\(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx\) [36]

Optimal result

Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {(B d-A e) (a+b x)^4}{5 e (b d-a e) (d+e x)^5}+\frac {(4 b B d+A b e-5 a B e) (a+b x)^4}{20 e (b d-a e)^2 (d+e x)^4} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(86)=172\).

Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.45 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {a^3 e^3 (4 A e+B (d+5 e x))+a^2 b e^2 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+a b^2 e \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )}{20 e^5 (d+e x)^5} \] Input:

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^6,x]
 

Output:

-1/20*(a^3*e^3*(4*A*e + B*(d + 5*e*x)) + a^2*b*e^2*(3*A*e*(d + 5*e*x) + 2* 
B*(d^2 + 5*d*e*x + 10*e^2*x^2)) + a*b^2*e*(2*A*e*(d^2 + 5*d*e*x + 10*e^2*x 
^2) + 3*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) + b^3*(A*e*(d^3 + 
 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*B*(d^4 + 5*d^3*e*x + 10*d^2*e^ 
2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)))/(e^5*(d + e*x)^5)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 86, 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 = {87, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 48

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

Input:

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^6,x]
 

Output:

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

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 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. \(263\) vs. \(2(82)=164\).

Time = 0.22 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.07

method	result	size
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{e}-\frac {b^{2} \left (A b e +3 B a e +4 B b d \right ) x^{3}}{2 e^{2}}-\frac {b \left (2 A a b \,e^{2}+A \,b^{2} d e +2 B \,a^{2} e^{2}+3 B a b d e +4 b^{2} B \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {\left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +B \,a^{3} e^{3}+2 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{4 e^{4}}-\frac {4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 e^{5}}}{\left (e x +d \right )^{5}}\)	\(264\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{e}-\frac {\left (A \,b^{3} e +3 B a \,b^{2} e +4 b^{3} B d \right ) x^{3}}{2 e^{2}}-\frac {\left (2 A a \,b^{2} e^{2}+A \,b^{3} d e +2 B \,a^{2} b \,e^{2}+3 B a \,b^{2} d e +4 b^{3} B \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {\left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +B \,a^{3} e^{3}+2 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{4 e^{4}}-\frac {4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 e^{5}}}{\left (e x +d \right )^{5}}\)	\(272\)
default	\(-\frac {b^{3} B}{e^{5} \left (e x +d \right )}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{2 e^{5} \left (e x +d \right )^{2}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{4 e^{5} \left (e x +d \right )^{4}}-\frac {b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{e^{5} \left (e x +d \right )^{3}}\)	\(281\)
gosper	\(-\frac {20 B \,x^{4} b^{3} e^{4}+10 A \,x^{3} b^{3} e^{4}+30 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+20 A \,x^{2} a \,b^{2} e^{4}+10 A \,x^{2} b^{3} d \,e^{3}+20 B \,x^{2} a^{2} b \,e^{4}+30 B \,x^{2} a \,b^{2} d \,e^{3}+40 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+10 A x a \,b^{2} d \,e^{3}+5 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+10 B x \,a^{2} b d \,e^{3}+15 B x a \,b^{2} d^{2} e^{2}+20 B x \,b^{3} d^{3} e +4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 \left (e x +d \right )^{5} e^{5}}\)	\(299\)
parallelrisch	\(-\frac {20 B \,x^{4} b^{3} e^{4}+10 A \,x^{3} b^{3} e^{4}+30 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+20 A \,x^{2} a \,b^{2} e^{4}+10 A \,x^{2} b^{3} d \,e^{3}+20 B \,x^{2} a^{2} b \,e^{4}+30 B \,x^{2} a \,b^{2} d \,e^{3}+40 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+10 A x a \,b^{2} d \,e^{3}+5 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+10 B x \,a^{2} b d \,e^{3}+15 B x a \,b^{2} d^{2} e^{2}+20 B x \,b^{3} d^{3} e +4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 \left (e x +d \right )^{5} e^{5}}\)	\(299\)
orering	\(-\frac {20 B \,x^{4} b^{3} e^{4}+10 A \,x^{3} b^{3} e^{4}+30 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+20 A \,x^{2} a \,b^{2} e^{4}+10 A \,x^{2} b^{3} d \,e^{3}+20 B \,x^{2} a^{2} b \,e^{4}+30 B \,x^{2} a \,b^{2} d \,e^{3}+40 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+10 A x a \,b^{2} d \,e^{3}+5 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+10 B x \,a^{2} b d \,e^{3}+15 B x a \,b^{2} d^{2} e^{2}+20 B x \,b^{3} d^{3} e +4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 \left (e x +d \right )^{5} e^{5}}\)	\(299\)

Input:

int((b*x+a)^3*(B*x+A)/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (82) = 164\).

Time = 0.07 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^6,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (75) = 150\).

