\(\int (A+B x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^2 \, dx\) [503]

Optimal result

Integrand size = 31, antiderivative size = 234 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {(b d-a e)^4 (B d-A e) (d+e x)^{1+m}}{e^6 (1+m)}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {b^4 B (d+e x)^{6+m}}{e^6 (6+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^4 (B d-A e)}{1+m}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)}{2+m}-\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^2}{3+m}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^3}{4+m}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^4}{5+m}+\frac {b^4 B (d+e x)^5}{6+m}\right )}{e^6} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 234, 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, 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.

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

\(\Big \downarrow \) 1184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}+\frac {2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac {(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac {(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac {2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac {b^4 B (d+e x)^{m+6}}{e^6 (m+6)}\)

Input:

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

Output:

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

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. \(1957\) vs. \(2(234)=468\).

Time = 1.41 (sec) , antiderivative size = 1958, normalized size of antiderivative = 8.37

method	result	size
norman	\(\text {Expression too large to display}\)	\(1958\)
gosper	\(\text {Expression too large to display}\)	\(2355\)
orering	\(\text {Expression too large to display}\)	\(2383\)
risch	\(\text {Expression too large to display}\)	\(3035\)
parallelrisch	\(\text {Expression too large to display}\)	\(4523\)

Input:

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

Output:

B*b^4/(6+m)*x^6*exp(m*ln(e*x+d))+d*(A*a^4*e^5*m^5+20*A*a^4*e^5*m^4-4*A*a^3 
*b*d*e^4*m^4-B*a^4*d*e^4*m^4+155*A*a^4*e^5*m^3-72*A*a^3*b*d*e^4*m^3+12*A*a 
^2*b^2*d^2*e^3*m^3-18*B*a^4*d*e^4*m^3+8*B*a^3*b*d^2*e^3*m^3+580*A*a^4*e^5* 
m^2-476*A*a^3*b*d*e^4*m^2+180*A*a^2*b^2*d^2*e^3*m^2-24*A*a*b^3*d^3*e^2*m^2 
-119*B*a^4*d*e^4*m^2+120*B*a^3*b*d^2*e^3*m^2-36*B*a^2*b^2*d^3*e^2*m^2+1044 
*A*a^4*e^5*m-1368*A*a^3*b*d*e^4*m+888*A*a^2*b^2*d^2*e^3*m-264*A*a*b^3*d^3* 
e^2*m+24*A*b^4*d^4*e*m-342*B*a^4*d*e^4*m+592*B*a^3*b*d^2*e^3*m-396*B*a^2*b 
^2*d^3*e^2*m+96*B*a*b^3*d^4*e*m+720*A*a^4*e^5-1440*A*a^3*b*d*e^4+1440*A*a^ 
2*b^2*d^2*e^3-720*A*a*b^3*d^3*e^2+144*A*b^4*d^4*e-360*B*a^4*d*e^4+960*B*a^ 
3*b*d^2*e^3-1080*B*a^2*b^2*d^3*e^2+576*B*a*b^3*d^4*e-120*B*b^4*d^5)/e^6/(m 
^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)*exp(m*ln(e*x+d))+(4*A*a^3*b 
*e^4*m^4+6*A*a^2*b^2*d*e^3*m^4+B*a^4*e^4*m^4+4*B*a^3*b*d*e^3*m^4+72*A*a^3* 
b*e^4*m^3+90*A*a^2*b^2*d*e^3*m^3-12*A*a*b^3*d^2*e^2*m^3+18*B*a^4*e^4*m^3+6 
0*B*a^3*b*d*e^3*m^3-18*B*a^2*b^2*d^2*e^2*m^3+476*A*a^3*b*e^4*m^2+444*A*a^2 
*b^2*d*e^3*m^2-132*A*a*b^3*d^2*e^2*m^2+12*A*b^4*d^3*e*m^2+119*B*a^4*e^4*m^ 
2+296*B*a^3*b*d*e^3*m^2-198*B*a^2*b^2*d^2*e^2*m^2+48*B*a*b^3*d^3*e*m^2+136 
8*A*a^3*b*e^4*m+720*A*a^2*b^2*d*e^3*m-360*A*a*b^3*d^2*e^2*m+72*A*b^4*d^3*e 
*m+342*B*a^4*e^4*m+480*B*a^3*b*d*e^3*m-540*B*a^2*b^2*d^2*e^2*m+288*B*a*b^3 
*d^3*e*m-60*B*b^4*d^4*m+1440*A*a^3*b*e^4+360*B*a^4*e^4)/e^4/(m^5+20*m^4+15 
5*m^3+580*m^2+1044*m+720)*x^2*exp(m*ln(e*x+d))+(A*a^4*e^5*m^5+4*A*a^3*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2274 vs. \(2 (234) = 468\).

