\(\int (a+b x) (d+e x)^2 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [29]

Optimal result
Mathematica [B] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [B] (verification not implemented)
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 31, antiderivative size = 65 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^2 (a+b x)^8}{8 b^3}+\frac {2 e (b d-a e) (a+b x)^9}{9 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3} \] Output:

1/8*(-a*e+b*d)^2*(b*x+a)^8/b^3+2/9*e*(-a*e+b*d)*(b*x+a)^9/b^3+1/10*e^2*(b* 
x+a)^10/b^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(65)=130\).

Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.52 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{360} x \left (120 a^7 \left (3 d^2+3 d e x+e^2 x^2\right )+210 a^6 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+252 a^5 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+210 a^4 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+120 a^3 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+45 a^2 b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+10 a b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )+b^7 x^7 \left (45 d^2+80 d e x+36 e^2 x^2\right )\right ) \] Input:

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

Output:

(x*(120*a^7*(3*d^2 + 3*d*e*x + e^2*x^2) + 210*a^6*b*x*(6*d^2 + 8*d*e*x + 3 
*e^2*x^2) + 252*a^5*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 210*a^4*b^3* 
x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 120*a^3*b^4*x^4*(21*d^2 + 35*d*e*x 
+ 15*e^2*x^2) + 45*a^2*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 10*a*b^6 
*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2) + b^7*x^7*(45*d^2 + 80*d*e*x + 36*e^ 
2*x^2)))/360
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 49, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^2dx}{b^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \int (a+b x)^7 (d+e x)^2dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (\frac {2 e (a+b x)^8 (b d-a e)}{b^2}+\frac {(a+b x)^7 (b d-a e)^2}{b^2}+\frac {e^2 (a+b x)^9}{b^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac {(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3}\)

Input:

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

Output:

((b*d - a*e)^2*(a + b*x)^8)/(8*b^3) + (2*e*(b*d - a*e)*(a + b*x)^9)/(9*b^3 
) + (e^2*(a + b*x)^10)/(10*b^3)
 

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 49
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

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. \(268\) vs. \(2(59)=118\).

