Integrand size = 29, antiderivative size = 111 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {(b d-a e)^3 (d+e x)^{1+m}}{e^4 (1+m)}+\frac {3 b (b d-a e)^2 (d+e x)^{2+m}}{e^4 (2+m)}-\frac {3 b^2 (b d-a e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {b^3 (d+e x)^{4+m}}{e^4 (4+m)} \] Output:
-(-a*e+b*d)^3*(e*x+d)^(1+m)/e^4/(1+m)+3*b*(-a*e+b*d)^2*(e*x+d)^(2+m)/e^4/( 2+m)-3*b^2*(-a*e+b*d)*(e*x+d)^(3+m)/e^4/(3+m)+b^3*(e*x+d)^(4+m)/e^4/(4+m)
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^3}{1+m}+\frac {3 b (b d-a e)^2 (d+e x)}{2+m}-\frac {3 b^2 (b d-a e) (d+e x)^2}{3+m}+\frac {b^3 (d+e x)^3}{4+m}\right )}{e^4} \] Input:
Integrate[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2),x]
Output:
((d + e*x)^(1 + m)*(-((b*d - a*e)^3/(1 + m)) + (3*b*(b*d - a*e)^2*(d + e*x ))/(2 + m) - (3*b^2*(b*d - a*e)*(d + e*x)^2)/(3 + m) + (b^3*(d + e*x)^3)/( 4 + m)))/e^4
Time = 0.47 (sec) , antiderivative size = 111, 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, 53, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right ) (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^2 (a+b x)^3 (d+e x)^mdx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^3 (d+e x)^mdx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {3 b^2 (b d-a e) (d+e x)^{m+2}}{e^3}+\frac {(a e-b d)^3 (d+e x)^m}{e^3}+\frac {3 b (b d-a e)^2 (d+e x)^{m+1}}{e^3}+\frac {b^3 (d+e x)^{m+3}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^2 (b d-a e) (d+e x)^{m+3}}{e^4 (m+3)}-\frac {(b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1)}+\frac {3 b (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2)}+\frac {b^3 (d+e x)^{m+4}}{e^4 (m+4)}\) |
Input:
Int[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2),x]
Output:
-(((b*d - a*e)^3*(d + e*x)^(1 + m))/(e^4*(1 + m))) + (3*b*(b*d - a*e)^2*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (3*b^2*(b*d - a*e)*(d + e*x)^(3 + m))/(e^ 4*(3 + m)) + (b^3*(d + e*x)^(4 + m))/(e^4*(4 + m))
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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
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. \(385\) vs. \(2(111)=222\).
Time = 0.82 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.48
| method | result | size |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (b^{3} e^{3} m^{3} x^{3}+3 a \,b^{2} e^{3} m^{3} x^{2}+6 b^{3} e^{3} m^{2} x^{3}+3 a^{2} b \,e^{3} m^{3} x +21 a \,b^{2} e^{3} m^{2} x^{2}-3 b^{3} d \,e^{2} m^{2} x^{2}+11 b^{3} e^{3} m \,x^{3}+a^{3} e^{3} m^{3}+24 a^{2} b \,e^{3} m^{2} x -6 a \,b^{2} d \,e^{2} m^{2} x +42 a \,b^{2} e^{3} m \,x^{2}-9 b^{3} d \,e^{2} m \,x^{2}+6 e^{3} x^{3} b^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+57 a^{2} b \,e^{3} m x -30 a \,b^{2} d \,e^{2} m x +24 x^{2} a \,b^{2} e^{3}+6 b^{3} d^{2} e m x -6 x^{2} b^{3} d \,e^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +36 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e m -24 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x +24 e^{3} a^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(386\) |
