Integrand size = 31, antiderivative size = 234 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {(b d-a e)^4 (B d-A e) (d+e x)^{1+m}}{e^6 (1+m)}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {b^4 B (d+e x)^{6+m}}{e^6 (6+m)} \]
-(-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)
Time = 0.07 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^4 (B d-A e)}{1+m}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)}{2+m}-\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^2}{3+m}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^3}{4+m}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^4}{5+m}+\frac {b^4 B (d+e x)^5}{6+m}\right )}{e^6} \]
((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
Time = 0.42 (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 |
\(\displaystyle -\frac {b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}+\frac {2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac {(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac {(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac {2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac {b^4 B (d+e x)^{m+6}}{e^6 (m+6)}\) |
-(((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))
3.19.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1957\) vs. \(2(234)=468\).
Time = 0.38 (sec) , antiderivative size = 1958, normalized size of antiderivative = 8.37
method | result | size |
norman | \(\text {Expression too large to display}\) | \(1958\) |
gosper | \(\text {Expression too large to display}\) | \(2355\) |
risch | \(\text {Expression too large to display}\) | \(3035\) |
parallelrisch | \(\text {Expression too large to display}\) | \(4523\) |
b^4*B/(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...
Leaf count of result is larger than twice the leaf count of optimal. 2274 vs. \(2 (234) = 468\).
Time = 0.38 (sec) , antiderivative size = 2274, normalized size of antiderivative = 9.72 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
(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*...
Leaf count of result is larger than twice the leaf count of optimal. 28048 vs. \(2 (224) = 448\).
Time = 5.10 (sec) , antiderivative size = 28048, normalized size of antiderivative = 119.86 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
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*...
Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (234) = 468\).
Time = 0.23 (sec) , antiderivative size = 957, normalized size of antiderivative = 4.09 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
(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...
Leaf count of result is larger than twice the leaf count of optimal. 4377 vs. \(2 (234) = 468\).
Time = 0.30 (sec) , antiderivative size = 4377, normalized size of antiderivative = 18.71 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
((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*...
Time = 11.94 (sec) , antiderivative size = 2117, normalized size of antiderivative = 9.05 \[ \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
((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...