\(\int (a+b x)^3 (A+B x) (d+e x)^2 \, dx\) [28]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^4}{4 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^6}{6 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.90 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=a^3 A d^2 x+\frac {1}{2} a^2 d (3 A b d+a B d+2 a A e) x^2+\frac {1}{3} a \left (a B d (3 b d+2 a e)+A \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (a B \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )+A b \left (b^2 d^2+6 a b d e+3 a^2 e^2\right )\right ) x^4+\frac {1}{5} b \left (3 a^2 B e^2+3 a b e (2 B d+A e)+b^2 d (B d+2 A e)\right ) x^5+\frac {1}{6} b^2 e (2 b B d+A b e+3 a B e) x^6+\frac {1}{7} b^3 B e^2 x^7 \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 118, 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.

\(\displaystyle \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(110)=220\).

Time = 0.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.07

method	result	size
default	\(\frac {b^{3} B \,e^{2} x^{7}}{7}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e^{2}+2 b^{3} B d e \right ) x^{6}}{6}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{2}+2 \left (b^{3} A +3 a \,b^{2} B \right ) d e +b^{3} B \,d^{2}\right ) x^{5}}{5}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e^{2}+2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d e +\left (b^{3} A +3 a \,b^{2} B \right ) d^{2}\right ) x^{4}}{4}+\frac {\left (a^{3} A \,e^{2}+2 \left (3 a^{2} b A +a^{3} B \right ) d e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} A d e +\left (3 a^{2} b A +a^{3} B \right ) d^{2}\right ) x^{2}}{2}+a^{3} A \,d^{2} x\)	\(244\)
norman	\(\frac {b^{3} B \,e^{2} x^{7}}{7}+\left (\frac {1}{6} A \,b^{3} e^{2}+\frac {1}{2} B a \,b^{2} e^{2}+\frac {1}{3} b^{3} B d e \right ) x^{6}+\left (\frac {3}{5} A a \,b^{2} e^{2}+\frac {2}{5} A \,b^{3} d e +\frac {3}{5} B \,a^{2} b \,e^{2}+\frac {6}{5} B a \,b^{2} d e +\frac {1}{5} b^{3} B \,d^{2}\right ) x^{5}+\left (\frac {3}{4} A \,a^{2} b \,e^{2}+\frac {3}{2} A a \,b^{2} d e +\frac {1}{4} A \,b^{3} d^{2}+\frac {1}{4} B \,a^{3} e^{2}+\frac {3}{2} B \,a^{2} b d e +\frac {3}{4} B a \,b^{2} d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{3} A \,e^{2}+2 A \,a^{2} b d e +A a \,b^{2} d^{2}+\frac {2}{3} B \,a^{3} d e +B \,a^{2} b \,d^{2}\right ) x^{3}+\left (a^{3} A d e +\frac {3}{2} A \,a^{2} b \,d^{2}+\frac {1}{2} B \,a^{3} d^{2}\right ) x^{2}+a^{3} A \,d^{2} x\)	\(247\)
gosper	\(\frac {1}{7} b^{3} B \,e^{2} x^{7}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+\frac {1}{2} x^{6} B a \,b^{2} e^{2}+\frac {1}{3} x^{6} b^{3} B d e +\frac {3}{5} x^{5} A a \,b^{2} e^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} B \,a^{2} b \,e^{2}+\frac {6}{5} x^{5} B a \,b^{2} d e +\frac {1}{5} x^{5} b^{3} B \,d^{2}+\frac {3}{4} x^{4} A \,a^{2} b \,e^{2}+\frac {3}{2} x^{4} A a \,b^{2} d e +\frac {1}{4} x^{4} A \,b^{3} d^{2}+\frac {1}{4} x^{4} B \,a^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b d e +\frac {3}{4} x^{4} B a \,b^{2} d^{2}+\frac {1}{3} x^{3} a^{3} A \,e^{2}+2 x^{3} A \,a^{2} b d e +x^{3} A a \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{3} d e +x^{3} B \,a^{2} b \,d^{2}+x^{2} a^{3} A d e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{2}+\frac {1}{2} x^{2} B \,a^{3} d^{2}+a^{3} A \,d^{2} x\)	\(288\)
risch	\(\frac {1}{7} b^{3} B \,e^{2} x^{7}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+\frac {1}{2} x^{6} B a \,b^{2} e^{2}+\frac {1}{3} x^{6} b^{3} B d e +\frac {3}{5} x^{5} A a \,b^{2} e^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} B \,a^{2} b \,e^{2}+\frac {6}{5} x^{5} B a \,b^{2} d e +\frac {1}{5} x^{5} b^{3} B \,d^{2}+\frac {3}{4} x^{4} A \,a^{2} b \,e^{2}+\frac {3}{2} x^{4} A a \,b^{2} d e +\frac {1}{4} x^{4} A \,b^{3} d^{2}+\frac {1}{4} x^{4} B \,a^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b d e +\frac {3}{4} x^{4} B a \,b^{2} d^{2}+\frac {1}{3} x^{3} a^{3} A \,e^{2}+2 x^{3} A \,a^{2} b d e +x^{3} A a \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{3} d e +x^{3} B \,a^{2} b \,d^{2}+x^{2} a^{3} A d e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{2}+\frac {1}{2} x^{2} B \,a^{3} d^{2}+a^{3} A \,d^{2} x\)	\(288\)
parallelrisch	\(\frac {1}{7} b^{3} B \,e^{2} x^{7}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+\frac {1}{2} x^{6} B a \,b^{2} e^{2}+\frac {1}{3} x^{6} b^{3} B d e +\frac {3}{5} x^{5} A a \,b^{2} e^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} B \,a^{2} b \,e^{2}+\frac {6}{5} x^{5} B a \,b^{2} d e +\frac {1}{5} x^{5} b^{3} B \,d^{2}+\frac {3}{4} x^{4} A \,a^{2} b \,e^{2}+\frac {3}{2} x^{4} A a \,b^{2} d e +\frac {1}{4} x^{4} A \,b^{3} d^{2}+\frac {1}{4} x^{4} B \,a^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b d e +\frac {3}{4} x^{4} B a \,b^{2} d^{2}+\frac {1}{3} x^{3} a^{3} A \,e^{2}+2 x^{3} A \,a^{2} b d e +x^{3} A a \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{3} d e +x^{3} B \,a^{2} b \,d^{2}+x^{2} a^{3} A d e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{2}+\frac {1}{2} x^{2} B \,a^{3} d^{2}+a^{3} A \,d^{2} x\)	\(288\)
orering	\(\frac {x \left (60 b^{3} B \,e^{2} x^{6}+70 A \,b^{3} e^{2} x^{5}+210 B a \,b^{2} e^{2} x^{5}+140 B \,b^{3} d e \,x^{5}+252 A a \,b^{2} e^{2} x^{4}+168 A \,b^{3} d e \,x^{4}+252 B \,a^{2} b \,e^{2} x^{4}+504 B a \,b^{2} d e \,x^{4}+84 B \,b^{3} d^{2} x^{4}+315 A \,a^{2} b \,e^{2} x^{3}+630 A a \,b^{2} d e \,x^{3}+105 A \,b^{3} d^{2} x^{3}+105 B \,a^{3} e^{2} x^{3}+630 B \,a^{2} b d e \,x^{3}+315 B a \,b^{2} d^{2} x^{3}+140 A \,a^{3} e^{2} x^{2}+840 A \,a^{2} b d e \,x^{2}+420 A a \,b^{2} d^{2} x^{2}+280 B \,a^{3} d e \,x^{2}+420 B \,a^{2} b \,d^{2} x^{2}+420 A \,a^{3} d e x +630 A \,a^{2} b \,d^{2} x +210 B \,a^{3} d^{2} x +420 a^{3} A \,d^{2}\right )}{420}\)	\(288\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (110) = 220\).

