Integrand size = 19, antiderivative size = 121 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {(3 c d-b e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c (d+e x)^{4+m}}{e^4 (4+m)} \] Output:
-d*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(1+m)/e^4/(1+m)+(3*c*d^2-e*(-a*e+2*b*d))*(e *x+d)^(2+m)/e^4/(2+m)-(-b*e+3*c*d)*(e*x+d)^(3+m)/e^4/(3+m)+c*(e*x+d)^(4+m) /e^4/(4+m)
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.17 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (c \left (-6 d^3+6 d^2 e (1+m) x-3 d e^2 \left (2+3 m+m^2\right ) x^2+e^3 \left (6+11 m+6 m^2+m^3\right ) x^3\right )+e (4+m) \left (a e (3+m) (-d+e (1+m) x)+b \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )\right )}{e^4 (1+m) (2+m) (3+m) (4+m)} \] Input:
Integrate[x*(d + e*x)^m*(a + b*x + c*x^2),x]
Output:
((d + e*x)^(1 + m)*(c*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2 )*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)*x^3) + e*(4 + m)*(a*e*(3 + m)*(-d + e *(1 + m)*x) + b*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2))))/(e^ 4*(1 + m)*(2 + m)*(3 + m)*(4 + m))
Time = 0.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1195, 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.
| \(\displaystyle \int x \left (a+b x+c x^2\right ) (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{m+1} \left (3 c d^2-e (2 b d-a e)\right )}{e^3}-\frac {d (d+e x)^m \left (a e^2-b d e+c d^2\right )}{e^3}+\frac {(b e-3 c d) (d+e x)^{m+2}}{e^3}+\frac {c (d+e x)^{m+3}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^{m+2} \left (3 c d^2-e (2 b d-a e)\right )}{e^4 (m+2)}-\frac {d (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(3 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {c (d+e x)^{m+4}}{e^4 (m+4)}\) |
Input:
Int[x*(d + e*x)^m*(a + b*x + c*x^2),x]
Output:
-((d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*c*d^2 - e*(2*b*d - a*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - ((3*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (c*(d + e*x)^(4 + m))/(e^4*(4 + m))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(121)=242\).
Time = 0.74 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.32
| method | result | size |
| gosper | \(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{3} m^{3} x^{3}-b \,e^{3} m^{3} x^{2}-6 c \,e^{3} m^{2} x^{3}-a \,e^{3} m^{3} x -7 b \,e^{3} m^{2} x^{2}+3 c d \,e^{2} m^{2} x^{2}-11 c \,e^{3} m \,x^{3}-8 a \,e^{3} m^{2} x +2 b d \,e^{2} m^{2} x -14 b \,e^{3} m \,x^{2}+9 c d \,e^{2} m \,x^{2}-6 c \,e^{3} x^{3}+a d \,e^{2} m^{2}-19 a \,e^{3} m x +10 b d \,e^{2} m x -8 x^{2} b \,e^{3}-6 c \,d^{2} e m x +6 c d \,e^{2} x^{2}+7 a d \,e^{2} m -12 a \,e^{3} x -2 b \,d^{2} e m +8 d x b \,e^{2}-6 c \,d^{2} e x +12 a d \,e^{2}-8 b \,d^{2} e +6 c \,d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(281\) |
| norman | \(\frac {c \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {\left (b e m +c d m +4 b e \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {\left (a \,e^{2} m^{2}+b d e \,m^{2}+7 a \,e^{2} m +4 b d e m -3 c \,d^{2} m +12 a \,e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {m d \left (a \,e^{2} m^{2}+7 a \,e^{2} m -2 b d e m +12 a \,e^{2}-8 b d e +6 c \,d^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {d^{2} \left (a \,e^{2} m^{2}+7 a \,e^{2} m -2 b d e m +12 a \,e^{2}-8 b d e +6 c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(284\) |
