Integrand size = 20, antiderivative size = 138 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {b^2 B (d+e x)^{4+m}}{e^4 (4+m)} \] Output:
-(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(1+m)/e^4/(1+m)+(-a*e+b*d)*(-2*A*b*e-B*a* e+3*B*b*d)*(e*x+d)^(2+m)/e^4/(2+m)-b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(3+m )/e^4/(3+m)+b^2*B*(e*x+d)^(4+m)/e^4/(4+m)
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.88 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^2 (B d-A e)}{1+m}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)}{2+m}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^2}{3+m}+\frac {b^2 B (d+e x)^3}{4+m}\right )}{e^4} \] Input:
Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]
Output:
((d + e*x)^(1 + m)*(-(((b*d - a*e)^2*(B*d - A*e))/(1 + m)) + ((b*d - a*e)* (3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x))/(2 + m) - (b*(3*b*B*d - A*b*e - 2*a *B*e)*(d + e*x)^2)/(3 + m) + (b^2*B*(d + e*x)^3)/(4 + m)))/e^4
Time = 0.28 (sec) , antiderivative size = 138, 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.
| \(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {(a e-b d)^2 (A e-B d) (d+e x)^m}{e^3}+\frac {(a e-b d) (d+e x)^{m+1} (a B e+2 A b e-3 b B d)}{e^3}+\frac {b (d+e x)^{m+2} (2 a B e+A b e-3 b B d)}{e^3}+\frac {b^2 B (d+e x)^{m+3}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b d-a e)^2 (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(b d-a e) (d+e x)^{m+2} (-a B e-2 A b e+3 b B d)}{e^4 (m+2)}-\frac {b (d+e x)^{m+3} (-2 a B e-A b e+3 b B d)}{e^4 (m+3)}+\frac {b^2 B (d+e x)^{m+4}}{e^4 (m+4)}\) |
Input:
Int[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]
Output:
-(((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((b*d - a *e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (b*(3*b *B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (b^2*B*(d + e*x )^(4 + m))/(e^4*(4 + m))
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]))))
Leaf count of result is larger than twice the leaf count of optimal. \(575\) vs. \(2(138)=276\).
Time = 0.27 (sec) , antiderivative size = 576, normalized size of antiderivative = 4.17
| method | result | size |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (B \,b^{2} e^{3} m^{3} x^{3}+A \,b^{2} e^{3} m^{3} x^{2}+2 B a b \,e^{3} m^{3} x^{2}+6 B \,b^{2} e^{3} m^{2} x^{3}+2 A a b \,e^{3} m^{3} x +7 A \,b^{2} e^{3} m^{2} x^{2}+B \,a^{2} e^{3} m^{3} x +14 B a b \,e^{3} m^{2} x^{2}-3 B \,b^{2} d \,e^{2} m^{2} x^{2}+11 B \,b^{2} e^{3} m \,x^{3}+A \,a^{2} e^{3} m^{3}+16 A a b \,e^{3} m^{2} x -2 A \,b^{2} d \,e^{2} m^{2} x +14 A \,b^{2} e^{3} m \,x^{2}+8 B \,a^{2} e^{3} m^{2} x -4 B a b d \,e^{2} m^{2} x +28 B a b \,e^{3} m \,x^{2}-9 B \,b^{2} d \,e^{2} m \,x^{2}+6 b^{2} B \,x^{3} e^{3}+9 A \,a^{2} e^{3} m^{2}-2 A a b d \,e^{2} m^{2}+38 A a b \,e^{3} m x -10 A \,b^{2} d \,e^{2} m x +8 A \,x^{2} b^{2} e^{3}-B \,a^{2} d \,e^{2} m^{2}+19 