| orering | \(\frac {\left (b^{3} e^{3} m^{3} x^{3}+3 a \,b^{2} e^{3} m^{3} x^{2}+6 b^{3} e^{3} m^{2} x^{3}+3 a^{2} b \,e^{3} m^{3} x +21 a \,b^{2} e^{3} m^{2} x^{2}-3 b^{3} d \,e^{2} m^{2} x^{2}+11 b^{3} e^{3} m \,x^{3}+a^{3} e^{3} m^{3}+24 a^{2} b \,e^{3} m^{2} x -6 a \,b^{2} d \,e^{2} m^{2} x +42 a \,b^{2} e^{3} m \,x^{2}-9 b^{3} d \,e^{2} m \,x^{2}+6 e^{3} x^{3} b^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+57 a^{2} b \,e^{3} m x -30 a \,b^{2} d \,e^{2} m x +24 x^{2} a \,b^{2} e^{3}+6 b^{3} d^{2} e m x -6 x^{2} b^{3} d \,e^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +36 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e m -24 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x +24 e^{3} a^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} d^{3}\right ) \left (e x +d \right ) \left (e x +d \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) \left (b x +a \right )^{2}}\) | \(412\) |
| norman | \(\frac {b^{3} x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {d \left (a^{3} e^{3} m^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +6 a \,b^{2} d^{2} e m +24 e^{3} a^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} d^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (a^{3} e^{3} m^{3}+3 a^{2} b d \,e^{2} m^{3}+9 a^{3} e^{3} m^{2}+21 a^{2} b d \,e^{2} m^{2}-6 a \,b^{2} d^{2} e \,m^{2}+26 a^{3} e^{3} m +36 a^{2} b d \,e^{2} m -24 a \,b^{2} d^{2} e m +6 b^{3} d^{3} m +24 e^{3} a^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (3 a e m +b d m +12 a e \right ) b^{2} x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {3 \left (a^{2} e^{2} m^{2}+a b d e \,m^{2}+7 a^{2} e^{2} m +4 a b d e m -b^{2} d^{2} m +12 e^{2} a^{2}\right ) b \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}\) | \(430\) |
| risch | \(\frac {\left (b^{3} e^{4} m^{3} x^{4}+3 a \,b^{2} e^{4} m^{3} x^{3}+b^{3} d \,e^{3} m^{3} x^{3}+6 b^{3} e^{4} m^{2} x^{4}+3 a^{2} b \,e^{4} m^{3} x^{2}+3 a \,b^{2} d \,e^{3} m^{3} x^{2}+21 a \,b^{2} e^{4} m^{2} x^{3}+3 b^{3} d \,e^{3} m^{2} x^{3}+11 b^{3} e^{4} m \,x^{4}+a^{3} e^{4} m^{3} x +3 a^{2} b d \,e^{3} m^{3} x +24 a^{2} b \,e^{4} m^{2} x^{2}+15 a \,b^{2} d \,e^{3} m^{2} x^{2}+42 a \,b^{2} e^{4} m \,x^{3}-3 b^{3} d^{2} e^{2} m^{2} x^{2}+2 b^{3} d \,e^{3} m \,x^{3}+6 b^{3} x^{4} e^{4}+a^{3} d \,e^{3} m^{3}+9 a^{3} e^{4} m^{2} x +21 a^{2} b d \,e^{3} m^{2} x +57 a^{2} b \,e^{4} m \,x^{2}-6 a \,b^{2} d^{2} e^{2} m^{2} x +12 a \,b^{2} d \,e^{3} m \,x^{2}+24 a \,b^{2} e^{4} x^{3}-3 b^{3} d^{2} e^{2} m \,x^{2}+9 a^{3} d \,e^{3} m^{2}+26 a^{3} e^{4} m x -3 a^{2} b \,d^{2} e^{2} m^{2}+36 a^{2} b d \,e^{3} m x +36 a^{2} b \,e^{4} x^{2}-24 a \,b^{2} d^{2} e^{2} m x +6 b^{3} d^{3} e m x +26 a^{3} d \,e^{3} m +24 e^{4} a^{3} x -21 a^{2} b \,d^{2} e^{2} m +6 a \,b^{2} d^{3} e m +24 a^{3} d \,e^{3}-36 d^{2} e^{2} a^{2} b +24 a \,b^{2} d^{3} e -6 b^{3} d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(547\) |
