\(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2) \, dx\) [279]

Optimal result

Integrand size = 29, antiderivative size = 120 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^5}{5 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^6}{6 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^7}{7 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(120)=240\).

Time = 0.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.36 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 A d^4 x+\frac {1}{2} a d^3 (2 A b d+a B d+4 a A e) x^2+\frac {1}{3} d^2 \left (2 a B d (b d+2 a e)+A \left (b^2 d^2+8 a b d e+6 a^2 e^2\right )\right ) x^3+\frac {1}{4} d \left (2 a^2 e^2 (3 B d+2 A e)+4 a b d e (2 B d+3 A e)+b^2 d^2 (B d+4 A e)\right ) x^4+\frac {1}{5} e \left (a^2 e^2 (4 B d+A e)+4 a b d e (3 B d+2 A e)+2 b^2 d^2 (2 B d+3 A e)\right ) x^5+\frac {1}{6} e^2 \left (a^2 B e^2+2 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x^6+\frac {1}{7} b e^3 (4 b B d+A b e+2 a B e) x^7+\frac {1}{8} b^2 B e^4 x^8 \] Input:

Integrate[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1184, 27, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right ) (A+B x) (d+e x)^4 \, dx\)

\(\Big \downarrow \) 1184

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

\(\Big \downarrow \) 27

\(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^4dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

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. \(318\) vs. \(2(112)=224\).