B \,a^{2} e^{3} m x -20 B a b d \,e^{2} m x +16 B \,x^{2} a b \,e^{3}+6 B \,b^{2} d^{2} e m x -6 B \,x^{2} b^{2} d \,e^{2}+26 A \,a^{2} e^{3} m -14 A a b d \,e^{2} m +24 A x a b \,e^{3}+2 A \,b^{2} d^{2} e m -8 A x \,b^{2} d \,e^{2}-7 B \,a^{2} d \,e^{2} m +12 B x \,a^{2} e^{3}+4 B a b \,d^{2} e m -16 B x a b d \,e^{2}+6 B x \,b^{2} d^{2} e +24 a^{2} A \,e^{3}-24 A a b d \,e^{2}+8 A \,b^{2} d^{2} e -12 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -6 b^{2} B \,d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(576\) |
| orering | \(\frac {\left (e x +d \right )^{m} \left (B \,b^{2} e^{3} m^{3} x^{3}+A \,b^{2} e^{3} m^{3} x^{2}+2 B a b \,e^{3} m^{3} x^{2}+6 B \,b^{2} e^{3} m^{2} x^{3}+2 A a b \,e^{3} m^{3} x +7 A \,b^{2} e^{3} m^{2} x^{2}+B \,a^{2} e^{3} m^{3} x +14 B a b \,e^{3} m^{2} x^{2}-3 B \,b^{2} d \,e^{2} m^{2} x^{2}+11 B \,b^{2} e^{3} m \,x^{3}+A \,a^{2} e^{3} m^{3}+16 A a b \,e^{3} m^{2} x -2 A \,b^{2} d \,e^{2} m^{2} x +14 A \,b^{2} e^{3} m \,x^{2}+8 B \,a^{2} e^{3} m^{2} x -4 B a b d \,e^{2} m^{2} x +28 B a b \,e^{3} m \,x^{2}-9 B \,b^{2} d \,e^{2} m \,x^{2}+6 b^{2} B \,x^{3} e^{3}+9 A \,a^{2} e^{3} m^{2}-2 A a b d \,e^{2} m^{2}+38 A a b \,e^{3} m x -10 A \,b^{2} d \,e^{2} m x +8 A \,x^{2} b^{2} e^{3}-B \,a^{2} d \,e^{2} m^{2}+19 B \,a^{2} e^{3} m x -20 B a b d \,e^{2} m x +16 B \,x^{2} a b \,e^{3}+6 B \,b^{2} d^{2} e m x -6 B \,x^{2} b^{2} d \,e^{2}+26 A \,a^{2} e^{3} m -14 A a b d \,e^{2} m +24 A x a b \,e^{3}+2 A \,b^{2} d^{2} e m -8 A x \,b^{2} d \,e^{2}-7 B \,a^{2} d \,e^{2} m +12 B x \,a^{2} e^{3}+4 B a b \,d^{2} e m -16 B x a b d \,e^{2}+6 B x \,b^{2} d^{2} e +24 a^{2} A \,e^{3}-24 A a b d \,e^{2}+8 A \,b^{2} d^{2} e -12 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -6 b^{2} B \,d^{3}\right ) \left (e x +d \right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(579\) |
| norman | \(\frac {b^{2} B \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {d \left (A \,a^{2} e^{3} m^{3}+9 A \,a^{2} e^{3} m^{2}-2 A a b d \,e^{2} m^{2}-B \,a^{2} d \,e^{2} m^{2}+26 A \,a^{2} e^{3} m -14 A a b d \,e^{2} m +2 A \,b^{2} d^{2} e m -7 B \,a^{2} d \,e^{2} m +4 B a b \,d^{2} e m +24 a^{2} A \,e^{3}-24 A a b d \,e^{2}+8 A \,b^{2} d^{2} e -12 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -6 b^{2} B \,d^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (2 A a b \,e^{2} m^{2}+A \,b^{2} d e \,m^{2}+B \,a^{2} e^{2} m^{2}+2 B a b d e \,m^{2}+14 A a b \,e^{2} m +4 A \,b^{2} d e m +7 B \,a^{2} e^{2} m +8 B a b d e m -3 B \,b^{2} d^{2} m +24 A a b \,e^{2}+12 B \,a^{2} e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (A \,a^{2} e^{3} m^{3}+2 A a b d \,e^{2} m^{3}+B \,a^{2} d \,e^{2} m^{3}+9 A \,a^{2} e^{3} m^{2}+14 A a b d \,e^{2} m^{2}-2 A \,b^{2} d^{2} e \,m^{2}+7 B \,a^{2} d \,e^{2} m^{2}-4 B a b \,d^{2} e \,m^{2}+26 A \,a^{2} e^{3} m +24 A a b d \,e^{2} m -8 A \,b^{2} d^{2} e m +12 B \,a^{2} d \,e^{2} m -16 B a b \,d^{2} e m +6 B \,b^{2} d^{3} m +24 a^{2} A \,e^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {b \left (A b e m +2 B a e m +B b d m +4 A b e +8 B a e \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}\) | \(609\) |