Time = 0.11 (sec) , antiderivative size = 2274, normalized size of antiderivative = 9.72 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

(A*a^4*d*e^5*m^5 - 120*B*b^4*d^6 + 720*A*a^4*d*e^5 + 144*(4*B*a*b^3 + A*b^ 
4)*d^5*e - 360*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 480*(2*B*a^3*b + 3*A*a^ 
2*b^2)*d^3*e^3 - 360*(B*a^4 + 4*A*a^3*b)*d^2*e^4 + (B*b^4*e^6*m^5 + 15*B*b 
^4*e^6*m^4 + 85*B*b^4*e^6*m^3 + 225*B*b^4*e^6*m^2 + 274*B*b^4*e^6*m + 120* 
B*b^4*e^6)*x^6 + (144*(4*B*a*b^3 + A*b^4)*e^6 + (B*b^4*d*e^5 + (4*B*a*b^3 
+ A*b^4)*e^6)*m^5 + 2*(5*B*b^4*d*e^5 + 8*(4*B*a*b^3 + A*b^4)*e^6)*m^4 + 5* 
(7*B*b^4*d*e^5 + 19*(4*B*a*b^3 + A*b^4)*e^6)*m^3 + 10*(5*B*b^4*d*e^5 + 26* 
(4*B*a*b^3 + A*b^4)*e^6)*m^2 + 12*(2*B*b^4*d*e^5 + 27*(4*B*a*b^3 + A*b^4)* 
e^6)*m)*x^5 + (20*A*a^4*d*e^5 - (B*a^4 + 4*A*a^3*b)*d^2*e^4)*m^4 + (360*(3 
*B*a^2*b^2 + 2*A*a*b^3)*e^6 + ((4*B*a*b^3 + A*b^4)*d*e^5 + 2*(3*B*a^2*b^2 
+ 2*A*a*b^3)*e^6)*m^5 - (5*B*b^4*d^2*e^4 - 12*(4*B*a*b^3 + A*b^4)*d*e^5 - 
34*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^4 - (30*B*b^4*d^2*e^4 - 47*(4*B*a*b^3 
+ A*b^4)*d*e^5 - 214*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^3 - (55*B*b^4*d^2*e^ 
4 - 72*(4*B*a*b^3 + A*b^4)*d*e^5 - 614*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^2 
- 6*(5*B*b^4*d^2*e^4 - 6*(4*B*a*b^3 + A*b^4)*d*e^5 - 132*(3*B*a^2*b^2 + 2* 
A*a*b^3)*e^6)*m)*x^4 + (155*A*a^4*d*e^5 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^3* 
e^3 - 18*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*m^3 + 2*(240*(2*B*a^3*b + 3*A*a^2*b^ 
2)*e^6 + ((3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + (2*B*a^3*b + 3*A*a^2*b^2)*e^6) 
*m^5 - 2*((4*B*a*b^3 + A*b^4)*d^2*e^4 - 7*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 
- 9*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m^4 + (10*B*b^4*d^3*e^3 - 18*(4*B*a*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28048 vs. \(2 (224) = 448\).

