{2} e^{3} x +12 B \,a^{3} e^{5} x +18 B \,a^{2} b d \,e^{4} x +18 B a \,b^{2} d^{2} e^{3} x +12 B \,b^{3} d^{3} e^{2} x +10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6} \left (e x +d \right )^{6}}\)	\(307\)

Input:

int((b*x+a)^3*(B*x+A)/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*b^3*B/e*x^4-1/3*b^2/e^2*(A*b*e+3*B*a*e+2*B*b*d)*x^3-1/4*b/e^3*(3*A*a 
*b*e^2+A*b^2*d*e+3*B*a^2*e^2+3*B*a*b*d*e+2*B*b^2*d^2)*x^2-1/10/e^4*(6*A*a^ 
2*b*e^3+3*A*a*b^2*d*e^2+A*b^3*d^2*e+2*B*a^3*e^3+3*B*a^2*b*d*e^2+3*B*a*b^2* 
d^2*e+2*B*b^3*d^3)*x-1/60/e^5*(10*A*a^3*e^4+6*A*a^2*b*d*e^3+3*A*a*b^2*d^2* 
e^2+A*b^3*d^3*e+2*B*a^3*d*e^3+3*B*a^2*b*d^2*e^2+3*B*a*b^2*d^3*e+2*B*b^3*d^ 
4))/(e*x+d)^6
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (127) = 254\).

Time = 0.07 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="fricas")
 

Output:

-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3) 
*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20* 
(2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3*B 
*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e 
+ (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3 
*A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x 
^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (122) = 244\).

Time = 100.28 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=\frac {- 10 A a^{3} e^{4} - 6 A a^{2} b d e^{3} - 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - 2 B a^{3} d e^{3} - 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e - 2 B b^{3} d^{4} - 30 B b^{3} e^{4} x^{4} + x^{3} \left (- 20 A b^{3} e^{4} - 60 B a b^{2} e^{4} - 40 B b^{3} d e^{3}\right ) + x^{2} \left (- 45 A a b^{2} e^{4} - 15 A b^{3} d e^{3} - 45 B a^{2} b e^{4} - 45 B a b^{2} d e^{3} - 30 B b^{3} d^{2} e^{2}\right ) + x \left (- 36 A a^{2} b e^{4} - 18 A a b^{2} d e^{3} - 6 A b^{3} d^{2} e^{2} - 12 B a^{3} e^{4} - 18 B a^{2} b d e^{3} - 18 B a b^{2} d^{2} e^{2} - 12 B b^{3} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \] Input:

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**7,x)
 

Output:

(-10*A*a**3*e**4 - 6*A*a**2*b*d*e**3 - 3*A*a*b**2*d**2*e**2 - A*b**3*d**3* 
e - 2*B*a**3*d*e**3 - 3*B*a**2*b*d**2*e**2 - 3*B*a*b**2*d**3*e - 2*B*b**3* 
d**4 - 30*B*b**3*e**4*x**4 + x**3*(-20*A*b**3*e**4 - 60*B*a*b**2*e**4 - 40 
*B*b**3*d*e**3) + x**2*(-45*A*a*b**2*e**4 - 15*A*b**3*d*e**3 - 45*B*a**2*b 
*e**4 - 45*B*a*b**2*d*e**3 - 30*B*b**3*d**2*e**2) + x*(-36*A*a**2*b*e**4 - 
 18*A*a*b**2*d*e**3 - 6*A*b**3*d**2*e**2 - 12*B*a**3*e**4 - 18*B*a**2*b*d* 
e**3 - 18*B*a*b**2*d**2*e**2 - 12*B*b**3*d**3*e))/(60*d**6*e**5 + 360*d**5 
*e**6*x + 900*d**4*e**7*x**2 + 1200*d**3*e**8*x**3 + 900*d**2*e**9*x**4 + 
360*d*e**10*x**5 + 60*e**11*x**6)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (127) = 254\).

Time = 0.04 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="maxima")
 

Output:

-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3) 
*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20* 
(2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3*B 
*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e 
+ (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3 
*A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x 
^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (127) = 254\).

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 40 \, B b^{3} d e^{3} x^{3} + 60 \, B a b^{2} e^{4} x^{3} + 20 \, A b^{3} e^{4} x^{3} + 30 \, B b^{3} d^{2} e^{2} x^{2} + 45 \, B a b^{2} d e^{3} x^{2} + 15 \, A b^{3} d e^{3} x^{2} + 45 \, B a^{2} b e^{4} x^{2} + 45 \, A a b^{2} e^{4} x^{2} + 12 \, B b^{3} d^{3} e x + 18 \, B a b^{2} d^{2} e^{2} x + 6 \, A b^{3} d^{2} e^{2} x + 18 \, B a^{2} b d e^{3} x + 18 \, A a b^{2} d e^{3} x + 12 \, B a^{3} e^{4} x + 36 \, A a^{2} b e^{4} x + 2 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} + 2 \, B a^{3} d e^{3} + 6 \, A a^{2} b d e^{3} + 10 \, A a^{3} e^{4}}{60 \, {\left (e x + d\right )}^{6} e^{5}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^7,x, algorithm="giac")
 