| risch | \(\frac {\left (B \,b^{2} e^{4} m^{3} x^{4}+A \,b^{2} e^{4} m^{3} x^{3}+2 B a b \,e^{4} m^{3} x^{3}+B \,b^{2} d \,e^{3} m^{3} x^{3}+6 B \,b^{2} e^{4} m^{2} x^{4}+2 A a b \,e^{4} m^{3} x^{2}+A \,b^{2} d \,e^{3} m^{3} x^{2}+7 A \,b^{2} e^{4} m^{2} x^{3}+B \,a^{2} e^{4} m^{3} x^{2}+2 B a b d \,e^{3} m^{3} x^{2}+14 B a b \,e^{4} m^{2} x^{3}+3 B \,b^{2} d \,e^{3} m^{2} x^{3}+11 B \,b^{2} e^{4} m \,x^{4}+A \,a^{2} e^{4} m^{3} x +2 A a b d \,e^{3} m^{3} x +16 A a b \,e^{4} m^{2} x^{2}+5 A \,b^{2} d \,e^{3} m^{2} x^{2}+14 A \,b^{2} e^{4} m \,x^{3}+B \,a^{2} d \,e^{3} m^{3} x +8 B \,a^{2} e^{4} m^{2} x^{2}+10 B a b d \,e^{3} m^{2} x^{2}+28 B a b \,e^{4} m \,x^{3}-3 B \,b^{2} d^{2} e^{2} m^{2} x^{2}+2 B \,b^{2} d \,e^{3} m \,x^{3}+6 b^{2} B \,x^{4} e^{4}+A \,a^{2} d \,e^{3} m^{3}+9 A \,a^{2} e^{4} m^{2} x +14 A a b d \,e^{3} m^{2} x +38 A a b \,e^{4} m \,x^{2}-2 A \,b^{2} d^{2} e^{2} m^{2} x +4 A \,b^{2} d \,e^{3} m \,x^{2}+8 A \,b^{2} e^{4} x^{3}+7 B \,a^{2} d \,e^{3} m^{2} x +19 B \,a^{2} e^{4} m \,x^{2}-4 B a b \,d^{2} e^{2} m^{2} x +8 B a b d \,e^{3} m \,x^{2}+16 B a b \,e^{4} x^{3}-3 B \,b^{2} d^{2} e^{2} m \,x^{2}+9 A \,a^{2} d \,e^{3} m^{2}+26 A \,a^{2} e^{4} m x -2 A a b \,d^{2} e^{2} m^{2}+24 A a b d \,e^{3} m x +24 A a b \,e^{4} x^{2}-8 A \,b^{2} d^{2} e^{2} m x -B \,a^{2} d^{2} e^{2} m^{2}+12 B \,a^{2} d \,e^{3} m x +12 B \,a^{2} e^{4} x^{2}-16 B a b \,d^{2} e^{2} m x +6 B \,b^{2} d^{3} e m x +26 A \,a^{2} d \,e^{3} m +24 A \,a^{2} e^{4} x -14 A a b \,d^{2} e^{2} m +2 A \,b^{2} d^{3} e m -7 B \,a^{2} d^{2} e^{2} m +4 B a b \,d^{3} e m +24 a^{2} A d \,e^{3}-24 A a b \,d^{2} e^{2}+8 A \,b^{2} d^{3} e -12 B \,a^{2} d^{2} e^{2}+16 B a b \,d^{3} e -6 b^{2} B \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(828\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1313\) |
Input:
int((b*x+a)^2*(B*x+A)*(e*x+d)^m,x,method=_RETURNVERBOSE)
Output:
1/e^4*(e*x+d)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(B*b^2*e^3*m^3*x^3+A*b^2*e ^3*m^3*x^2+2*B*a*b*e^3*m^3*x^2+6*B*b^2*e^3*m^2*x^3+2*A*a*b*e^3*m^3*x+7*A*b ^2*e^3*m^2*x^2+B*a^2*e^3*m^3*x+14*B*a*b*e^3*m^2*x^2-3*B*b^2*d*e^2*m^2*x^2+ 11*B*b^2*e^3*m*x^3+A*a^2*e^3*m^3+16*A*a*b*e^3*m^2*x-2*A*b^2*d*e^2*m^2*x+14 *A*b^2*e^3*m*x^2+8*B*a^2*e^3*m^2*x-4*B*a*b*d*e^2*m^2*x+28*B*a*b*e^3*m*x^2- 9*B*b^2*d*e^2*m*x^2+6*B*b^2*e^3*x^3+9*A*a^2*e^3*m^2-2*A*a*b*d*e^2*m^2+38*A *a*b*e^3*m*x-10*A*b^2*d*e^2*m*x+8*A*b^2*e^3*x^2-B*a^2*d*e^2*m^2+19*B*a^2*e ^3*m*x-20*B*a*b*d*e^2*m*x+16*B*a*b*e^3*x^2+6*B*b^2*d^2*e*m*x-6*B*b^2*d*e^2 *x^2+26*A*a^2*e^3*m-14*A*a*b*d*e^2*m+24*A*a*b*e^3*x+2*A*b^2*d^2*e*m-8*A*b^ 2*d*e^2*x-7*B*a^2*d*e^2*m+12*B*a^2*e^3*x+4*B*a*b*d^2*e*m-16*B*a*b*d*e^2*x+ 6*B*b^2*d^2*e*x+24*A*a^2*e^3-24*A*a*b*d*e^2+8*A*b^2*d^2*e-12*B*a^2*d*e^2+1 6*B*a*b*d^2*e-6*B*b^2*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (138) = 276\).