| parallelrisch | \(\frac {26 x \left (e x +d \right )^{m} a^{3} d \,e^{4} m +6 x \left (e x +d \right )^{m} b^{3} d^{4} e m -3 \left (e x +d \right )^{m} a^{2} b \,d^{3} e^{2} m^{2}-21 \left (e x +d \right )^{m} a^{2} b \,d^{3} e^{2} m +6 \left (e x +d \right )^{m} a \,b^{2} d^{4} e m +2 x^{3} \left (e x +d \right )^{m} b^{3} d^{2} e^{3} m -3 x^{2} \left (e x +d \right )^{m} b^{3} d^{3} e^{2} m^{2}+x \left (e x +d \right )^{m} a^{3} d \,e^{4} m^{3}+24 x^{3} \left (e x +d \right )^{m} a \,b^{2} d \,e^{4}+21 x \left (e x +d \right )^{m} a^{2} b \,d^{2} e^{3} m^{2}-6 x \left (e x +d \right )^{m} a \,b^{2} d^{3} e^{2} m^{2}+21 x^{3} \left (e x +d \right )^{m} a \,b^{2} d \,e^{4} m^{2}+3 x^{2} \left (e x +d \right )^{m} a^{2} b d \,e^{4} m^{3}+3 x^{2} \left (e x +d \right )^{m} a \,b^{2} d^{2} e^{3} m^{3}+42 x^{3} \left (e x +d \right )^{m} a \,b^{2} d \,e^{4} m +24 x^{2} \left (e x +d \right )^{m} a^{2} b d \,e^{4} m^{2}+15 x^{2} \left (e x +d \right )^{m} a \,b^{2} d^{2} e^{3} m^{2}+3 x \left (e x +d \right )^{m} a^{2} b \,d^{2} e^{3} m^{3}+57 x^{2} \left (e x +d \right )^{m} a^{2} b d \,e^{4} m +12 x^{2} \left (e x +d \right )^{m} a \,b^{2} d^{2} e^{3} m +3 x^{3} \left (e x +d \right )^{m} a \,b^{2} d \,e^{4} m^{3}+36 x \left (e x +d \right )^{m} a^{2} b \,d^{2} e^{3} m -24 x \left (e x +d \right )^{m} a \,b^{2} d^{3} e^{2} m +24 \left (e x +d \right )^{m} a^{3} d^{2} e^{3}+6 x^{4} \left (e x +d \right )^{m} b^{3} d \,e^{4}+\left (e x +d \right )^{m} a^{3} d^{2} e^{3} m^{3}+9 \left (e x +d \right )^{m} a^{3} d^{2} e^{3} m^{2}+24 x \left (e x +d \right )^{m} a^{3} d \,e^{4}+26 \left (e x +d \right )^{m} a^{3} d^{2} e^{3} m -36 \left (e x +d \right )^{m} a^{2} b \,d^{3} e^{2}+24 \left (e x +d \right )^{m} a \,b^{2} d^{4} e +x^{4} \left (e x +d \right )^{m} b^{3} d \,e^{4} m^{3}+6 x^{4} \left (e x +d \right )^{m} b^{3} d \,e^{4} m^{2}+x^{3} \left (e x +d \right )^{m} b^{3} d^{2} e^{3} m^{3}+11 x^{4} \left (e x +d \right )^{m} b^{3} d \,e^{4} m +3 x^{3} \left (e x +d \right )^{m} b^{3} d^{2} e^{3} m^{2}-6 \left (e x +d \right )^{m} b^{3} d^{5}-3 x^{2} \left (e x +d \right )^{m} b^{3} d^{3} e^{2} m +9 x \left (e x +d \right )^{m} a^{3} d \,e^{4} m^{2}+36 x^{2} \left (e x +d \right )^{m} a^{2} b d \,e^{4}}{d \left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(865\) |
Input:
int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)
Output:
1/e^4*(e*x+d)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(b^3*e^3*m^3*x^3+3*a*b^2*e ^3*m^3*x^2+6*b^3*e^3*m^2*x^3+3*a^2*b*e^3*m^3*x+21*a*b^2*e^3*m^2*x^2-3*b^3* d*e^2*m^2*x^2+11*b^3*e^3*m*x^3+a^3*e^3*m^3+24*a^2*b*e^3*m^2*x-6*a*b^2*d*e^ 2*m^2*x+42*a*b^2*e^3*m*x^2-9*b^3*d*e^2*m*x^2+6*b^3*e^3*x^3+9*a^3*e^3*m^2-3 *a^2*b*d*e^2*m^2+57*a^2*b*e^3*m*x-30*a*b^2*d*e^2*m*x+24*a*b^2*e^3*x^2+6*b^ 3*d^2*e*m*x-6*b^3*d*e^2*x^2+26*a^3*e^3*m-21*a^2*b*d*e^2*m+36*a^2*b*e^3*x+6 *a*b^2*d^2*e*m-24*a*b^2*d*e^2*x+6*b^3*d^2*e*x+24*a^3*e^3-36*a^2*b*d*e^2+24 *a*b^2*d^2*e-6*b^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (111) = 222\).