Time = 1.16 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.14

method result size
norman \(\frac {b^{7} e^{2} x^{10}}{10}+\left (\frac {7}{9} e^{2} a \,b^{6}+\frac {2}{9} d e \,b^{7}\right ) x^{9}+\left (\frac {21}{8} e^{2} a^{2} b^{5}+\frac {7}{4} d e a \,b^{6}+\frac {1}{8} d^{2} b^{7}\right ) x^{8}+\left (5 e^{2} a^{3} b^{4}+6 d e \,a^{2} b^{5}+a \,b^{6} d^{2}\right ) x^{7}+\left (\frac {35}{6} a^{4} b^{3} e^{2}+\frac {35}{3} a^{3} b^{4} d e +\frac {7}{2} a^{2} b^{5} d^{2}\right ) x^{6}+\left (\frac {21}{5} e^{2} a^{5} b^{2}+14 d e \,a^{4} b^{3}+7 d^{2} a^{3} b^{4}\right ) x^{5}+\left (\frac {7}{4} e^{2} a^{6} b +\frac {21}{2} d e \,a^{5} b^{2}+\frac {35}{4} d^{2} a^{4} b^{3}\right ) x^{4}+\left (\frac {1}{3} e^{2} a^{7}+\frac {14}{3} d e \,a^{6} b +7 d^{2} a^{5} b^{2}\right ) x^{3}+\left (d e \,a^{7}+\frac {7}{2} d^{2} a^{6} b \right ) x^{2}+d^{2} a^{7} x\) \(269\)
risch \(\frac {1}{10} b^{7} e^{2} x^{10}+\frac {7}{9} x^{9} e^{2} a \,b^{6}+\frac {2}{9} x^{9} d e \,b^{7}+\frac {21}{8} x^{8} e^{2} a^{2} b^{5}+\frac {7}{4} x^{8} d e a \,b^{6}+\frac {1}{8} x^{8} d^{2} b^{7}+5 a^{3} b^{4} e^{2} x^{7}+6 a^{2} b^{5} d e \,x^{7}+a \,b^{6} d^{2} x^{7}+\frac {35}{6} x^{6} a^{4} b^{3} e^{2}+\frac {35}{3} x^{6} a^{3} b^{4} d e +\frac {7}{2} x^{6} a^{2} b^{5} d^{2}+\frac {21}{5} x^{5} e^{2} a^{5} b^{2}+14 x^{5} d e \,a^{4} b^{3}+7 x^{5} d^{2} a^{3} b^{4}+\frac {7}{4} x^{4} e^{2} a^{6} b +\frac {21}{2} x^{4} d e \,a^{5} b^{2}+\frac {35}{4} x^{4} d^{2} a^{4} b^{3}+\frac {1}{3} x^{3} e^{2} a^{7}+\frac {14}{3} x^{3} d e \,a^{6} b +7 x^{3} d^{2} a^{5} b^{2}+x^{2} d e \,a^{7}+\frac {7}{2} x^{2} d^{2} a^{6} b +d^{2} a^{7} x\) \(295\)
parallelrisch \(\frac {1}{10} b^{7} e^{2} x^{10}+\frac {7}{9} x^{9} e^{2} a \,b^{6}+\frac {2}{9} x^{9} d e \,b^{7}+\frac {21}{8} x^{8} e^{2} a^{2} b^{5}+\frac {7}{4} x^{8} d e a \,b^{6}+\frac {1}{8} x^{8} d^{2} b^{7}+5 a^{3} b^{4} e^{2} x^{7}+6 a^{2} b^{5} d e \,x^{7}+a \,b^{6} d^{2} x^{7}+\frac {35}{6} x^{6} a^{4} b^{3} e^{2}+\frac {35}{3} x^{6} a^{3} b^{4} d e +\frac {7}{2} x^{6} a^{2} b^{5} d^{2}+\frac {21}{5} x^{5} e^{2} a^{5} b^{2}+14 x^{5} d e \,a^{4} b^{3}+7 x^{5} d^{2} a^{3} b^{4}+\frac {7}{4} x^{4} e^{2} a^{6} b +\frac {21}{2} x^{4} d e \,a^{5} b^{2}+\frac {35}{4} x^{4} d^{2} a^{4} b^{3}+\frac {1}{3} x^{3} e^{2} a^{7}+\frac {14}{3} x^{3} d e \,a^{6} b +7 x^{3} d^{2} a^{5} b^{2}+x^{2} d e \,a^{7}+\frac {7}{2} x^{2} d^{2} a^{6} b +d^{2} a^{7} x\) \(295\)
gosper \(\frac {x \left (36 b^{7} e^{2} x^{9}+280 x^{8} e^{2} a \,b^{6}+80 x^{8} d e \,b^{7}+945 x^{7} e^{2} a^{2} b^{5}+630 x^{7} d e a \,b^{6}+45 x^{7} d^{2} b^{7}+1800 a^{3} b^{4} e^{2} x^{6}+2160 a^{2} b^{5} d e \,x^{6}+360 a \,b^{6} d^{2} x^{6}+2100 x^{5} a^{4} b^{3} e^{2}+4200 x^{5} a^{3} b^{4} d e +1260 x^{5} a^{2} b^{5} d^{2}+1512 x^{4} e^{2} a^{5} b^{2}+5040 x^{4} d e \,a^{4} b^{3}+2520 x^{4} d^{2} a^{3} b^{4}+630 x^{3} e^{2} a^{6} b +3780 x^{3} d e \,a^{5} b^{2}+3150 x^{3} d^{2} a^{4} b^{3}+120 x^{2} e^{2} a^{7}+1680 x^{2} d e \,a^{6} b +2520 x^{2} d^{2} a^{5} b^{2}+360 x d e \,a^{7}+1260 x \,d^{2} a^{6} b +360 d^{2} a^{7}\right )}{360}\) \(296\)
orering \(\frac {x \left (36 b^{7} e^{2} x^{9}+280 x^{8} e^{2} a \,b^{6}+80 x^{8} d e \,b^{7}+945 x^{7} e^{2} a^{2} b^{5}+630 x^{7} d e a \,b^{6}+45 x^{7} d^{2} b^{7}+1800 a^{3} b^{4} e^{2} x^{6}+2160 a^{2} b^{5} d e \,x^{6}+360 a \,b^{6} d^{2} x^{6}+2100 x^{5} a^{4} b^{3} e^{2}+4200 x^{5} a^{3} b^{4} d e +1260 x^{5} a^{2} b^{5} d^{2}+1512 x^{4} e^{2} a^{5} b^{2}+5040 x^{4} d e \,a^{4} b^{3}+2520 x^{4} d^{2} a^{3} b^{4}+630 x^{3} e^{2} a^{6} b +3780 x^{3} d e \,a^{5} b^{2}+3150 x^{3} d^{2} a^{4} b^{3}+120 x^{2} e^{2} a^{7}+1680 x^{2} d e \,a^{6} b +2520 x^{2} d^{2} a^{5} b^{2}+360 x d e \,a^{7}+1260 x \,d^{2} a^{6} b +360 d^{2} a^{7}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{360 \left (b x +a \right )^{6}}\) \(321\)
default \(\frac {b^{7} e^{2} x^{10}}{10}+\frac {\left (\left (a \,e^{2}+2 b d e \right ) b^{6}+6 e^{2} a \,b^{6}\right ) x^{9}}{9}+\frac {\left (\left (2 a d e +b \,d^{2}\right ) b^{6}+6 \left (a \,e^{2}+2 b d e \right ) a \,b^{5}+15 e^{2} a^{2} b^{5}\right ) x^{8}}{8}+\frac {\left (a \,b^{6} d^{2}+6 \left (2 a d e +b \,d^{2}\right ) a \,b^{5}+15 \left (a \,e^{2}+2 b d e \right ) a^{2} b^{4}+20 e^{2} a^{3} b^{4}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{5} d^{2}+15 \left (2 a d e +b \,d^{2}\right ) a^{2} b^{4}+20 \left (a \,e^{2}+2 b d e \right ) a^{3} b^{3}+15 a^{4} b^{3} e^{2}\right ) x^{6}}{6}+\frac {\left (15 d^{2} a^{3} b^{4}+20 \left (2 a d e +b \,d^{2}\right ) a^{3} b^{3}+15 \left (a \,e^{2}+2 b d e \right ) a^{4} b^{2}+6 e^{2} a^{5} b^{2}\right ) x^{5}}{5}+\frac {\left (20 d^{2} a^{4} b^{3}+15 \left (2 a d e +b \,d^{2}\right ) a^{4} b^{2}+6 \left (a \,e^{2}+2 b d e \right ) a^{5} b +e^{2} a^{6} b \right ) x^{4}}{4}+\frac {\left (15 d^{2} a^{5} b^{2}+6 \left (2 a d e +b \,d^{2}\right ) a^{5} b +\left (a \,e^{2}+2 b d e \right ) a^{6}\right ) x^{3}}{3}+\frac {\left (6 d^{2} a^{6} b +\left (2 a d e +b \,d^{2}\right ) a^{6}\right ) x^{2}}{2}+d^{2} a^{7} x\) \(433\)