Time = 0.13 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.78 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\frac {{\left (A a^{2} d e^{3} m^{3} - 6 \, B b^{2} d^{4} + 24 \, A a^{2} d e^{3} + 8 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e - 12 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + {\left (B b^{2} e^{4} m^{3} + 6 \, B b^{2} e^{4} m^{2} + 11 \, B b^{2} e^{4} m + 6 \, B b^{2} e^{4}\right )} x^{4} + {\left (8 \, {\left (2 \, B a b + A b^{2}\right )} e^{4} + {\left (B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m^{3} + {\left (3 \, B b^{2} d e^{3} + 7 \, {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m^{2} + 2 \, {\left (B b^{2} d e^{3} + 7 \, {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m\right )} x^{3} + {\left (9 \, A a^{2} d e^{3} - {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} m^{2} + {\left (12 \, {\left (B a^{2} + 2 \, A a b\right )} e^{4} + {\left ({\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m^{3} - {\left (3 \, B b^{2} d^{2} e^{2} - 5 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} - 8 \, {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m^{2} - {\left (3 \, B b^{2} d^{2} e^{2} - 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} - 19 \, {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m\right )} x^{2} + {\left (26 \, A a^{2} d e^{3} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e - 7 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} m + {\left (24 \, A a^{2} e^{4} + {\left (A a^{2} e^{4} + {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m^{3} + {\left (9 \, A a^{2} e^{4} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 7 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m^{2} + 2 \, {\left (3 \, B b^{2} d^{3} e + 13 \, A a^{2} e^{4} - 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="fricas")
Output:
(A*a^2*d*e^3*m^3 - 6*B*b^2*d^4 + 24*A*a^2*d*e^3 + 8*(2*B*a*b + A*b^2)*d^3* e - 12*(B*a^2 + 2*A*a*b)*d^2*e^2 + (B*b^2*e^4*m^3 + 6*B*b^2*e^4*m^2 + 11*B *b^2*e^4*m + 6*B*b^2*e^4)*x^4 + (8*(2*B*a*b + A*b^2)*e^4 + (B*b^2*d*e^3 + (2*B*a*b + A*b^2)*e^4)*m^3 + (3*B*b^2*d*e^3 + 7*(2*B*a*b + A*b^2)*e^4)*m^2 + 2*(B*b^2*d*e^3 + 7*(2*B*a*b + A*b^2)*e^4)*m)*x^3 + (9*A*a^2*d*e^3 - (B* a^2 + 2*A*a*b)*d^2*e^2)*m^2 + (12*(B*a^2 + 2*A*a*b)*e^4 + ((2*B*a*b + A*b^ 2)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)*m^3 - (3*B*b^2*d^2*e^2 - 5*(2*B*a*b + A* b^2)*d*e^3 - 8*(B*a^2 + 2*A*a*b)*e^4)*m^2 - (3*B*b^2*d^2*e^2 - 4*(2*B*a*b + A*b^2)*d*e^3 - 19*(B*a^2 + 2*A*a*b)*e^4)*m)*x^2 + (26*A*a^2*d*e^3 + 2*(2 *B*a*b + A*b^2)*d^3*e - 7*(B*a^2 + 2*A*a*b)*d^2*e^2)*m + (24*A*a^2*e^4 + ( A*a^2*e^4 + (B*a^2 + 2*A*a*b)*d*e^3)*m^3 + (9*A*a^2*e^4 - 2*(2*B*a*b + A*b ^2)*d^2*e^2 + 7*(B*a^2 + 2*A*a*b)*d*e^3)*m^2 + 2*(3*B*b^2*d^3*e + 13*A*a^2 *e^4 - 4*(2*B*a*b + A*b^2)*d^2*e^2 + 6*(B*a^2 + 2*A*a*b)*d*e^3)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50*e^4*m + 24*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 6186 vs. \(2 (126) = 252\).