Time = 0.09 (sec) , antiderivative size = 496, normalized size of antiderivative = 4.47 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {{\left (a^{3} d e^{3} m^{3} - 6 \, b^{3} d^{4} + 24 \, a b^{2} d^{3} e - 36 \, a^{2} b d^{2} e^{2} + 24 \, a^{3} d e^{3} + {\left (b^{3} e^{4} m^{3} + 6 \, b^{3} e^{4} m^{2} + 11 \, b^{3} e^{4} m + 6 \, b^{3} e^{4}\right )} x^{4} + {\left (24 \, a b^{2} e^{4} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} m^{3} + 3 \, {\left (b^{3} d e^{3} + 7 \, a b^{2} e^{4}\right )} m^{2} + 2 \, {\left (b^{3} d e^{3} + 21 \, a b^{2} e^{4}\right )} m\right )} x^{3} - 3 \, {\left (a^{2} b d^{2} e^{2} - 3 \, a^{3} d e^{3}\right )} m^{2} + 3 \, {\left (12 \, a^{2} b e^{4} + {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} m^{3} - {\left (b^{3} d^{2} e^{2} - 5 \, a b^{2} d e^{3} - 8 \, a^{2} b e^{4}\right )} m^{2} - {\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} - 19 \, a^{2} b e^{4}\right )} m\right )} x^{2} + {\left (6 \, a b^{2} d^{3} e - 21 \, a^{2} b d^{2} e^{2} + 26 \, a^{3} d e^{3}\right )} m + {\left (24 \, a^{3} e^{4} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} m^{3} - 3 \, {\left (2 \, a b^{2} d^{2} e^{2} - 7 \, a^{2} b d e^{3} - 3 \, a^{3} e^{4}\right )} m^{2} + 2 \, {\left (3 \, b^{3} d^{3} e - 12 \, a b^{2} d^{2} e^{2} + 18 \, a^{2} b d e^{3} + 13 \, a^{3} e^{4}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \] Input:
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")
Output:
(a^3*d*e^3*m^3 - 6*b^3*d^4 + 24*a*b^2*d^3*e - 36*a^2*b*d^2*e^2 + 24*a^3*d* e^3 + (b^3*e^4*m^3 + 6*b^3*e^4*m^2 + 11*b^3*e^4*m + 6*b^3*e^4)*x^4 + (24*a *b^2*e^4 + (b^3*d*e^3 + 3*a*b^2*e^4)*m^3 + 3*(b^3*d*e^3 + 7*a*b^2*e^4)*m^2 + 2*(b^3*d*e^3 + 21*a*b^2*e^4)*m)*x^3 - 3*(a^2*b*d^2*e^2 - 3*a^3*d*e^3)*m ^2 + 3*(12*a^2*b*e^4 + (a*b^2*d*e^3 + a^2*b*e^4)*m^3 - (b^3*d^2*e^2 - 5*a* b^2*d*e^3 - 8*a^2*b*e^4)*m^2 - (b^3*d^2*e^2 - 4*a*b^2*d*e^3 - 19*a^2*b*e^4 )*m)*x^2 + (6*a*b^2*d^3*e - 21*a^2*b*d^2*e^2 + 26*a^3*d*e^3)*m + (24*a^3*e ^4 + (3*a^2*b*d*e^3 + a^3*e^4)*m^3 - 3*(2*a*b^2*d^2*e^2 - 7*a^2*b*d*e^3 - 3*a^3*e^4)*m^2 + 2*(3*b^3*d^3*e - 12*a*b^2*d^2*e^2 + 18*a^2*b*d*e^3 + 13*a ^3*e^4)*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. 4058 vs. \(2 (95) = 190\).
Time = 1.44 (sec) , antiderivative size = 4058, normalized size of antiderivative = 36.56 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2),x)
Output:
Piecewise((d**m*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq (e, 0)), (-2*a**3*e**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6* e**7*x**3) - 3*a**2*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) - 9*a**2*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) - 6*a*b**2*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 1 8*d*e**6*x**2 + 6*e**7*x**3) - 18*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) - 18*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) + 6*b**3*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**3*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** 3*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**3*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**3*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**3*d*e**2*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**3*e**3*x**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), E q(m, -4)), (-a**3*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 3*a**2*b *d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*a**2*b*e**3*x/(2*d**2 *e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*a*b**2*d**2*e*log(d/e + x)/(2*d**2*e **4 + 4*d*e**5*x + 2*e**6*x**2) + 9*a*b**2*d**2*e/(2*d**2*e**4 + 4*d*e*...
