3.32.81 \(\int (a+b x)^3 (A+B x) (d+e x)^m \, dx\) [3181]

3.32.81.1 Optimal result

Integrand size = 20, antiderivative size = 186 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\frac {(b d-a e)^3 (B d-A e) (d+e x)^{1+m}}{e^5 (1+m)}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {b^3 B (d+e x)^{5+m}}{e^5 (5+m)} \]

output

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

3.32.81.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.89 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\frac {(d+e x)^{1+m} \left (\frac {(b d-a e)^3 (B d-A e)}{1+m}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)}{2+m}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^2}{3+m}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^3}{4+m}+\frac {b^3 B (d+e x)^4}{5+m}\right )}{e^5} \]

input

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

output

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

3.32.81.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 186, 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.

\(\displaystyle \int (a+b x)^3 (A+B x) (d+e x)^m \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^2 (d+e x)^{m+3} (3 a B e+A b e-4 b B d)}{e^4}+\frac {(a e-b d)^3 (A e-B d) (d+e x)^m}{e^4}+\frac {(a e-b d)^2 (d+e x)^{m+1} (a B e+3 A b e-4 b B d)}{e^4}-\frac {3 b (b d-a e) (d+e x)^{m+2} (a B e+A b e-2 b B d)}{e^4}+\frac {b^3 B (d+e x)^{m+4}}{e^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^2 (d+e x)^{m+4} (-3 a B e-A b e+4 b B d)}{e^5 (m+4)}+\frac {(b d-a e)^3 (B d-A e) (d+e x)^{m+1}}{e^5 (m+1)}-\frac {(b d-a e)^2 (d+e x)^{m+2} (-a B e-3 A b e+4 b B d)}{e^5 (m+2)}+\frac {3 b (b d-a e) (d+e x)^{m+3} (-a B e-A b e+2 b B d)}{e^5 (m+3)}+\frac {b^3 B (d+e x)^{m+5}}{e^5 (m+5)}\)

input

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

output

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

3.32.81.3.1 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]
 

3.32.81.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1164\) vs. \(2(186)=372\).

Time = 1.54 (sec) , antiderivative size = 1165, normalized size of antiderivative = 6.26

method	result	size
norman	\(\text {Expression too large to display}\)	\(1165\)
gosper	\(\text {Expression too large to display}\)	\(1270\)
risch	\(\text {Expression too large to display}\)	\(1712\)
parallelrisch	\(\text {Expression too large to display}\)	\(2612\)