Time = 0.73 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.66

method	result	size
default	\(\frac {B \,b^{2} e^{4} x^{8}}{8}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) b^{2}+2 B a b \,e^{4}\right ) x^{7}}{7}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b^{2}+2 \left (A \,e^{4}+4 B d \,e^{3}\right ) a b +B \,e^{4} a^{2}\right ) x^{6}}{6}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 d^{3} B e \right ) b^{2}+2 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a b +\left (A \,e^{4}+4 B d \,e^{3}\right ) a^{2}\right ) x^{5}}{5}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) b^{2}+2 \left (6 A \,d^{2} e^{2}+4 d^{3} B e \right ) a b +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (A \,b^{2} d^{4}+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) a b +\left (6 A \,d^{2} e^{2}+4 d^{3} B e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2}\right ) x^{2}}{2}+A \,d^{4} a^{2} x\)	\(319\)
norman	\(\frac {B \,b^{2} e^{4} x^{8}}{8}+\left (\frac {1}{7} A \,b^{2} e^{4}+\frac {2}{7} B a b \,e^{4}+\frac {4}{7} B \,b^{2} d \,e^{3}\right ) x^{7}+\left (\frac {1}{3} A a b \,e^{4}+\frac {2}{3} A \,b^{2} d \,e^{3}+\frac {1}{6} B \,e^{4} a^{2}+\frac {4}{3} B a b d \,e^{3}+B \,b^{2} d^{2} e^{2}\right ) x^{6}+\left (\frac {1}{5} A \,a^{2} e^{4}+\frac {8}{5} A a b d \,e^{3}+\frac {6}{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} B \,a^{2} d \,e^{3}+\frac {12}{5} B a b \,d^{2} e^{2}+\frac {4}{5} B \,b^{2} d^{3} e \right ) x^{5}+\left (a^{2} A d \,e^{3}+3 A a b \,d^{2} e^{2}+A \,b^{2} d^{3} e +\frac {3}{2} B \,a^{2} d^{2} e^{2}+2 B a b \,d^{3} e +\frac {1}{4} B \,b^{2} d^{4}\right ) x^{4}+\left (2 a^{2} A \,d^{2} e^{2}+\frac {8}{3} A a b \,d^{3} e +\frac {1}{3} A \,b^{2} d^{4}+\frac {4}{3} B \,a^{2} d^{3} e +\frac {2}{3} B a b \,d^{4}\right ) x^{3}+\left (2 a^{2} A \,d^{3} e +A a b \,d^{4}+\frac {1}{2} a^{2} B \,d^{4}\right ) x^{2}+A \,d^{4} a^{2} x\)	\(321\)
risch	\(2 x^{2} a^{2} A \,d^{3} e +x^{2} A a b \,d^{4}+2 x^{3} a^{2} A \,d^{2} e^{2}+x^{4} A \,b^{2} d^{3} e +\frac {1}{4} B \,b^{2} x^{4} d^{4}+\frac {1}{3} A \,b^{2} d^{4} x^{3}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} d \,e^{3}+\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {4}{3} x^{6} B a b d \,e^{3}+A \,d^{4} a^{2} x +\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e +x^{4} a^{2} A d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} a^{2} B \,d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+\frac {8}{3} x^{3} A a b \,d^{3} e\)	\(375\)
parallelrisch	\(2 x^{2} a^{2} A \,d^{3} e +x^{2} A a b \,d^{4}+2 x^{3} a^{2} A \,d^{2} e^{2}+x^{4} A \,b^{2} d^{3} e +\frac {1}{4} B \,b^{2} x^{4} d^{4}+\frac {1}{3} A \,b^{2} d^{4} x^{3}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} d \,e^{3}+\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {4}{3} x^{6} B a b d \,e^{3}+A \,d^{4} a^{2} x +\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e +x^{4} a^{2} A d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} a^{2} B \,d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+\frac {8}{3} x^{3} A a b \,d^{3} e\)	\(375\)
gosper	\(\frac {x \left (105 B \,b^{2} e^{4} x^{7}+120 x^{6} A \,b^{2} e^{4}+240 x^{6} B a b \,e^{4}+480 x^{6} B \,b^{2} d \,e^{3}+280 x^{5} A a b \,e^{4}+560 x^{5} A \,b^{2} d \,e^{3}+140 x^{5} B \,e^{4} a^{2}+1120 x^{5} B a b d \,e^{3}+840 x^{5} B \,b^{2} d^{2} e^{2}+168 x^{4} A \,a^{2} e^{4}+1344 x^{4} A a b d \,e^{3}+1008 x^{4} A \,b^{2} d^{2} e^{2}+672 x^{4} B \,a^{2} d \,e^{3}+2016 x^{4} B a b \,d^{2} e^{2}+672 x^{4} B \,b^{2} d^{3} e +840 x^{3} a^{2} A d \,e^{3}+2520 x^{3} A a b \,d^{2} e^{2}+840 x^{3} A \,b^{2} d^{3} e +1260 x^{3} B \,a^{2} d^{2} e^{2}+1680 x^{3} B a b \,d^{3} e +210 B \,b^{2} d^{4} x^{3}+1680 x^{2} a^{2} A \,d^{2} e^{2}+2240 x^{2} A a b \,d^{3} e +280 A \,b^{2} d^{4} x^{2}+1120 x^{2} B \,a^{2} d^{3} e +560 B a b \,d^{4} x^{2}+1680 x \,a^{2} A \,d^{3} e +840 A a b \,d^{4} x +420 x \,a^{2} B \,d^{4}+840 A \,a^{2} d^{4}\right )}{840}\)	\(376\)
orering	\(\frac {x \left (105 B \,b^{2} e^{4} x^{7}+120 x^{6} A \,b^{2} e^{4}+240 x^{6} B a b \,e^{4}+480 x^{6} B \,b^{2} d \,e^{3}+280 x^{5} A a b \,e^{4}+560 x^{5} A \,b^{2} d \,e^{3}+140 x^{5} B \,e^{4} a^{2}+1120 x^{5} B a b d \,e^{3}+840 x^{5} B \,b^{2} d^{2} e^{2}+168 x^{4} A \,a^{2} e^{4}+1344 x^{4} A a b d \,e^{3}+1008 x^{4} A \,b^{2} d^{2} e^{2}+672 x^{4} B \,a^{2} d \,e^{3}+2016 x^{4} B a b \,d^{2} e^{2}+672 x^{4} B \,b^{2} d^{3} e +840 x^{3} a^{2} A d \,e^{3}+2520 x^{3} A a b \,d^{2} e^{2}+840 x^{3} A \,b^{2} d^{3} e +1260 x^{3} B \,a^{2} d^{2} e^{2}+1680 x^{3} B a b \,d^{3} e +210 B \,b^{2} d^{4} x^{3}+1680 x^{2} a^{2} A \,d^{2} e^{2}+2240 x^{2} A a b \,d^{3} e +280 A \,b^{2} d^{4} x^{2}+1120 x^{2} B \,a^{2} d^{3} e +560 B a b \,d^{4} x^{2}+1680 x \,a^{2} A \,d^{3} e +840 A a b \,d^{4} x +420 x \,a^{2} B \,d^{4}+840 A \,a^{2} d^{4}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}{840 \left (b x +a \right )^{2}}\)	\(399\)

Input:

int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (112) = 224\).

Time = 0.07 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B b^{2} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (114) = 228\).