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (111) = 222\).
Time = 0.05 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.22 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {3 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{2} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{3}}{e {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \] Input:
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")
Output:
3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^2*b/((m^2 + 3*m + 2)*e^2 ) + (e*x + d)^(m + 1)*a^3/(e*(m + 1)) + 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^3/(( 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. 833 vs. \(2 (111) = 222\).
Time = 0.16 (sec) , antiderivative size = 833, normalized size of antiderivative = 7.50 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")
Output:
((e*x + d)^m*b^3*e^4*m^3*x^4 + (e*x + d)^m*b^3*d*e^3*m^3*x^3 + 3*(e*x + d) ^m*a*b^2*e^4*m^3*x^3 + 6*(e*x + d)^m*b^3*e^4*m^2*x^4 + 3*(e*x + d)^m*a*b^2 *d*e^3*m^3*x^2 + 3*(e*x + d)^m*a^2*b*e^4*m^3*x^2 + 3*(e*x + d)^m*b^3*d*e^3 *m^2*x^3 + 21*(e*x + d)^m*a*b^2*e^4*m^2*x^3 + 11*(e*x + d)^m*b^3*e^4*m*x^4 + 3*(e*x + d)^m*a^2*b*d*e^3*m^3*x + (e*x + d)^m*a^3*e^4*m^3*x - 3*(e*x + d)^m*b^3*d^2*e^2*m^2*x^2 + 15*(e*x + d)^m*a*b^2*d*e^3*m^2*x^2 + 24*(e*x + d)^m*a^2*b*e^4*m^2*x^2 + 2*(e*x + d)^m*b^3*d*e^3*m*x^3 + 42*(e*x + d)^m*a* b^2*e^4*m*x^3 + 6*(e*x + d)^m*b^3*e^4*x^4 + (e*x + d)^m*a^3*d*e^3*m^3 - 6* (e*x + d)^m*a*b^2*d^2*e^2*m^2*x + 21*(e*x + d)^m*a^2*b*d*e^3*m^2*x + 9*(e* x + d)^m*a^3*e^4*m^2*x - 3*(e*x + d)^m*b^3*d^2*e^2*m*x^2 + 12*(e*x + d)^m* a*b^2*d*e^3*m*x^2 + 57*(e*x + d)^m*a^2*b*e^4*m*x^2 + 24*(e*x + d)^m*a*b^2* e^4*x^3 - 3*(e*x + d)^m*a^2*b*d^2*e^2*m^2 + 9*(e*x + d)^m*a^3*d*e^3*m^2 + 6*(e*x + d)^m*b^3*d^3*e*m*x - 24*(e*x + d)^m*a*b^2*d^2*e^2*m*x + 36*(e*x + d)^m*a^2*b*d*e^3*m*x + 26*(e*x + d)^m*a^3*e^4*m*x + 36*(e*x + d)^m*a^2*b* e^4*x^2 + 6*(e*x + d)^m*a*b^2*d^3*e*m - 21*(e*x + d)^m*a^2*b*d^2*e^2*m + 2 6*(e*x + d)^m*a^3*d*e^3*m + 24*(e*x + d)^m*a^3*e^4*x - 6*(e*x + d)^m*b^3*d ^4 + 24*(e*x + d)^m*a*b^2*d^3*e - 36*(e*x + d)^m*a^2*b*d^2*e^2 + 24*(e*x + d)^m*a^3*d*e^3)/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50*e^4*m + 24*e^4)
Time = 11.44 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.31 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^3\,e^4\,m^3+9\,a^3\,e^4\,m^2+26\,a^3\,e^4\,m+24\,a^3\,e^4+3\,a^2\,b\,d\,e^3\,m^3+21\,a^2\,b\,d\,e^3\,m^2+36\,a^2\,b\,d\,e^3\,m-6\,a\,b^2\,d^2\,e^2\,m^2-24\,a\,b^2\,d^2\,e^2\,m+6\,b^3\,d^3\,e\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {d\,{\left (d+e\,x\right )}^m\,\left (a^3\,e^3\,m^3+9\,a^3\,e^3\,m^2+26\,a^3\,e^3\,m+24\,a^3\,e^3-3\,a^2\,b\,d\,e^2\,m^2-21\,a^2\,b\,d\,e^2\,m-36\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e\,m+24\,a\,b^2\,d^2\,e-6\,b^3\,d^3\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {3\,b\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^2\,m^2+7\,a^2\,e^2\,m+12\,a^2\,e^2+a\,b\,d\,e\,m^2+4\,a\,b\,d\,e\,m-b^2\,d^2\,m\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {b^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (12\,a\,e+3\,a\,e\,m+b\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \] Input:
int((a + b*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x),x)
Output:
(x*(d + e*x)^m*(24*a^3*e^4 + 26*a^3*e^4*m + 9*a^3*e^4*m^2 + a^3*e^4*m^3 + 6*b^3*d^3*e*m + 36*a^2*b*d*e^3*m - 24*a*b^2*d^2*e^2*m + 21*a^2*b*d*e^3*m^2 + 3*a^2*b*d*e^3*m^3 - 6*a*b^2*d^2*e^2*m^2))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b^3*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) + (d*(d + e*x)^m*(24*a^3*e^3 - 6*b^3*d^3 + 26*a^3* e^3*m + 9*a^3*e^3*m^2 + a^3*e^3*m^3 + 24*a*b^2*d^2*e - 36*a^2*b*d*e^2 + 6* a*b^2*d^2*e*m - 21*a^2*b*d*e^2*m - 3*a^2*b*d*e^2*m^2))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (3*b*x^2*(m + 1)*(d + e*x)^m*(12*a^2*e^2 + 7*a^2* e^2*m - b^2*d^2*m + a^2*e^2*m^2 + 4*a*b*d*e*m + a*b*d*e*m^2))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b^2*x^3*(d + e*x)^m*(12*a*e + 3*a*e*m + b *d*m)*(3*m + m^2 + 2))/(e*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))
Time = 0.28 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.92 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {\left (e x +d \right )^{m} \left (b^{3} e^{4} m^{3} x^{4}+3 a \,b^{2} e^{4} m^{3} x^{3}+b^{3} d \,e^{3} m^{3} x^{3}+6 b^{3} e^{4} m^{2} x^{4}+3 a^{2} b \,e^{4} m^{3} x^{2}+3 a \,b^{2} d \,e^{3} m^{3} x^{2}+21 a \,b^{2} e^{4} m^{2} x^{3}+3 b^{3} d \,e^{3} m^{2} x^{3}+11 b^{3} e^{4} m \,x^{4}+a^{3} e^{4} m^{3} x +3 a^{2} b d \,e^{3} m^{3} x +24 a^{2} b \,e^{4} m^{2} x^{2}+15 a \,b^{2} d \,e^{3} m^{2} x^{2}+42 a \,b^{2} e^{4} m \,x^{3}-3 b^{3} d^{2} e^{2} m^{2} x^{2}+2 b^{3} d \,e^{3} m \,x^{3}+6 b^{3} e^{4} x^{4}+a^{3} d \,e^{3} m^{3}+9 a^{3} e^{4} m^{2} x +21 a^{2} b d \,e^{3} m^{2} x +57 a^{2} b \,e^{4} m \,x^{2}-6 a \,b^{2} d^{2} e^{2} m^{2} x +12 a \,b^{2} d \,e^{3} m \,x^{2}+24 a \,b^{2} e^{4} x^{3}-3 b^{3} d^{2} e^{2} m \,x^{2}+9 a^{3} d \,e^{3} m^{2}+26 a^{3} e^{4} m x -3 a^{2} b \,d^{2} e^{2} m^{2}+36 a^{2} b d \,e^{3} m x +36 a^{2} b \,e^{4} x^{2}-24 a \,b^{2} d^{2} e^{2} m x +6 b^{3} d^{3} e m x +26 a^{3} d \,e^{3} m +24 a^{3} e^{4} x -21 a^{2} b \,d^{2} e^{2} m +6 a \,b^{2} d^{3} e m +24 a^{3} d \,e^{3}-36 a^{2} b \,d^{2} e^{2}+24 a \,b^{2} d^{3} e -6 b^{3} d^{4}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \] Input:
int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2),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))