Output:

-1/60*(30*B*b^3*e^4*x^4 + 40*B*b^3*d*e^3*x^3 + 60*B*a*b^2*e^4*x^3 + 20*A*b 
^3*e^4*x^3 + 30*B*b^3*d^2*e^2*x^2 + 45*B*a*b^2*d*e^3*x^2 + 15*A*b^3*d*e^3* 
x^2 + 45*B*a^2*b*e^4*x^2 + 45*A*a*b^2*e^4*x^2 + 12*B*b^3*d^3*e*x + 18*B*a* 
b^2*d^2*e^2*x + 6*A*b^3*d^2*e^2*x + 18*B*a^2*b*d*e^3*x + 18*A*a*b^2*d*e^3* 
x + 12*B*a^3*e^4*x + 36*A*a^2*b*e^4*x + 2*B*b^3*d^4 + 3*B*a*b^2*d^3*e + A* 
b^3*d^3*e + 3*B*a^2*b*d^2*e^2 + 3*A*a*b^2*d^2*e^2 + 2*B*a^3*d*e^3 + 6*A*a^ 
2*b*d*e^3 + 10*A*a^3*e^4)/((e*x + d)^6*e^5)
 

Mupad [B] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {\frac {2\,B\,a^3\,d\,e^3+10\,A\,a^3\,e^4+3\,B\,a^2\,b\,d^2\,e^2+6\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+3\,A\,a\,b^2\,d^2\,e^2+2\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{60\,e^5}+\frac {x\,\left (2\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2+6\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+3\,A\,a\,b^2\,d\,e^2+2\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{10\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{3\,e^2}+\frac {b\,x^2\,\left (3\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+3\,A\,a\,b\,e^2+2\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^3\,x^4}{2\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^7,x)
 

Output:

-((10*A*a^3*e^4 + 2*B*b^3*d^4 + A*b^3*d^3*e + 2*B*a^3*d*e^3 + 3*A*a*b^2*d^ 
2*e^2 + 3*B*a^2*b*d^2*e^2 + 6*A*a^2*b*d*e^3 + 3*B*a*b^2*d^3*e)/(60*e^5) + 
(x*(2*B*a^3*e^3 + 2*B*b^3*d^3 + 6*A*a^2*b*e^3 + A*b^3*d^2*e + 3*A*a*b^2*d* 
e^2 + 3*B*a*b^2*d^2*e + 3*B*a^2*b*d*e^2))/(10*e^4) + (b^2*x^3*(A*b*e + 3*B 
*a*e + 2*B*b*d))/(3*e^2) + (b*x^2*(3*B*a^2*e^2 + 2*B*b^2*d^2 + 3*A*a*b*e^2 
 + A*b^2*d*e + 3*B*a*b*d*e))/(4*e^3) + (B*b^3*x^4)/(2*e))/(d^6 + e^6*x^6 + 
 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e* 
x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=\frac {-15 b^{4} e^{4} x^{4}-40 a \,b^{3} e^{4} x^{3}-20 b^{4} d \,e^{3} x^{3}-45 a^{2} b^{2} e^{4} x^{2}-30 a \,b^{3} d \,e^{3} x^{2}-15 b^{4} d^{2} e^{2} x^{2}-24 a^{3} b \,e^{4} x -18 a^{2} b^{2} d \,e^{3} x -12 a \,b^{3} d^{2} e^{2} x -6 b^{4} d^{3} e x -5 a^{4} e^{4}-4 a^{3} b d \,e^{3}-3 a^{2} b^{2} d^{2} e^{2}-2 a \,b^{3} d^{3} e -b^{4} d^{4}}{30 e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:

int((b*x+a)^3*(B*x+A)/(e*x+d)^7,x)
 

Output:

( - 5*a**4*e**4 - 4*a**3*b*d*e**3 - 24*a**3*b*e**4*x - 3*a**2*b**2*d**2*e* 
*2 - 18*a**2*b**2*d*e**3*x - 45*a**2*b**2*e**4*x**2 - 2*a*b**3*d**3*e - 12 
*a*b**3*d**2*e**2*x - 30*a*b**3*d*e**3*x**2 - 40*a*b**3*e**4*x**3 - b**4*d 
**4 - 6*b**4*d**3*e*x - 15*b**4*d**2*e**2*x**2 - 20*b**4*d*e**3*x**3 - 15* 
b**4*e**4*x**4)/(30*e**5*(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3* 
e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6))