Time = 5.94 (sec) , antiderivative size = 28048, normalized size of antiderivative = 119.86 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((d**m*(A*a**4*x + 2*A*a**3*b*x**2 + 2*A*a**2*b**2*x**3 + A*a*b** 
3*x**4 + A*b**4*x**5/5 + B*a**4*x**2/2 + 4*B*a**3*b*x**3/3 + 3*B*a**2*b**2 
*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x**6/6), Eq(e, 0)), (-12*A*a**4*e**5/ 
(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 
+ 300*d*e**10*x**4 + 60*e**11*x**5) - 12*A*a**3*b*d*e**4/(60*d**5*e**6 + 3 
00*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x** 
4 + 60*e**11*x**5) - 60*A*a**3*b*e**5*x/(60*d**5*e**6 + 300*d**4*e**7*x + 
600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5 
) - 12*A*a**2*b**2*d**2*e**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e* 
*8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*a* 
*2*b**2*d*e**4*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 60 
0*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*a**2*b**2*e** 
5*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e** 
9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 12*A*a*b**3*d**3*e**2/(60*d** 
5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d 
*e**10*x**4 + 60*e**11*x**5) - 60*A*a*b**3*d**2*e**3*x/(60*d**5*e**6 + 300 
*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 
+ 60*e**11*x**5) - 120*A*a*b**3*d*e**4*x**2/(60*d**5*e**6 + 300*d**4*e**7* 
x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11* 
x**5) - 120*A*a*b**3*e**5*x**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (234) = 468\).

Time = 0.09 (sec) , antiderivative size = 957, normalized size of antiderivative = 4.09 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^4/((m^2 + 3*m + 2)*e^2) 
+ 4*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*A*a^3*b/((m^2 + 3*m + 2) 
*e^2) + (e*x + d)^(m + 1)*A*a^4/(e*(m + 1)) + 4*((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^3*b/((m^3 + 6* 
m^2 + 11*m + 6)*e^3) + 6*((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^2*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) 
+ 6*((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^2*b^2/((m^4 + 
 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 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*a*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 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*a*b^3/((m^5 + 15*m^4 + 85*m^3 + 225* 
m^2 + 274*m + 120)*e^5) + ((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*A*b^4/((m^ 
5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^5 + 15*m^4 + 85*m^ 
3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24* 
m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4377 vs. \(2 (234) = 468\).

Time = 0.84 (sec) , antiderivative size = 4377, normalized size of antiderivative = 18.71 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

((e*x + d)^m*B*b^4*e^6*m^5*x^6 + (e*x + d)^m*B*b^4*d*e^5*m^5*x^5 + 4*(e*x 
+ d)^m*B*a*b^3*e^6*m^5*x^5 + (e*x + d)^m*A*b^4*e^6*m^5*x^5 + 15*(e*x + d)^ 
m*B*b^4*e^6*m^4*x^6 + 4*(e*x + d)^m*B*a*b^3*d*e^5*m^5*x^4 + (e*x + d)^m*A* 
b^4*d*e^5*m^5*x^4 + 6*(e*x + d)^m*B*a^2*b^2*e^6*m^5*x^4 + 4*(e*x + d)^m*A* 
a*b^3*e^6*m^5*x^4 + 10*(e*x + d)^m*B*b^4*d*e^5*m^4*x^5 + 64*(e*x + d)^m*B* 
a*b^3*e^6*m^4*x^5 + 16*(e*x + d)^m*A*b^4*e^6*m^4*x^5 + 85*(e*x + d)^m*B*b^ 
4*e^6*m^3*x^6 + 6*(e*x + d)^m*B*a^2*b^2*d*e^5*m^5*x^3 + 4*(e*x + d)^m*A*a* 
b^3*d*e^5*m^5*x^3 + 4*(e*x + d)^m*B*a^3*b*e^6*m^5*x^3 + 6*(e*x + d)^m*A*a^ 
2*b^2*e^6*m^5*x^3 - 5*(e*x + d)^m*B*b^4*d^2*e^4*m^4*x^4 + 48*(e*x + d)^m*B 
*a*b^3*d*e^5*m^4*x^4 + 12*(e*x + d)^m*A*b^4*d*e^5*m^4*x^4 + 102*(e*x + d)^ 
m*B*a^2*b^2*e^6*m^4*x^4 + 68*(e*x + d)^m*A*a*b^3*e^6*m^4*x^4 + 35*(e*x + d 
)^m*B*b^4*d*e^5*m^3*x^5 + 380*(e*x + d)^m*B*a*b^3*e^6*m^3*x^5 + 95*(e*x + 
d)^m*A*b^4*e^6*m^3*x^5 + 225*(e*x + d)^m*B*b^4*e^6*m^2*x^6 + 4*(e*x + d)^m 
*B*a^3*b*d*e^5*m^5*x^2 + 6*(e*x + d)^m*A*a^2*b^2*d*e^5*m^5*x^2 + (e*x + d) 
^m*B*a^4*e^6*m^5*x^2 + 4*(e*x + d)^m*A*a^3*b*e^6*m^5*x^2 - 16*(e*x + d)^m* 
B*a*b^3*d^2*e^4*m^4*x^3 - 4*(e*x + d)^m*A*b^4*d^2*e^4*m^4*x^3 + 84*(e*x + 
d)^m*B*a^2*b^2*d*e^5*m^4*x^3 + 56*(e*x + d)^m*A*a*b^3*d*e^5*m^4*x^3 + 72*( 
e*x + d)^m*B*a^3*b*e^6*m^4*x^3 + 108*(e*x + d)^m*A*a^2*b^2*e^6*m^4*x^3 - 3 
0*(e*x + d)^m*B*b^4*d^2*e^4*m^3*x^4 + 188*(e*x + d)^m*B*a*b^3*d*e^5*m^3*x^ 
4 + 47*(e*x + d)^m*A*b^4*d*e^5*m^3*x^4 + 642*(e*x + d)^m*B*a^2*b^2*e^6*...
 