input

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

output

b^3*B/(5+m)*x^5*exp(m*ln(e*x+d))+d*(A*a^3*e^4*m^4+14*A*a^3*e^4*m^3-3*A*a^2 
*b*d*e^3*m^3-B*a^3*d*e^3*m^3+71*A*a^3*e^4*m^2-36*A*a^2*b*d*e^3*m^2+6*A*a*b 
^2*d^2*e^2*m^2-12*B*a^3*d*e^3*m^2+6*B*a^2*b*d^2*e^2*m^2+154*A*a^3*e^4*m-14 
1*A*a^2*b*d*e^3*m+54*A*a*b^2*d^2*e^2*m-6*A*b^3*d^3*e*m-47*B*a^3*d*e^3*m+54 
*B*a^2*b*d^2*e^2*m-18*B*a*b^2*d^3*e*m+120*A*a^3*e^4-180*A*a^2*b*d*e^3+120* 
A*a*b^2*d^2*e^2-30*A*b^3*d^3*e-60*B*a^3*d*e^3+120*B*a^2*b*d^2*e^2-90*B*a*b 
^2*d^3*e+24*B*b^3*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*exp(m*ln( 
e*x+d))+(3*A*a^2*b*e^3*m^3+3*A*a*b^2*d*e^2*m^3+B*a^3*e^3*m^3+3*B*a^2*b*d*e 
^2*m^3+36*A*a^2*b*e^3*m^2+27*A*a*b^2*d*e^2*m^2-3*A*b^3*d^2*e*m^2+12*B*a^3* 
e^3*m^2+27*B*a^2*b*d*e^2*m^2-9*B*a*b^2*d^2*e*m^2+141*A*a^2*b*e^3*m+60*A*a* 
b^2*d*e^2*m-15*A*b^3*d^2*e*m+47*B*a^3*e^3*m+60*B*a^2*b*d*e^2*m-45*B*a*b^2* 
d^2*e*m+12*B*b^3*d^3*m+180*A*a^2*b*e^3+60*B*a^3*e^3)/e^3/(m^4+14*m^3+71*m^ 
2+154*m+120)*x^2*exp(m*ln(e*x+d))+(A*a^3*e^4*m^4+3*A*a^2*b*d*e^3*m^4+B*a^3 
*d*e^3*m^4+14*A*a^3*e^4*m^3+36*A*a^2*b*d*e^3*m^3-6*A*a*b^2*d^2*e^2*m^3+12* 
B*a^3*d*e^3*m^3-6*B*a^2*b*d^2*e^2*m^3+71*A*a^3*e^4*m^2+141*A*a^2*b*d*e^3*m 
^2-54*A*a*b^2*d^2*e^2*m^2+6*A*b^3*d^3*e*m^2+47*B*a^3*d*e^3*m^2-54*B*a^2*b* 
d^2*e^2*m^2+18*B*a*b^2*d^3*e*m^2+154*A*a^3*e^4*m+180*A*a^2*b*d*e^3*m-120*A 
*a*b^2*d^2*e^2*m+30*A*b^3*d^3*e*m+60*B*a^3*d*e^3*m-120*B*a^2*b*d^2*e^2*m+9 
0*B*a*b^2*d^3*e*m-24*B*b^3*d^4*m+120*A*a^3*e^4)/e^4/(m^5+15*m^4+85*m^3+225 
*m^2+274*m+120)*x*exp(m*ln(e*x+d))+(A*b*e*m+3*B*a*e*m+B*b*d*m+5*A*b*e+1...
 

3.32.81.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (186) = 372\).

Time = 0.26 (sec) , antiderivative size = 1282, normalized size of antiderivative = 6.89 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\text {Too large to display} \]