Time = 1.37 (sec) , antiderivative size = 6186, normalized size of antiderivative = 44.83 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**2*(B*x+A)*(e*x+d)**m,x)
Output:
Piecewise((d**m*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2 *B*a*b*x**3/3 + B*b**2*x**4/4), Eq(e, 0)), (-2*A*a**2*e**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*A*a*b*d*e**2/(6*d**3*e* *4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*a*b*e**3*x/(6*d* *3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*A*b**2*d**2*e /(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*b**2* d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6 *A*b**2*e**3*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7* x**3) - B*a**2*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e **7*x**3) - 3*B*a**2*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 4*B*a*b*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6 *x**2 + 6*e**7*x**3) - 12*B*a*b*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 1 8*d*e**6*x**2 + 6*e**7*x**3) - 12*B*a*b*e**3*x**2/(6*d**3*e**4 + 18*d**2*e **5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*b**2*d**3*log(d/e + x)/(6*d**3 *e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*B*b**2*d**3/(6 *d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*b**2*d* *2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e** 7*x**3) + 27*B*b**2*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x** 2 + 6*e**7*x**3) + 18*B*b**2*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d* *2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*b**2*d*e**2*x**2/(6*d*...
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (138) = 276\).
Time = 0.05 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.64 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A a b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a^{2}}{e {\left (m + 1\right )}} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} B a b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B b^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="maxima")
Output:
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^2/((m^2 + 3*m + 2)*e^2) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*A*a*b/((m^2 + 3*m + 2)*e ^2) + (e*x + d)^(m + 1)*A*a^2/(e*(m + 1)) + 2*((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*b/((m^3 + 6*m^2 + 11*m + 6)*e^3) + ((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*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + ((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*b^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 1261 vs. \(2 (138) = 276\).