Mupad [B] (verification not implemented)

Time = 12.49 (sec) , antiderivative size = 2117, normalized size of antiderivative = 9.05 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

((d + e*x)^m*(720*A*a^4*d*e^5 - 120*B*b^4*d^6 + 144*A*b^4*d^5*e - 360*B*a^ 
4*d^2*e^4 - 720*A*a*b^3*d^4*e^2 - 1440*A*a^3*b*d^2*e^4 + 960*B*a^3*b*d^3*e 
^3 + 580*A*a^4*d*e^5*m^2 + 155*A*a^4*d*e^5*m^3 + 20*A*a^4*d*e^5*m^4 + A*a^ 
4*d*e^5*m^5 - 342*B*a^4*d^2*e^4*m + 1440*A*a^2*b^2*d^3*e^3 - 1080*B*a^2*b^ 
2*d^4*e^2 - 119*B*a^4*d^2*e^4*m^2 - 18*B*a^4*d^2*e^4*m^3 - B*a^4*d^2*e^4*m 
^4 + 576*B*a*b^3*d^5*e + 1044*A*a^4*d*e^5*m + 24*A*b^4*d^5*e*m + 888*A*a^2 
*b^2*d^3*e^3*m - 24*A*a*b^3*d^4*e^2*m^2 - 476*A*a^3*b*d^2*e^4*m^2 - 72*A*a 
^3*b*d^2*e^4*m^3 - 4*A*a^3*b*d^2*e^4*m^4 - 396*B*a^2*b^2*d^4*e^2*m + 120*B 
*a^3*b*d^3*e^3*m^2 + 8*B*a^3*b*d^3*e^3*m^3 + 96*B*a*b^3*d^5*e*m + 180*A*a^ 
2*b^2*d^3*e^3*m^2 + 12*A*a^2*b^2*d^3*e^3*m^3 - 36*B*a^2*b^2*d^4*e^2*m^2 - 
264*A*a*b^3*d^4*e^2*m - 1368*A*a^3*b*d^2*e^4*m + 592*B*a^3*b*d^3*e^3*m))/( 
e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x*(d 
+ e*x)^m*(720*A*a^4*e^6 + 1044*A*a^4*e^6*m + 580*A*a^4*e^6*m^2 + 155*A*a^4 
*e^6*m^3 + 20*A*a^4*e^6*m^4 + A*a^4*e^6*m^5 - 144*A*b^4*d^4*e^2*m + 342*B* 
a^4*d*e^5*m^2 + 119*B*a^4*d*e^5*m^3 + 18*B*a^4*d*e^5*m^4 + B*a^4*d*e^5*m^5 
 - 24*A*b^4*d^4*e^2*m^2 + 360*B*a^4*d*e^5*m + 120*B*b^4*d^5*e*m - 1440*A*a 
^2*b^2*d^2*e^4*m + 264*A*a*b^3*d^3*e^3*m^2 + 24*A*a*b^3*d^3*e^3*m^3 + 1080 
*B*a^2*b^2*d^3*e^3*m - 96*B*a*b^3*d^4*e^2*m^2 - 592*B*a^3*b*d^2*e^4*m^2 - 
120*B*a^3*b*d^2*e^4*m^3 - 8*B*a^3*b*d^2*e^4*m^4 + 1440*A*a^3*b*d*e^5*m - 8 
88*A*a^2*b^2*d^2*e^4*m^2 - 180*A*a^2*b^2*d^2*e^4*m^3 - 12*A*a^2*b^2*d^2...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1728, normalized size of antiderivative = 7.38 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