input

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

output

(A*a^3*d*e^4*m^4 + 24*B*b^3*d^5 + 120*A*a^3*d*e^4 - 30*(3*B*a*b^2 + A*b^3) 
*d^4*e + 120*(B*a^2*b + A*a*b^2)*d^3*e^2 - 60*(B*a^3 + 3*A*a^2*b)*d^2*e^3 
+ (B*b^3*e^5*m^4 + 10*B*b^3*e^5*m^3 + 35*B*b^3*e^5*m^2 + 50*B*b^3*e^5*m + 
24*B*b^3*e^5)*x^5 + (30*(3*B*a*b^2 + A*b^3)*e^5 + (B*b^3*d*e^4 + (3*B*a*b^ 
2 + A*b^3)*e^5)*m^4 + (6*B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*m^3 + ( 
11*B*b^3*d*e^4 + 41*(3*B*a*b^2 + A*b^3)*e^5)*m^2 + (6*B*b^3*d*e^4 + 61*(3* 
B*a*b^2 + A*b^3)*e^5)*m)*x^4 + (14*A*a^3*d*e^4 - (B*a^3 + 3*A*a^2*b)*d^2*e 
^3)*m^3 + (120*(B*a^2*b + A*a*b^2)*e^5 + ((3*B*a*b^2 + A*b^3)*d*e^4 + 3*(B 
*a^2*b + A*a*b^2)*e^5)*m^4 - 4*(B*b^3*d^2*e^3 - 2*(3*B*a*b^2 + A*b^3)*d*e^ 
4 - 9*(B*a^2*b + A*a*b^2)*e^5)*m^3 - (12*B*b^3*d^2*e^3 - 17*(3*B*a*b^2 + A 
*b^3)*d*e^4 - 147*(B*a^2*b + A*a*b^2)*e^5)*m^2 - 2*(4*B*b^3*d^2*e^3 - 5*(3 
*B*a*b^2 + A*b^3)*d*e^4 - 117*(B*a^2*b + A*a*b^2)*e^5)*m)*x^3 + (71*A*a^3* 
d*e^4 + 6*(B*a^2*b + A*a*b^2)*d^3*e^2 - 12*(B*a^3 + 3*A*a^2*b)*d^2*e^3)*m^ 
2 + (60*(B*a^3 + 3*A*a^2*b)*e^5 + (3*(B*a^2*b + A*a*b^2)*d*e^4 + (B*a^3 + 
3*A*a^2*b)*e^5)*m^4 - (3*(3*B*a*b^2 + A*b^3)*d^2*e^3 - 30*(B*a^2*b + A*a*b 
^2)*d*e^4 - 13*(B*a^3 + 3*A*a^2*b)*e^5)*m^3 + (12*B*b^3*d^3*e^2 - 18*(3*B* 
a*b^2 + A*b^3)*d^2*e^3 + 87*(B*a^2*b + A*a*b^2)*d*e^4 + 59*(B*a^3 + 3*A*a^ 
2*b)*e^5)*m^2 + (12*B*b^3*d^3*e^2 - 15*(3*B*a*b^2 + A*b^3)*d^2*e^3 + 60*(B 
*a^2*b + A*a*b^2)*d*e^4 + 107*(B*a^3 + 3*A*a^2*b)*e^5)*m)*x^2 + (154*A*a^3 
*d*e^4 - 6*(3*B*a*b^2 + A*b^3)*d^4*e + 54*(B*a^2*b + A*a*b^2)*d^3*e^2 -...
 

3.32.81.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14256 vs. \(2 (172) = 344\).

Time = 2.73 (sec) , antiderivative size = 14256, normalized size of antiderivative = 76.65 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\text {Too large to display} \]

input

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

output

Piecewise((d**m*(A*a**3*x + 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x** 
4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b**2*x**4/4 + B*b**3*x**5/5), 
Eq(e, 0)), (-3*A*a**3*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x 
**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 3*A*a**2*b*d*e**3/(12*d**4*e**5 + 4 
8*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*A* 
a**2*b*e**4*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e* 
*8*x**3 + 12*e**9*x**4) - 3*A*a*b**2*d**2*e**2/(12*d**4*e**5 + 48*d**3*e** 
6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*A*a*b**2*d*e 
**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 
+ 12*e**9*x**4) - 18*A*a*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 7 
2*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 3*A*b**3*d**3*e/(12*d* 
*4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x* 
*4) - 12*A*b**3*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7* 
x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 18*A*b**3*d*e**3*x**2/(12*d**4*e** 
5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 
12*A*b**3*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 4 
8*d*e**8*x**3 + 12*e**9*x**4) - B*a**3*d*e**3/(12*d**4*e**5 + 48*d**3*e**6 
*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*B*a**3*e**4*x/ 
(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e 
**9*x**4) - 3*B*a**2*b*d**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d*...
 

3.32.81.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (186) = 372\).

Time = 0.23 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.33 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a^{3}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {3 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A a^{2} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a^{3}}{e {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} B a^{2} b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A a b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {3 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B a b^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} A b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} B b^{3}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \]

input

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

output

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^3/((m^2 + 3*m + 2)*e^2) 
+ 3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*A*a^2*b/((m^2 + 3*m + 2) 
*e^2) + (e*x + d)^(m + 1)*A*a^3/(e*(m + 1)) + 3*((m^2 + 3*m + 2)*e^3*x^3 + 
 (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*B*a^2*b/((m^3 + 6* 
m^2 + 11*m + 6)*e^3) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 
2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*a*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 
3*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m 
^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*B*a*b^2/((m^4 + 10* 
m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 
+ 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)* 
(e*x + d)^m*A*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m 
^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 
- 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e* 
m*x + 24*d^5)*(e*x + d)^m*B*b^3/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m 
+ 120)*e^5)
 

3.32.81.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2519 vs. \(2 (186) = 372\).