Time = 0.13 (sec) , antiderivative size = 1261, normalized size of antiderivative = 9.14 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="giac")
Output:
((e*x + d)^m*B*b^2*e^4*m^3*x^4 + (e*x + d)^m*B*b^2*d*e^3*m^3*x^3 + 2*(e*x + d)^m*B*a*b*e^4*m^3*x^3 + (e*x + d)^m*A*b^2*e^4*m^3*x^3 + 6*(e*x + d)^m*B *b^2*e^4*m^2*x^4 + 2*(e*x + d)^m*B*a*b*d*e^3*m^3*x^2 + (e*x + d)^m*A*b^2*d *e^3*m^3*x^2 + (e*x + d)^m*B*a^2*e^4*m^3*x^2 + 2*(e*x + d)^m*A*a*b*e^4*m^3 *x^2 + 3*(e*x + d)^m*B*b^2*d*e^3*m^2*x^3 + 14*(e*x + d)^m*B*a*b*e^4*m^2*x^ 3 + 7*(e*x + d)^m*A*b^2*e^4*m^2*x^3 + 11*(e*x + d)^m*B*b^2*e^4*m*x^4 + (e* x + d)^m*B*a^2*d*e^3*m^3*x + 2*(e*x + d)^m*A*a*b*d*e^3*m^3*x + (e*x + d)^m *A*a^2*e^4*m^3*x - 3*(e*x + d)^m*B*b^2*d^2*e^2*m^2*x^2 + 10*(e*x + d)^m*B* a*b*d*e^3*m^2*x^2 + 5*(e*x + d)^m*A*b^2*d*e^3*m^2*x^2 + 8*(e*x + d)^m*B*a^ 2*e^4*m^2*x^2 + 16*(e*x + d)^m*A*a*b*e^4*m^2*x^2 + 2*(e*x + d)^m*B*b^2*d*e ^3*m*x^3 + 28*(e*x + d)^m*B*a*b*e^4*m*x^3 + 14*(e*x + d)^m*A*b^2*e^4*m*x^3 + 6*(e*x + d)^m*B*b^2*e^4*x^4 + (e*x + d)^m*A*a^2*d*e^3*m^3 - 4*(e*x + d) ^m*B*a*b*d^2*e^2*m^2*x - 2*(e*x + d)^m*A*b^2*d^2*e^2*m^2*x + 7*(e*x + d)^m *B*a^2*d*e^3*m^2*x + 14*(e*x + d)^m*A*a*b*d*e^3*m^2*x + 9*(e*x + d)^m*A*a^ 2*e^4*m^2*x - 3*(e*x + d)^m*B*b^2*d^2*e^2*m*x^2 + 8*(e*x + d)^m*B*a*b*d*e^ 3*m*x^2 + 4*(e*x + d)^m*A*b^2*d*e^3*m*x^2 + 19*(e*x + d)^m*B*a^2*e^4*m*x^2 + 38*(e*x + d)^m*A*a*b*e^4*m*x^2 + 16*(e*x + d)^m*B*a*b*e^4*x^3 + 8*(e*x + d)^m*A*b^2*e^4*x^3 - (e*x + d)^m*B*a^2*d^2*e^2*m^2 - 2*(e*x + d)^m*A*a*b *d^2*e^2*m^2 + 9*(e*x + d)^m*A*a^2*d*e^3*m^2 + 6*(e*x + d)^m*B*b^2*d^3*e*m *x - 16*(e*x + d)^m*B*a*b*d^2*e^2*m*x - 8*(e*x + d)^m*A*b^2*d^2*e^2*m*x...