Input:

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

Output:

1/10*b^7*e^2*x^10+(7/9*e^2*a*b^6+2/9*d*e*b^7)*x^9+(21/8*e^2*a^2*b^5+7/4*d* 
e*a*b^6+1/8*d^2*b^7)*x^8+(5*a^3*b^4*e^2+6*a^2*b^5*d*e+a*b^6*d^2)*x^7+(35/6 
*a^4*b^3*e^2+35/3*a^3*b^4*d*e+7/2*a^2*b^5*d^2)*x^6+(21/5*e^2*a^5*b^2+14*d* 
e*a^4*b^3+7*d^2*a^3*b^4)*x^5+(7/4*e^2*a^6*b+21/2*d*e*a^5*b^2+35/4*d^2*a^4* 
b^3)*x^4+(1/3*e^2*a^7+14/3*d*e*a^6*b+7*d^2*a^5*b^2)*x^3+(d*e*a^7+7/2*d^2*a 
^6*b)*x^2+d^2*a^7*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (59) = 118\).

Time = 0.07 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.20 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, b^{7} e^{2} x^{10} + a^{7} d^{2} x + \frac {1}{9} \, {\left (2 \, b^{7} d e + 7 \, a b^{6} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{2} + 14 \, a b^{6} d e + 21 \, a^{2} b^{5} e^{2}\right )} x^{8} + {\left (a b^{6} d^{2} + 6 \, a^{2} b^{5} d e + 5 \, a^{3} b^{4} e^{2}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d^{2} + 10 \, a^{3} b^{4} d e + 5 \, a^{4} b^{3} e^{2}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{2} + 10 \, a^{4} b^{3} d e + 3 \, a^{5} b^{2} e^{2}\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d^{2} + 6 \, a^{5} b^{2} d e + a^{6} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{2} + 14 \, a^{6} b d e + a^{7} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{2} + 2 \, a^{7} d e\right )} x^{2} \] Input:

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

Output:

1/10*b^7*e^2*x^10 + a^7*d^2*x + 1/9*(2*b^7*d*e + 7*a*b^6*e^2)*x^9 + 1/8*(b 
^7*d^2 + 14*a*b^6*d*e + 21*a^2*b^5*e^2)*x^8 + (a*b^6*d^2 + 6*a^2*b^5*d*e + 
 5*a^3*b^4*e^2)*x^7 + 7/6*(3*a^2*b^5*d^2 + 10*a^3*b^4*d*e + 5*a^4*b^3*e^2) 
*x^6 + 7/5*(5*a^3*b^4*d^2 + 10*a^4*b^3*d*e + 3*a^5*b^2*e^2)*x^5 + 7/4*(5*a 
^4*b^3*d^2 + 6*a^5*b^2*d*e + a^6*b*e^2)*x^4 + 1/3*(21*a^5*b^2*d^2 + 14*a^6 
*b*d*e + a^7*e^2)*x^3 + 1/2*(7*a^6*b*d^2 + 2*a^7*d*e)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (56) = 112\).

Time = 0.05 (sec) , antiderivative size = 303, normalized size of antiderivative = 4.66 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{7} d^{2} x + \frac {b^{7} e^{2} x^{10}}{10} + x^{9} \cdot \left (\frac {7 a b^{6} e^{2}}{9} + \frac {2 b^{7} d e}{9}\right ) + x^{8} \cdot \left (\frac {21 a^{2} b^{5} e^{2}}{8} + \frac {7 a b^{6} d e}{4} + \frac {b^{7} d^{2}}{8}\right ) + x^{7} \cdot \left (5 a^{3} b^{4} e^{2} + 6 a^{2} b^{5} d e + a b^{6} d^{2}\right ) + x^{6} \cdot \left (\frac {35 a^{4} b^{3} e^{2}}{6} + \frac {35 a^{3} b^{4} d e}{3} + \frac {7 a^{2} b^{5} d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {21 a^{5} b^{2} e^{2}}{5} + 14 a^{4} b^{3} d e + 7 a^{3} b^{4} d^{2}\right ) + x^{4} \cdot \left (\frac {7 a^{6} b e^{2}}{4} + \frac {21 a^{5} b^{2} d e}{2} + \frac {35 a^{4} b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{7} e^{2}}{3} + \frac {14 a^{6} b d e}{3} + 7 a^{5} b^{2} d^{2}\right ) + x^{2} \left (a^{7} d e + \frac {7 a^{6} b d^{2}}{2}\right ) \] Input:

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

Output:

a**7*d**2*x + b**7*e**2*x**10/10 + x**9*(7*a*b**6*e**2/9 + 2*b**7*d*e/9) + 
 x**8*(21*a**2*b**5*e**2/8 + 7*a*b**6*d*e/4 + b**7*d**2/8) + x**7*(5*a**3* 
b**4*e**2 + 6*a**2*b**5*d*e + a*b**6*d**2) + x**6*(35*a**4*b**3*e**2/6 + 3 
5*a**3*b**4*d*e/3 + 7*a**2*b**5*d**2/2) + x**5*(21*a**5*b**2*e**2/5 + 14*a 
**4*b**3*d*e + 7*a**3*b**4*d**2) + x**4*(7*a**6*b*e**2/4 + 21*a**5*b**2*d* 
e/2 + 35*a**4*b**3*d**2/4) + x**3*(a**7*e**2/3 + 14*a**6*b*d*e/3 + 7*a**5* 
b**2*d**2) + x**2*(a**7*d*e + 7*a**6*b*d**2/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (59) = 118\).