Time = 0.30 (sec) , antiderivative size = 2519, normalized size of antiderivative = 13.54 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\text {Too large to display} \]

input

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

output

((e*x + d)^m*B*b^3*e^5*m^4*x^5 + (e*x + d)^m*B*b^3*d*e^4*m^4*x^4 + 3*(e*x 
+ d)^m*B*a*b^2*e^5*m^4*x^4 + (e*x + d)^m*A*b^3*e^5*m^4*x^4 + 10*(e*x + d)^ 
m*B*b^3*e^5*m^3*x^5 + 3*(e*x + d)^m*B*a*b^2*d*e^4*m^4*x^3 + (e*x + d)^m*A* 
b^3*d*e^4*m^4*x^3 + 3*(e*x + d)^m*B*a^2*b*e^5*m^4*x^3 + 3*(e*x + d)^m*A*a* 
b^2*e^5*m^4*x^3 + 6*(e*x + d)^m*B*b^3*d*e^4*m^3*x^4 + 33*(e*x + d)^m*B*a*b 
^2*e^5*m^3*x^4 + 11*(e*x + d)^m*A*b^3*e^5*m^3*x^4 + 35*(e*x + d)^m*B*b^3*e 
^5*m^2*x^5 + 3*(e*x + d)^m*B*a^2*b*d*e^4*m^4*x^2 + 3*(e*x + d)^m*A*a*b^2*d 
*e^4*m^4*x^2 + (e*x + d)^m*B*a^3*e^5*m^4*x^2 + 3*(e*x + d)^m*A*a^2*b*e^5*m 
^4*x^2 - 4*(e*x + d)^m*B*b^3*d^2*e^3*m^3*x^3 + 24*(e*x + d)^m*B*a*b^2*d*e^ 
4*m^3*x^3 + 8*(e*x + d)^m*A*b^3*d*e^4*m^3*x^3 + 36*(e*x + d)^m*B*a^2*b*e^5 
*m^3*x^3 + 36*(e*x + d)^m*A*a*b^2*e^5*m^3*x^3 + 11*(e*x + d)^m*B*b^3*d*e^4 
*m^2*x^4 + 123*(e*x + d)^m*B*a*b^2*e^5*m^2*x^4 + 41*(e*x + d)^m*A*b^3*e^5* 
m^2*x^4 + 50*(e*x + d)^m*B*b^3*e^5*m*x^5 + (e*x + d)^m*B*a^3*d*e^4*m^4*x + 
 3*(e*x + d)^m*A*a^2*b*d*e^4*m^4*x + (e*x + d)^m*A*a^3*e^5*m^4*x - 9*(e*x 
+ d)^m*B*a*b^2*d^2*e^3*m^3*x^2 - 3*(e*x + d)^m*A*b^3*d^2*e^3*m^3*x^2 + 30* 
(e*x + d)^m*B*a^2*b*d*e^4*m^3*x^2 + 30*(e*x + d)^m*A*a*b^2*d*e^4*m^3*x^2 + 
 13*(e*x + d)^m*B*a^3*e^5*m^3*x^2 + 39*(e*x + d)^m*A*a^2*b*e^5*m^3*x^2 - 1 
2*(e*x + d)^m*B*b^3*d^2*e^3*m^2*x^3 + 51*(e*x + d)^m*B*a*b^2*d*e^4*m^2*x^3 
 + 17*(e*x + d)^m*A*b^3*d*e^4*m^2*x^3 + 147*(e*x + d)^m*B*a^2*b*e^5*m^2*x^ 
3 + 147*(e*x + d)^m*A*a*b^2*e^5*m^2*x^3 + 6*(e*x + d)^m*B*b^3*d*e^4*m*x...
 