Time = 1.37 (sec) , antiderivative size = 676, normalized size of antiderivative = 4.90 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (-B\,a^2\,d^2\,e^2\,m^2-7\,B\,a^2\,d^2\,e^2\,m-12\,B\,a^2\,d^2\,e^2+A\,a^2\,d\,e^3\,m^3+9\,A\,a^2\,d\,e^3\,m^2+26\,A\,a^2\,d\,e^3\,m+24\,A\,a^2\,d\,e^3+4\,B\,a\,b\,d^3\,e\,m+16\,B\,a\,b\,d^3\,e-2\,A\,a\,b\,d^2\,e^2\,m^2-14\,A\,a\,b\,d^2\,e^2\,m-24\,A\,a\,b\,d^2\,e^2-6\,B\,b^2\,d^4+2\,A\,b^2\,d^3\,e\,m+8\,A\,b^2\,d^3\,e\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (B\,a^2\,d\,e^3\,m^3+7\,B\,a^2\,d\,e^3\,m^2+12\,B\,a^2\,d\,e^3\,m+A\,a^2\,e^4\,m^3+9\,A\,a^2\,e^4\,m^2+26\,A\,a^2\,e^4\,m+24\,A\,a^2\,e^4-4\,B\,a\,b\,d^2\,e^2\,m^2-16\,B\,a\,b\,d^2\,e^2\,m+2\,A\,a\,b\,d\,e^3\,m^3+14\,A\,a\,b\,d\,e^3\,m^2+24\,A\,a\,b\,d\,e^3\,m+6\,B\,b^2\,d^3\,e\,m-2\,A\,b^2\,d^2\,e^2\,m^2-8\,A\,b^2\,d^2\,e^2\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (B\,a^2\,e^2\,m^2+7\,B\,a^2\,e^2\,m+12\,B\,a^2\,e^2+2\,B\,a\,b\,d\,e\,m^2+8\,B\,a\,b\,d\,e\,m+2\,A\,a\,b\,e^2\,m^2+14\,A\,a\,b\,e^2\,m+24\,A\,a\,b\,e^2-3\,B\,b^2\,d^2\,m+A\,b^2\,d\,e\,m^2+4\,A\,b^2\,d\,e\,m\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {B\,b^2\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,A\,b\,e+8\,B\,a\,e+A\,b\,e\,m+2\,B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \] Input:
int((A + B*x)*(a + b*x)^2*(d + e*x)^m,x)
Output:
((d + e*x)^m*(24*A*a^2*d*e^3 - 6*B*b^2*d^4 + 8*A*b^2*d^3*e - 12*B*a^2*d^2* e^2 + 9*A*a^2*d*e^3*m^2 + A*a^2*d*e^3*m^3 - 7*B*a^2*d^2*e^2*m + 16*B*a*b*d ^3*e - B*a^2*d^2*e^2*m^2 - 24*A*a*b*d^2*e^2 + 26*A*a^2*d*e^3*m + 2*A*b^2*d ^3*e*m - 14*A*a*b*d^2*e^2*m - 2*A*a*b*d^2*e^2*m^2 + 4*B*a*b*d^3*e*m))/(e^4 *(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x*(d + e*x)^m*(24*A*a^2*e^4 + 26* A*a^2*e^4*m + 9*A*a^2*e^4*m^2 + A*a^2*e^4*m^3 - 8*A*b^2*d^2*e^2*m + 7*B*a^ 2*d*e^3*m^2 + B*a^2*d*e^3*m^3 - 2*A*b^2*d^2*e^2*m^2 + 12*B*a^2*d*e^3*m + 6 *B*b^2*d^3*e*m + 14*A*a*b*d*e^3*m^2 + 2*A*a*b*d*e^3*m^3 - 16*B*a*b*d^2*e^2 *m - 4*B*a*b*d^2*e^2*m^2 + 24*A*a*b*d*e^3*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^2*(m + 1)*(d + e*x)^m*(12*B*a^2*e^2 + 24*A*a*b*e^2 + 7* B*a^2*e^2*m - 3*B*b^2*d^2*m + B*a^2*e^2*m^2 + 14*A*a*b*e^2*m + 4*A*b^2*d*e *m + 2*A*a*b*e^2*m^2 + A*b^2*d*e*m^2 + 8*B*a*b*d*e*m + 2*B*a*b*d*e*m^2))/( e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (B*b^2*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (b*x^3*(d + e*x)^m *(3*m + m^2 + 2)*(4*A*b*e + 8*B*a*e + A*b*e*m + 2*B*a*e*m + B*b*d*m))/(e*( 50*m + 35*m^2 + 10*m^3 + m^4 + 24))