Time = 0.08 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.03 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac {1}{6} \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{3} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^2,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (116) = 232\).

Time = 0.05 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.51 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=A a^{3} d^{2} x + \frac {B b^{3} e^{2} x^{7}}{7} + x^{6} \left (\frac {A b^{3} e^{2}}{6} + \frac {B a b^{2} e^{2}}{2} + \frac {B b^{3} d e}{3}\right ) + x^{5} \cdot \left (\frac {3 A a b^{2} e^{2}}{5} + \frac {2 A b^{3} d e}{5} + \frac {3 B a^{2} b e^{2}}{5} + \frac {6 B a b^{2} d e}{5} + \frac {B b^{3} d^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b e^{2}}{4} + \frac {3 A a b^{2} d e}{2} + \frac {A b^{3} d^{2}}{4} + \frac {B a^{3} e^{2}}{4} + \frac {3 B a^{2} b d e}{2} + \frac {3 B a b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{3} e^{2}}{3} + 2 A a^{2} b d e + A a b^{2} d^{2} + \frac {2 B a^{3} d e}{3} + B a^{2} b d^{2}\right ) + x^{2} \left (A a^{3} d e + \frac {3 A a^{2} b d^{2}}{2} + \frac {B a^{3} d^{2}}{2}\right ) \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (110) = 220\).