3.32.81.9 Mupad [B] (verification not implemented)

Time = 3.51 (sec) , antiderivative size = 1273, normalized size of antiderivative = 6.84 \[ \int (a+b x)^3 (A+B x) (d+e x)^m \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (-B\,a^3\,d^2\,e^3\,m^3-12\,B\,a^3\,d^2\,e^3\,m^2-47\,B\,a^3\,d^2\,e^3\,m-60\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4\,m^4+14\,A\,a^3\,d\,e^4\,m^3+71\,A\,a^3\,d\,e^4\,m^2+154\,A\,a^3\,d\,e^4\,m+120\,A\,a^3\,d\,e^4+6\,B\,a^2\,b\,d^3\,e^2\,m^2+54\,B\,a^2\,b\,d^3\,e^2\,m+120\,B\,a^2\,b\,d^3\,e^2-3\,A\,a^2\,b\,d^2\,e^3\,m^3-36\,A\,a^2\,b\,d^2\,e^3\,m^2-141\,A\,a^2\,b\,d^2\,e^3\,m-180\,A\,a^2\,b\,d^2\,e^3-18\,B\,a\,b^2\,d^4\,e\,m-90\,B\,a\,b^2\,d^4\,e+6\,A\,a\,b^2\,d^3\,e^2\,m^2+54\,A\,a\,b^2\,d^3\,e^2\,m+120\,A\,a\,b^2\,d^3\,e^2+24\,B\,b^3\,d^5-6\,A\,b^3\,d^4\,e\,m-30\,A\,b^3\,d^4\,e\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (B\,a^3\,d\,e^4\,m^4+12\,B\,a^3\,d\,e^4\,m^3+47\,B\,a^3\,d\,e^4\,m^2+60\,B\,a^3\,d\,e^4\,m+A\,a^3\,e^5\,m^4+14\,A\,a^3\,e^5\,m^3+71\,A\,a^3\,e^5\,m^2+154\,A\,a^3\,e^5\,m+120\,A\,a^3\,e^5-6\,B\,a^2\,b\,d^2\,e^3\,m^3-54\,B\,a^2\,b\,d^2\,e^3\,m^2-120\,B\,a^2\,b\,d^2\,e^3\,m+3\,A\,a^2\,b\,d\,e^4\,m^4+36\,A\,a^2\,b\,d\,e^4\,m^3+141\,A\,a^2\,b\,d\,e^4\,m^2+180\,A\,a^2\,b\,d\,e^4\,m+18\,B\,a\,b^2\,d^3\,e^2\,m^2+90\,B\,a\,b^2\,d^3\,e^2\,m-6\,A\,a\,b^2\,d^2\,e^3\,m^3-54\,A\,a\,b^2\,d^2\,e^3\,m^2-120\,A\,a\,b^2\,d^2\,e^3\,m-24\,B\,b^3\,d^4\,e\,m+6\,A\,b^3\,d^3\,e^2\,m^2+30\,A\,b^3\,d^3\,e^2\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (B\,a^3\,e^3\,m^3+12\,B\,a^3\,e^3\,m^2+47\,B\,a^3\,e^3\,m+60\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2\,m^3+27\,B\,a^2\,b\,d\,e^2\,m^2+60\,B\,a^2\,b\,d\,e^2\,m+3\,A\,a^2\,b\,e^3\,m^3+36\,A\,a^2\,b\,e^3\,m^2+141\,A\,a^2\,b\,e^3\,m+180\,A\,a^2\,b\,e^3-9\,B\,a\,b^2\,d^2\,e\,m^2-45\,B\,a\,b^2\,d^2\,e\,m+3\,A\,a\,b^2\,d\,e^2\,m^3+27\,A\,a\,b^2\,d\,e^2\,m^2+60\,A\,a\,b^2\,d\,e^2\,m+12\,B\,b^3\,d^3\,m-3\,A\,b^3\,d^2\,e\,m^2-15\,A\,b^3\,d^2\,e\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {B\,b^3\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {b^2\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (5\,A\,b\,e+15\,B\,a\,e+A\,b\,e\,m+3\,B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (3\,B\,a^2\,e^2\,m^2+27\,B\,a^2\,e^2\,m+60\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e\,m^2+15\,B\,a\,b\,d\,e\,m+3\,A\,a\,b\,e^2\,m^2+27\,A\,a\,b\,e^2\,m+60\,A\,a\,b\,e^2-4\,B\,b^2\,d^2\,m+A\,b^2\,d\,e\,m^2+5\,A\,b^2\,d\,e\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \]