Time = 0.16 (sec) , antiderivative size = 546, normalized size of antiderivative = 3.96 \[ \int (a+b x)^2 (A+B x) (d+e x)^m \, dx=\frac {\left (e x +d \right )^{m} \left (b^{3} e^{4} m^{3} x^{4}+3 a \,b^{2} e^{4} m^{3} x^{3}+b^{3} d \,e^{3} m^{3} x^{3}+6 b^{3} e^{4} m^{2} x^{4}+3 a^{2} b \,e^{4} m^{3} x^{2}+3 a \,b^{2} d \,e^{3} m^{3} x^{2}+21 a \,b^{2} e^{4} m^{2} x^{3}+3 b^{3} d \,e^{3} m^{2} x^{3}+11 b^{3} e^{4} m \,x^{4}+a^{3} e^{4} m^{3} x +3 a^{2} b d \,e^{3} m^{3} x +24 a^{2} b \,e^{4} m^{2} x^{2}+15 a \,b^{2} d \,e^{3} m^{2} x^{2}+42 a \,b^{2} e^{4} m \,x^{3}-3 b^{3} d^{2} e^{2} m^{2} x^{2}+2 b^{3} d \,e^{3} m \,x^{3}+6 b^{3} e^{4} x^{4}+a^{3} d \,e^{3} m^{3}+9 a^{3} e^{4} m^{2} x +21 a^{2} b d \,e^{3} m^{2} x +57 a^{2} b \,e^{4} m \,x^{2}-6 a \,b^{2} d^{2} e^{2} m^{2} x +12 a \,b^{2} d \,e^{3} m \,x^{2}+24 a \,b^{2} e^{4} x^{3}-3 b^{3} d^{2} e^{2} m \,x^{2}+9 a^{3} d \,e^{3} m^{2}+26 a^{3} e^{4} m x -3 a^{2} b \,d^{2} e^{2} m^{2}+36 a^{2} b d \,e^{3} m x +36 a^{2} b \,e^{4} x^{2}-24 a \,b^{2} d^{2} e^{2} m x +6 b^{3} d^{3} e m x +26 a^{3} d \,e^{3} m +24 a^{3} e^{4} x -21 a^{2} b \,d^{2} e^{2} m +6 a \,b^{2} d^{3} e m +24 a^{3} d \,e^{3}-36 a^{2} b \,d^{2} e^{2}+24 a \,b^{2} d^{3} e -6 b^{3} d^{4}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \] Input:
int((b*x+a)^2*(B*x+A)*(e*x+d)^m,x)
Output:
((d + e*x)**m*(a**3*d*e**3*m**3 + 9*a**3*d*e**3*m**2 + 26*a**3*d*e**3*m + 24*a**3*d*e**3 + a**3*e**4*m**3*x + 9*a**3*e**4*m**2*x + 26*a**3*e**4*m*x + 24*a**3*e**4*x - 3*a**2*b*d**2*e**2*m**2 - 21*a**2*b*d**2*e**2*m - 36*a* *2*b*d**2*e**2 + 3*a**2*b*d*e**3*m**3*x + 21*a**2*b*d*e**3*m**2*x + 36*a** 2*b*d*e**3*m*x + 3*a**2*b*e**4*m**3*x**2 + 24*a**2*b*e**4*m**2*x**2 + 57*a **2*b*e**4*m*x**2 + 36*a**2*b*e**4*x**2 + 6*a*b**2*d**3*e*m + 24*a*b**2*d* *3*e - 6*a*b**2*d**2*e**2*m**2*x - 24*a*b**2*d**2*e**2*m*x + 3*a*b**2*d*e* *3*m**3*x**2 + 15*a*b**2*d*e**3*m**2*x**2 + 12*a*b**2*d*e**3*m*x**2 + 3*a* b**2*e**4*m**3*x**3 + 21*a*b**2*e**4*m**2*x**3 + 42*a*b**2*e**4*m*x**3 + 2 4*a*b**2*e**4*x**3 - 6*b**3*d**4 + 6*b**3*d**3*e*m*x - 3*b**3*d**2*e**2*m* *2*x**2 - 3*b**3*d**2*e**2*m*x**2 + b**3*d*e**3*m**3*x**3 + 3*b**3*d*e**3* m**2*x**3 + 2*b**3*d*e**3*m*x**3 + b**3*e**4*m**3*x**4 + 6*b**3*e**4*m**2* x**4 + 11*b**3*e**4*m*x**4 + 6*b**3*e**4*x**4))/(e**4*(m**4 + 10*m**3 + 35 *m**2 + 50*m + 24))