Time = 35.07 (sec) , antiderivative size = 372, normalized size of antiderivative = 4.33 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=\frac {- 4 A a^{3} e^{4} - 3 A a^{2} b d e^{3} - 2 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} - 2 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e - 4 B b^{3} d^{4} - 20 B b^{3} e^{4} x^{4} + x^{3} \left (- 10 A b^{3} e^{4} - 30 B a b^{2} e^{4} - 40 B b^{3} d e^{3}\right ) + x^{2} \left (- 20 A a b^{2} e^{4} - 10 A b^{3} d e^{3} - 20 B a^{2} b e^{4} - 30 B a b^{2} d e^{3} - 40 B b^{3} d^{2} e^{2}\right ) + x \left (- 15 A a^{2} b e^{4} - 10 A a b^{2} d e^{3} - 5 A b^{3} d^{2} e^{2} - 5 B a^{3} e^{4} - 10 B a^{2} b d e^{3} - 15 B a b^{2} d^{2} e^{2} - 20 B b^{3} d^{3} e\right )}{20 d^{5} e^{5} + 100 d^{4} e^{6} x + 200 d^{3} e^{7} x^{2} + 200 d^{2} e^{8} x^{3} + 100 d e^{9} x^{4} + 20 e^{10} x^{5}} \] Input:

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**6,x)
 

Output:

(-4*A*a**3*e**4 - 3*A*a**2*b*d*e**3 - 2*A*a*b**2*d**2*e**2 - A*b**3*d**3*e 
 - B*a**3*d*e**3 - 2*B*a**2*b*d**2*e**2 - 3*B*a*b**2*d**3*e - 4*B*b**3*d** 
4 - 20*B*b**3*e**4*x**4 + x**3*(-10*A*b**3*e**4 - 30*B*a*b**2*e**4 - 40*B* 
b**3*d*e**3) + x**2*(-20*A*a*b**2*e**4 - 10*A*b**3*d*e**3 - 20*B*a**2*b*e* 
*4 - 30*B*a*b**2*d*e**3 - 40*B*b**3*d**2*e**2) + x*(-15*A*a**2*b*e**4 - 10 
*A*a*b**2*d*e**3 - 5*A*b**3*d**2*e**2 - 5*B*a**3*e**4 - 10*B*a**2*b*d*e**3 
 - 15*B*a*b**2*d**2*e**2 - 20*B*b**3*d**3*e))/(20*d**5*e**5 + 100*d**4*e** 
6*x + 200*d**3*e**7*x**2 + 200*d**2*e**8*x**3 + 100*d*e**9*x**4 + 20*e**10 
*x**5)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (82) = 164\).

Time = 0.04 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^6,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (82) = 164\).

Time = 0.12 (sec) , antiderivative size = 298, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b^{3} e^{4} x^{4} + 40 \, B b^{3} d e^{3} x^{3} + 30 \, B a b^{2} e^{4} x^{3} + 10 \, A b^{3} e^{4} x^{3} + 40 \, B b^{3} d^{2} e^{2} x^{2} + 30 \, B a b^{2} d e^{3} x^{2} + 10 \, A b^{3} d e^{3} x^{2} + 20 \, B a^{2} b e^{4} x^{2} + 20 \, A a b^{2} e^{4} x^{2} + 20 \, B b^{3} d^{3} e x + 15 \, B a b^{2} d^{2} e^{2} x + 5 \, A b^{3} d^{2} e^{2} x + 10 \, B a^{2} b d e^{3} x + 10 \, A a b^{2} d e^{3} x + 5 \, B a^{3} e^{4} x + 15 \, A a^{2} b e^{4} x + 4 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 2 \, B a^{2} b d^{2} e^{2} + 2 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} + 4 \, A a^{3} e^{4}}{20 \, {\left (e x + d\right )}^{5} e^{5}} \] Input:

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^6,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.57 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {\frac {B\,a^3\,d\,e^3+4\,A\,a^3\,e^4+2\,B\,a^2\,b\,d^2\,e^2+3\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+2\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{20\,e^5}+\frac {x\,\left (B\,a^3\,e^3+2\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+4\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{2\,e^3}+\frac {B\,b^3\,x^4}{e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \] Input:

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^6,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.23 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=\frac {b^{4} e^{4} x^{5}-10 a \,b^{3} d \,e^{3} x^{3}-10 a^{2} b^{2} d \,e^{3} x^{2}-10 a \,b^{3} d^{2} e^{2} x^{2}-5 a^{3} b d \,e^{3} x -5 a^{2} b^{2} d^{2} e^{2} x -5 a \,b^{3} d^{3} e x -a^{4} d \,e^{3}-a^{3} b \,d^{2} e^{2}-a^{2} b^{2} d^{3} e -a \,b^{3} d^{4}}{5 d \,e^{4} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right )} \] Input:

int((b*x+a)^3*(B*x+A)/(e*x+d)^6,x)
 

Output:

( - a**4*d*e**3 - a**3*b*d**2*e**2 - 5*a**3*b*d*e**3*x - a**2*b**2*d**3*e 
- 5*a**2*b**2*d**2*e**2*x - 10*a**2*b**2*d*e**3*x**2 - a*b**3*d**4 - 5*a*b 
**3*d**3*e*x - 10*a*b**3*d**2*e**2*x**2 - 10*a*b**3*d*e**3*x**3 + b**4*e** 
4*x**5)/(5*d*e**4*(d**5 + 5*d**4*e*x + 10*d**3*e**2*x**2 + 10*d**2*e**3*x* 
*3 + 5*d*e**4*x**4 + e**5*x**5))