Time = 0.04 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.03 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac {1}{6} \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{3} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (110) = 220\).

Time = 0.12 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.43 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {1}{7} \, B b^{3} e^{2} x^{7} + \frac {1}{3} \, B b^{3} d e x^{6} + \frac {1}{2} \, B a b^{2} e^{2} x^{6} + \frac {1}{6} \, A b^{3} e^{2} x^{6} + \frac {1}{5} \, B b^{3} d^{2} x^{5} + \frac {6}{5} \, B a b^{2} d e x^{5} + \frac {2}{5} \, A b^{3} d e x^{5} + \frac {3}{5} \, B a^{2} b e^{2} x^{5} + \frac {3}{5} \, A a b^{2} e^{2} x^{5} + \frac {3}{4} \, B a b^{2} d^{2} x^{4} + \frac {1}{4} \, A b^{3} d^{2} x^{4} + \frac {3}{2} \, B a^{2} b d e x^{4} + \frac {3}{2} \, A a b^{2} d e x^{4} + \frac {1}{4} \, B a^{3} e^{2} x^{4} + \frac {3}{4} \, A a^{2} b e^{2} x^{4} + B a^{2} b d^{2} x^{3} + A a b^{2} d^{2} x^{3} + \frac {2}{3} \, B a^{3} d e x^{3} + 2 \, A a^{2} b d e x^{3} + \frac {1}{3} \, A a^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{2} x^{2} + \frac {3}{2} \, A a^{2} b d^{2} x^{2} + A a^{3} d e x^{2} + A a^{3} d^{2} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.96 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=x^4\,\left (\frac {B\,a^3\,e^2}{4}+\frac {3\,B\,a^2\,b\,d\,e}{2}+\frac {3\,A\,a^2\,b\,e^2}{4}+\frac {3\,B\,a\,b^2\,d^2}{4}+\frac {3\,A\,a\,b^2\,d\,e}{2}+\frac {A\,b^3\,d^2}{4}\right )+x^3\,\left (\frac {2\,B\,a^3\,d\,e}{3}+\frac {A\,a^3\,e^2}{3}+B\,a^2\,b\,d^2+2\,A\,a^2\,b\,d\,e+A\,a\,b^2\,d^2\right )+x^5\,\left (\frac {3\,B\,a^2\,b\,e^2}{5}+\frac {6\,B\,a\,b^2\,d\,e}{5}+\frac {3\,A\,a\,b^2\,e^2}{5}+\frac {B\,b^3\,d^2}{5}+\frac {2\,A\,b^3\,d\,e}{5}\right )+A\,a^3\,d^2\,x+\frac {a^2\,d\,x^2\,\left (2\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e\,x^6\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{6}+\frac {B\,b^3\,e^2\,x^7}{7} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.46 \[ \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx=\frac {x \left (15 b^{4} e^{2} x^{6}+70 a \,b^{3} e^{2} x^{5}+35 b^{4} d e \,x^{5}+126 a^{2} b^{2} e^{2} x^{4}+168 a \,b^{3} d e \,x^{4}+21 b^{4} d^{2} x^{4}+105 a^{3} b \,e^{2} x^{3}+315 a^{2} b^{2} d e \,x^{3}+105 a \,b^{3} d^{2} x^{3}+35 a^{4} e^{2} x^{2}+280 a^{3} b d e \,x^{2}+210 a^{2} b^{2} d^{2} x^{2}+105 a^{4} d e x +210 a^{3} b \,d^{2} x +105 a^{4} d^{2}\right )}{105} \] Input:

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

Output:

(x*(105*a**4*d**2 + 105*a**4*d*e*x + 35*a**4*e**2*x**2 + 210*a**3*b*d**2*x 
 + 280*a**3*b*d*e*x**2 + 105*a**3*b*e**2*x**3 + 210*a**2*b**2*d**2*x**2 + 
315*a**2*b**2*d*e*x**3 + 126*a**2*b**2*e**2*x**4 + 105*a*b**3*d**2*x**3 + 
168*a*b**3*d*e*x**4 + 70*a*b**3*e**2*x**5 + 21*b**4*d**2*x**4 + 35*b**4*d* 
e*x**5 + 15*b**4*e**2*x**6))/105