Time = 0.03 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.20 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, b^{7} e^{2} x^{10} + a^{7} d^{2} x + \frac {1}{9} \, {\left (2 \, b^{7} d e + 7 \, a b^{6} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{2} + 14 \, a b^{6} d e + 21 \, a^{2} b^{5} e^{2}\right )} x^{8} + {\left (a b^{6} d^{2} + 6 \, a^{2} b^{5} d e + 5 \, a^{3} b^{4} e^{2}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d^{2} + 10 \, a^{3} b^{4} d e + 5 \, a^{4} b^{3} e^{2}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{2} + 10 \, a^{4} b^{3} d e + 3 \, a^{5} b^{2} e^{2}\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d^{2} + 6 \, a^{5} b^{2} d e + a^{6} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{2} + 14 \, a^{6} b d e + a^{7} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{2} + 2 \, a^{7} d e\right )} x^{2} \] Input:

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

Output:

1/10*b^7*e^2*x^10 + a^7*d^2*x + 1/9*(2*b^7*d*e + 7*a*b^6*e^2)*x^9 + 1/8*(b 
^7*d^2 + 14*a*b^6*d*e + 21*a^2*b^5*e^2)*x^8 + (a*b^6*d^2 + 6*a^2*b^5*d*e + 
 5*a^3*b^4*e^2)*x^7 + 7/6*(3*a^2*b^5*d^2 + 10*a^3*b^4*d*e + 5*a^4*b^3*e^2) 
*x^6 + 7/5*(5*a^3*b^4*d^2 + 10*a^4*b^3*d*e + 3*a^5*b^2*e^2)*x^5 + 7/4*(5*a 
^4*b^3*d^2 + 6*a^5*b^2*d*e + a^6*b*e^2)*x^4 + 1/3*(21*a^5*b^2*d^2 + 14*a^6 
*b*d*e + a^7*e^2)*x^3 + 1/2*(7*a^6*b*d^2 + 2*a^7*d*e)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (59) = 118\).

Time = 0.19 (sec) , antiderivative size = 294, normalized size of antiderivative = 4.52 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, b^{7} e^{2} x^{10} + \frac {2}{9} \, b^{7} d e x^{9} + \frac {7}{9} \, a b^{6} e^{2} x^{9} + \frac {1}{8} \, b^{7} d^{2} x^{8} + \frac {7}{4} \, a b^{6} d e x^{8} + \frac {21}{8} \, a^{2} b^{5} e^{2} x^{8} + a b^{6} d^{2} x^{7} + 6 \, a^{2} b^{5} d e x^{7} + 5 \, a^{3} b^{4} e^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} d^{2} x^{6} + \frac {35}{3} \, a^{3} b^{4} d e x^{6} + \frac {35}{6} \, a^{4} b^{3} e^{2} x^{6} + 7 \, a^{3} b^{4} d^{2} x^{5} + 14 \, a^{4} b^{3} d e x^{5} + \frac {21}{5} \, a^{5} b^{2} e^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} d^{2} x^{4} + \frac {21}{2} \, a^{5} b^{2} d e x^{4} + \frac {7}{4} \, a^{6} b e^{2} x^{4} + 7 \, a^{5} b^{2} d^{2} x^{3} + \frac {14}{3} \, a^{6} b d e x^{3} + \frac {1}{3} \, a^{7} e^{2} x^{3} + \frac {7}{2} \, a^{6} b d^{2} x^{2} + a^{7} d e x^{2} + a^{7} d^{2} x \] Input:

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

Output:

1/10*b^7*e^2*x^10 + 2/9*b^7*d*e*x^9 + 7/9*a*b^6*e^2*x^9 + 1/8*b^7*d^2*x^8 
+ 7/4*a*b^6*d*e*x^8 + 21/8*a^2*b^5*e^2*x^8 + a*b^6*d^2*x^7 + 6*a^2*b^5*d*e 
*x^7 + 5*a^3*b^4*e^2*x^7 + 7/2*a^2*b^5*d^2*x^6 + 35/3*a^3*b^4*d*e*x^6 + 35 
/6*a^4*b^3*e^2*x^6 + 7*a^3*b^4*d^2*x^5 + 14*a^4*b^3*d*e*x^5 + 21/5*a^5*b^2 
*e^2*x^5 + 35/4*a^4*b^3*d^2*x^4 + 21/2*a^5*b^2*d*e*x^4 + 7/4*a^6*b*e^2*x^4 
 + 7*a^5*b^2*d^2*x^3 + 14/3*a^6*b*d*e*x^3 + 1/3*a^7*e^2*x^3 + 7/2*a^6*b*d^ 
2*x^2 + a^7*d*e*x^2 + a^7*d^2*x
 

Mupad [B] (verification not implemented)