input

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

output

((d + e*x)^m*(24*B*b^3*d^5 + 120*A*a^3*d*e^4 - 30*A*b^3*d^4*e - 60*B*a^3*d 
^2*e^3 + 120*A*a*b^2*d^3*e^2 - 180*A*a^2*b*d^2*e^3 + 120*B*a^2*b*d^3*e^2 + 
 71*A*a^3*d*e^4*m^2 + 14*A*a^3*d*e^4*m^3 + A*a^3*d*e^4*m^4 - 47*B*a^3*d^2* 
e^3*m - 12*B*a^3*d^2*e^3*m^2 - B*a^3*d^2*e^3*m^3 - 90*B*a*b^2*d^4*e + 154* 
A*a^3*d*e^4*m - 6*A*b^3*d^4*e*m + 6*A*a*b^2*d^3*e^2*m^2 - 36*A*a^2*b*d^2*e 
^3*m^2 - 3*A*a^2*b*d^2*e^3*m^3 + 6*B*a^2*b*d^3*e^2*m^2 - 18*B*a*b^2*d^4*e* 
m + 54*A*a*b^2*d^3*e^2*m - 141*A*a^2*b*d^2*e^3*m + 54*B*a^2*b*d^3*e^2*m))/ 
(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(12 
0*A*a^3*e^5 + 154*A*a^3*e^5*m + 71*A*a^3*e^5*m^2 + 14*A*a^3*e^5*m^3 + A*a^ 
3*e^5*m^4 + 30*A*b^3*d^3*e^2*m + 47*B*a^3*d*e^4*m^2 + 12*B*a^3*d*e^4*m^3 + 
 B*a^3*d*e^4*m^4 + 6*A*b^3*d^3*e^2*m^2 + 60*B*a^3*d*e^4*m - 24*B*b^3*d^4*e 
*m - 54*A*a*b^2*d^2*e^3*m^2 - 6*A*a*b^2*d^2*e^3*m^3 + 18*B*a*b^2*d^3*e^2*m 
^2 - 54*B*a^2*b*d^2*e^3*m^2 - 6*B*a^2*b*d^2*e^3*m^3 + 180*A*a^2*b*d*e^4*m 
- 120*A*a*b^2*d^2*e^3*m + 141*A*a^2*b*d*e^4*m^2 + 36*A*a^2*b*d*e^4*m^3 + 3 
*A*a^2*b*d*e^4*m^4 + 90*B*a*b^2*d^3*e^2*m - 120*B*a^2*b*d^2*e^3*m))/(e^5*( 
274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x^2*(m + 1)*(d + e*x)^m 
*(60*B*a^3*e^3 + 180*A*a^2*b*e^3 + 47*B*a^3*e^3*m + 12*B*b^3*d^3*m + 12*B* 
a^3*e^3*m^2 + B*a^3*e^3*m^3 + 36*A*a^2*b*e^3*m^2 + 3*A*a^2*b*e^3*m^3 - 3*A 
*b^3*d^2*e*m^2 + 141*A*a^2*b*e^3*m - 15*A*b^3*d^2*e*m + 60*A*a*b^2*d*e^2*m 
 - 45*B*a*b^2*d^2*e*m + 60*B*a^2*b*d*e^2*m + 27*A*a*b^2*d*e^2*m^2 + 3*A...