Time = 0.04 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.20 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=A a^{2} d^{4} x + \frac {B b^{2} e^{4} x^{8}}{8} + x^{7} \left (\frac {A b^{2} e^{4}}{7} + \frac {2 B a b e^{4}}{7} + \frac {4 B b^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {2 A b^{2} d e^{3}}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + B b^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {4 B b^{2} d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + A b^{2} d^{3} e + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \] Input:

integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2),x)
 

Output:

A*a**2*d**4*x + B*b**2*e**4*x**8/8 + x**7*(A*b**2*e**4/7 + 2*B*a*b*e**4/7 
+ 4*B*b**2*d*e**3/7) + x**6*(A*a*b*e**4/3 + 2*A*b**2*d*e**3/3 + B*a**2*e** 
4/6 + 4*B*a*b*d*e**3/3 + B*b**2*d**2*e**2) + x**5*(A*a**2*e**4/5 + 8*A*a*b 
*d*e**3/5 + 6*A*b**2*d**2*e**2/5 + 4*B*a**2*d*e**3/5 + 12*B*a*b*d**2*e**2/ 
5 + 4*B*b**2*d**3*e/5) + x**4*(A*a**2*d*e**3 + 3*A*a*b*d**2*e**2 + A*b**2* 
d**3*e + 3*B*a**2*d**2*e**2/2 + 2*B*a*b*d**3*e + B*b**2*d**4/4) + x**3*(2* 
A*a**2*d**2*e**2 + 8*A*a*b*d**3*e/3 + A*b**2*d**4/3 + 4*B*a**2*d**3*e/3 + 
2*B*a*b*d**4/3) + x**2*(2*A*a**2*d**3*e + A*a*b*d**4 + B*a**2*d**4/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (112) = 224\).

Time = 0.04 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B b^{2} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (112) = 224\).

Time = 0.17 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.12 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{8} \, B b^{2} e^{4} x^{8} + \frac {4}{7} \, B b^{2} d e^{3} x^{7} + \frac {2}{7} \, B a b e^{4} x^{7} + \frac {1}{7} \, A b^{2} e^{4} x^{7} + B b^{2} d^{2} e^{2} x^{6} + \frac {4}{3} \, B a b d e^{3} x^{6} + \frac {2}{3} \, A b^{2} d e^{3} x^{6} + \frac {1}{6} \, B a^{2} e^{4} x^{6} + \frac {1}{3} \, A a b e^{4} x^{6} + \frac {4}{5} \, B b^{2} d^{3} e x^{5} + \frac {12}{5} \, B a b d^{2} e^{2} x^{5} + \frac {6}{5} \, A b^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{2} d e^{3} x^{5} + \frac {8}{5} \, A a b d e^{3} x^{5} + \frac {1}{5} \, A a^{2} e^{4} x^{5} + \frac {1}{4} \, B b^{2} d^{4} x^{4} + 2 \, B a b d^{3} e x^{4} + A b^{2} d^{3} e x^{4} + \frac {3}{2} \, B a^{2} d^{2} e^{2} x^{4} + 3 \, A a b d^{2} e^{2} x^{4} + A a^{2} d e^{3} x^{4} + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {4}{3} \, B a^{2} d^{3} e x^{3} + \frac {8}{3} \, A a b d^{3} e x^{3} + 2 \, A a^{2} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + 2 \, A a^{2} d^{3} e x^{2} + A a^{2} d^{4} x \] Input:

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")
 

Output:

1/8*B*b^2*e^4*x^8 + 4/7*B*b^2*d*e^3*x^7 + 2/7*B*a*b*e^4*x^7 + 1/7*A*b^2*e^ 
4*x^7 + B*b^2*d^2*e^2*x^6 + 4/3*B*a*b*d*e^3*x^6 + 2/3*A*b^2*d*e^3*x^6 + 1/ 
6*B*a^2*e^4*x^6 + 1/3*A*a*b*e^4*x^6 + 4/5*B*b^2*d^3*e*x^5 + 12/5*B*a*b*d^2 
*e^2*x^5 + 6/5*A*b^2*d^2*e^2*x^5 + 4/5*B*a^2*d*e^3*x^5 + 8/5*A*a*b*d*e^3*x 
^5 + 1/5*A*a^2*e^4*x^5 + 1/4*B*b^2*d^4*x^4 + 2*B*a*b*d^3*e*x^4 + A*b^2*d^3 
*e*x^4 + 3/2*B*a^2*d^2*e^2*x^4 + 3*A*a*b*d^2*e^2*x^4 + A*a^2*d*e^3*x^4 + 2 
/3*B*a*b*d^4*x^3 + 1/3*A*b^2*d^4*x^3 + 4/3*B*a^2*d^3*e*x^3 + 8/3*A*a*b*d^3 
*e*x^3 + 2*A*a^2*d^2*e^2*x^3 + 1/2*B*a^2*d^4*x^2 + A*a*b*d^4*x^2 + 2*A*a^2 
*d^3*e*x^2 + A*a^2*d^4*x
 