Time = 11.04 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.83 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^3\,\left (\frac {a^7\,e^2}{3}+\frac {14\,a^6\,b\,d\,e}{3}+7\,a^5\,b^2\,d^2\right )+x^8\,\left (\frac {21\,a^2\,b^5\,e^2}{8}+\frac {7\,a\,b^6\,d\,e}{4}+\frac {b^7\,d^2}{8}\right )+a^7\,d^2\,x+\frac {b^7\,e^2\,x^{10}}{10}+\frac {a^6\,d\,x^2\,\left (2\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e\,x^9\,\left (7\,a\,e+2\,b\,d\right )}{9}+\frac {7\,a^4\,b\,x^4\,\left (a^2\,e^2+6\,a\,b\,d\,e+5\,b^2\,d^2\right )}{4}+a\,b^4\,x^7\,\left (5\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )+\frac {7\,a^3\,b^2\,x^5\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{5}+\frac {7\,a^2\,b^3\,x^6\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{6} \] Input:

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

Output:

x^3*((a^7*e^2)/3 + 7*a^5*b^2*d^2 + (14*a^6*b*d*e)/3) + x^8*((b^7*d^2)/8 + 
(21*a^2*b^5*e^2)/8 + (7*a*b^6*d*e)/4) + a^7*d^2*x + (b^7*e^2*x^10)/10 + (a 
^6*d*x^2*(2*a*e + 7*b*d))/2 + (b^6*e*x^9*(7*a*e + 2*b*d))/9 + (7*a^4*b*x^4 
*(a^2*e^2 + 5*b^2*d^2 + 6*a*b*d*e))/4 + a*b^4*x^7*(5*a^2*e^2 + b^2*d^2 + 6 
*a*b*d*e) + (7*a^3*b^2*x^5*(3*a^2*e^2 + 5*b^2*d^2 + 10*a*b*d*e))/5 + (7*a^ 
2*b^3*x^6*(5*a^2*e^2 + 3*b^2*d^2 + 10*a*b*d*e))/6
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 295, normalized size of antiderivative = 4.54 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x \left (36 b^{7} e^{2} x^{9}+280 a \,b^{6} e^{2} x^{8}+80 b^{7} d e \,x^{8}+945 a^{2} b^{5} e^{2} x^{7}+630 a \,b^{6} d e \,x^{7}+45 b^{7} d^{2} x^{7}+1800 a^{3} b^{4} e^{2} x^{6}+2160 a^{2} b^{5} d e \,x^{6}+360 a \,b^{6} d^{2} x^{6}+2100 a^{4} b^{3} e^{2} x^{5}+4200 a^{3} b^{4} d e \,x^{5}+1260 a^{2} b^{5} d^{2} x^{5}+1512 a^{5} b^{2} e^{2} x^{4}+5040 a^{4} b^{3} d e \,x^{4}+2520 a^{3} b^{4} d^{2} x^{4}+630 a^{6} b \,e^{2} x^{3}+3780 a^{5} b^{2} d e \,x^{3}+3150 a^{4} b^{3} d^{2} x^{3}+120 a^{7} e^{2} x^{2}+1680 a^{6} b d e \,x^{2}+2520 a^{5} b^{2} d^{2} x^{2}+360 a^{7} d e x +1260 a^{6} b \,d^{2} x +360 a^{7} d^{2}\right )}{360} \] Input:

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

Output:

(x*(360*a**7*d**2 + 360*a**7*d*e*x + 120*a**7*e**2*x**2 + 1260*a**6*b*d**2 
*x + 1680*a**6*b*d*e*x**2 + 630*a**6*b*e**2*x**3 + 2520*a**5*b**2*d**2*x** 
2 + 3780*a**5*b**2*d*e*x**3 + 1512*a**5*b**2*e**2*x**4 + 3150*a**4*b**3*d* 
*2*x**3 + 5040*a**4*b**3*d*e*x**4 + 2100*a**4*b**3*e**2*x**5 + 2520*a**3*b 
**4*d**2*x**4 + 4200*a**3*b**4*d*e*x**5 + 1800*a**3*b**4*e**2*x**6 + 1260* 
a**2*b**5*d**2*x**5 + 2160*a**2*b**5*d*e*x**6 + 945*a**2*b**5*e**2*x**7 + 
360*a*b**6*d**2*x**6 + 630*a*b**6*d*e*x**7 + 280*a*b**6*e**2*x**8 + 45*b** 
7*d**2*x**7 + 80*b**7*d*e*x**8 + 36*b**7*e**2*x**9))/360