((d + e*x)**m*(a**5*d*e**5*m**5 + 20*a**5*d*e**5*m**4 + 155*a**5*d*e**5*m* 
*3 + 580*a**5*d*e**5*m**2 + 1044*a**5*d*e**5*m + 720*a**5*d*e**5 + a**5*e* 
*6*m**5*x + 20*a**5*e**6*m**4*x + 155*a**5*e**6*m**3*x + 580*a**5*e**6*m** 
2*x + 1044*a**5*e**6*m*x + 720*a**5*e**6*x - 5*a**4*b*d**2*e**4*m**4 - 90* 
a**4*b*d**2*e**4*m**3 - 595*a**4*b*d**2*e**4*m**2 - 1710*a**4*b*d**2*e**4* 
m - 1800*a**4*b*d**2*e**4 + 5*a**4*b*d*e**5*m**5*x + 90*a**4*b*d*e**5*m**4 
*x + 595*a**4*b*d*e**5*m**3*x + 1710*a**4*b*d*e**5*m**2*x + 1800*a**4*b*d* 
e**5*m*x + 5*a**4*b*e**6*m**5*x**2 + 95*a**4*b*e**6*m**4*x**2 + 685*a**4*b 
*e**6*m**3*x**2 + 2305*a**4*b*e**6*m**2*x**2 + 3510*a**4*b*e**6*m*x**2 + 1 
800*a**4*b*e**6*x**2 + 20*a**3*b**2*d**3*e**3*m**3 + 300*a**3*b**2*d**3*e* 
*3*m**2 + 1480*a**3*b**2*d**3*e**3*m + 2400*a**3*b**2*d**3*e**3 - 20*a**3* 
b**2*d**2*e**4*m**4*x - 300*a**3*b**2*d**2*e**4*m**3*x - 1480*a**3*b**2*d* 
*2*e**4*m**2*x - 2400*a**3*b**2*d**2*e**4*m*x + 10*a**3*b**2*d*e**5*m**5*x 
**2 + 160*a**3*b**2*d*e**5*m**4*x**2 + 890*a**3*b**2*d*e**5*m**3*x**2 + 19 
40*a**3*b**2*d*e**5*m**2*x**2 + 1200*a**3*b**2*d*e**5*m*x**2 + 10*a**3*b** 
2*e**6*m**5*x**3 + 180*a**3*b**2*e**6*m**4*x**3 + 1210*a**3*b**2*e**6*m**3 
*x**3 + 3720*a**3*b**2*e**6*m**2*x**3 + 5080*a**3*b**2*e**6*m*x**3 + 2400* 
a**3*b**2*e**6*x**3 - 60*a**2*b**3*d**4*e**2*m**2 - 660*a**2*b**3*d**4*e** 
2*m - 1800*a**2*b**3*d**4*e**2 + 60*a**2*b**3*d**3*e**3*m**3*x + 660*a**2* 
b**3*d**3*e**3*m**2*x + 1800*a**2*b**3*d**3*e**3*m*x - 30*a**2*b**3*d**...