Mupad [B] (verification not implemented)

Time = 11.59 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.54 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}\right )+x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^3\,x^7\,\left (A\,b\,e+2\,B\,a\,e+4\,B\,b\,d\right )}{7}+\frac {B\,b^2\,e^4\,x^8}{8} \] Input:

int((A + B*x)*(d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x),x)
 

Output:

x^4*((B*b^2*d^4)/4 + A*a^2*d*e^3 + A*b^2*d^3*e + (3*B*a^2*d^2*e^2)/2 + 2*B 
*a*b*d^3*e + 3*A*a*b*d^2*e^2) + x^5*((A*a^2*e^4)/5 + (4*B*a^2*d*e^3)/5 + ( 
4*B*b^2*d^3*e)/5 + (6*A*b^2*d^2*e^2)/5 + (8*A*a*b*d*e^3)/5 + (12*B*a*b*d^2 
*e^2)/5) + x^3*((A*b^2*d^4)/3 + (2*B*a*b*d^4)/3 + (4*B*a^2*d^3*e)/3 + 2*A* 
a^2*d^2*e^2 + (8*A*a*b*d^3*e)/3) + x^6*((B*a^2*e^4)/6 + (A*a*b*e^4)/3 + (2 
*A*b^2*d*e^3)/3 + B*b^2*d^2*e^2 + (4*B*a*b*d*e^3)/3) + A*a^2*d^4*x + (a*d^ 
3*x^2*(4*A*a*e + 2*A*b*d + B*a*d))/2 + (b*e^3*x^7*(A*b*e + 2*B*a*e + 4*B*b 
*d))/7 + (B*b^2*e^4*x^8)/8
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.06 \[ \int (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {x \left (35 b^{3} e^{4} x^{7}+120 a \,b^{2} e^{4} x^{6}+160 b^{3} d \,e^{3} x^{6}+140 a^{2} b \,e^{4} x^{5}+560 a \,b^{2} d \,e^{3} x^{5}+280 b^{3} d^{2} e^{2} x^{5}+56 a^{3} e^{4} x^{4}+672 a^{2} b d \,e^{3} x^{4}+1008 a \,b^{2} d^{2} e^{2} x^{4}+224 b^{3} d^{3} e \,x^{4}+280 a^{3} d \,e^{3} x^{3}+1260 a^{2} b \,d^{2} e^{2} x^{3}+840 a \,b^{2} d^{3} e \,x^{3}+70 b^{3} d^{4} x^{3}+560 a^{3} d^{2} e^{2} x^{2}+1120 a^{2} b \,d^{3} e \,x^{2}+280 a \,b^{2} d^{4} x^{2}+560 a^{3} d^{3} e x +420 a^{2} b \,d^{4} x +280 a^{3} d^{4}\right )}{280} \] Input:

int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x)
 

Output:

(x*(280*a**3*d**4 + 560*a**3*d**3*e*x + 560*a**3*d**2*e**2*x**2 + 280*a**3 
*d*e**3*x**3 + 56*a**3*e**4*x**4 + 420*a**2*b*d**4*x + 1120*a**2*b*d**3*e* 
x**2 + 1260*a**2*b*d**2*e**2*x**3 + 672*a**2*b*d*e**3*x**4 + 140*a**2*b*e* 
*4*x**5 + 280*a*b**2*d**4*x**2 + 840*a*b**2*d**3*e*x**3 + 1008*a*b**2*d**2 
*e**2*x**4 + 560*a*b**2*d*e**3*x**5 + 120*a*b**2*e**4*x**6 + 70*b**3*d**4* 
x**3 + 224*b**3*d**3*e*x**4 + 280*b**3*d**2*e**2*x**5 + 160*b**3*d*e**3*x* 
*6 + 35*b**3*e**4*x**7))/280