| orering | \(-\frac {\left (e x +d \right )^{m} \left (-c \,e^{3} m^{3} x^{3}-b \,e^{3} m^{3} x^{2}-6 c \,e^{3} m^{2} x^{3}-a \,e^{3} m^{3} x -7 b \,e^{3} m^{2} x^{2}+3 c d \,e^{2} m^{2} x^{2}-11 c \,e^{3} m \,x^{3}-8 a \,e^{3} m^{2} x +2 b d \,e^{2} m^{2} x -14 b \,e^{3} m \,x^{2}+9 c d \,e^{2} m \,x^{2}-6 c \,e^{3} x^{3}+a d \,e^{2} m^{2}-19 a \,e^{3} m x +10 b d \,e^{2} m x -8 x^{2} b \,e^{3}-6 c \,d^{2} e m x +6 c d \,e^{2} x^{2}+7 a d \,e^{2} m -12 a \,e^{3} x -2 b \,d^{2} e m +8 d x b \,e^{2}-6 c \,d^{2} e x +12 a d \,e^{2}-8 b \,d^{2} e +6 c \,d^{3}\right ) \left (e x +d \right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(284\) |
| risch | \(-\frac {\left (-c \,e^{4} m^{3} x^{4}-b \,e^{4} m^{3} x^{3}-c d \,e^{3} m^{3} x^{3}-6 c \,e^{4} m^{2} x^{4}-a \,e^{4} m^{3} x^{2}-b d \,e^{3} m^{3} x^{2}-7 b \,e^{4} m^{2} x^{3}-3 c d \,e^{3} m^{2} x^{3}-11 c \,e^{4} m \,x^{4}-a d \,e^{3} m^{3} x -8 a \,e^{4} m^{2} x^{2}-5 b d \,e^{3} m^{2} x^{2}-14 b \,e^{4} m \,x^{3}+3 c \,d^{2} e^{2} m^{2} x^{2}-2 c d \,e^{3} m \,x^{3}-6 c \,x^{4} e^{4}-7 a d \,e^{3} m^{2} x -19 a \,e^{4} m \,x^{2}+2 b \,d^{2} e^{2} m^{2} x -4 b d \,e^{3} m \,x^{2}-8 b \,e^{4} x^{3}+3 c \,d^{2} e^{2} m \,x^{2}+a \,d^{2} e^{2} m^{2}-12 a d \,e^{3} m x -12 a \,e^{4} x^{2}+8 b \,d^{2} e^{2} m x -6 c \,d^{3} e m x +7 a \,d^{2} e^{2} m -2 b \,d^{3} e m +12 a \,d^{2} e^{2}-8 b \,d^{3} e +6 c \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(380\) |
| parallelrisch | \(\frac {8 x^{2} \left (e x +d \right )^{m} a d \,e^{4} m^{2}+5 x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{3} m^{2}-3 x^{2} \left (e x +d \right )^{m} c \,d^{3} e^{2} m^{2}+x \left (e x +d \right )^{m} a \,d^{2} e^{3} m^{3}+19 x^{2} \left (e x +d \right )^{m} a d \,e^{4} m +4 x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{3} m -3 x^{2} \left (e x +d \right )^{m} c \,d^{3} e^{2} m +7 x \left (e x +d \right )^{m} a \,d^{2} e^{3} m^{2}-2 x \left (e x +d \right )^{m} b \,d^{3} e^{2} m^{2}+12 x \left (e x +d \right )^{m} a \,d^{2} e^{3} m +3 x^{3} \left (e x +d \right )^{m} c \,d^{2} e^{3} m^{2}+x^{2} \left (e x +d \right )^{m} a d \,e^{4} m^{3}+x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{3} m^{3}+6 x^{4} \left (e x +d \right )^{m} c d \,e^{4}+8 x^{3} \left (e x +d \right )^{m} b d \,e^{4}+12 x^{2} \left (e x +d \right )^{m} a d \,e^{4}-8 x \left (e x +d \right )^{m} b \,d^{3} e^{2} m +6 x \left (e x +d \right )^{m} c \,d^{4} e m +14 x^{3} \left (e x +d \right )^{m} b d \,e^{4} m -\left (e x +d \right )^{m} a \,d^{3} e^{2} m^{2}-7 \left (e x +d \right )^{m} a \,d^{3} e^{2} m +2 \left (e x +d \right )^{m} b \,d^{4} e m -12 \left (e x +d \right )^{m} a \,d^{3} e^{2}+8 \left (e x +d \right )^{m} b \,d^{4} e +x^{3} \left (e x +d \right )^{m} c \,d^{2} e^{3} m^{3}+11 x^{4} \left (e x +d \right )^{m} c d \,e^{4} m +7 x^{3} \left (e x +d \right )^{m} b d \,e^{4} m^{2}+x^{4} \left (e x +d \right )^{m} c d \,e^{4} m^{3}+6 x^{4} \left (e x +d \right )^{m} c d \,e^{4} m^{2}+x^{3} \left (e x +d \right )^{m} b d \,e^{4} m^{3}+2 x^{3} \left (e x +d \right )^{m} c \,d^{2} e^{3} m -6 \left (e x +d \right )^{m} c \,d^{5}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) d \,e^{4}}\) | \(624\) |
Input:
int(x*(e*x+d)^m*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/e^4*(e*x+d)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(-c*e^3*m^3*x^3-b*e^3*m^3 *x^2-6*c*e^3*m^2*x^3-a*e^3*m^3*x-7*b*e^3*m^2*x^2+3*c*d*e^2*m^2*x^2-11*c*e^ 3*m*x^3-8*a*e^3*m^2*x+2*b*d*e^2*m^2*x-14*b*e^3*m*x^2+9*c*d*e^2*m*x^2-6*c*e ^3*x^3+a*d*e^2*m^2-19*a*e^3*m*x+10*b*d*e^2*m*x-8*b*e^3*x^2-6*c*d^2*e*m*x+6 *c*d*e^2*x^2+7*a*d*e^2*m-12*a*e^3*x-2*b*d^2*e*m+8*b*d*e^2*x-6*c*d^2*e*x+12 *a*d*e^2-8*b*d^2*e+6*c*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (121) = 242\).
Time = 0.08 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.83 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=-\frac {{\left (a d^{2} e^{2} m^{2} + 6 \, c d^{4} - 8 \, b d^{3} e + 12 \, a d^{2} e^{2} - {\left (c e^{4} m^{3} + 6 \, c e^{4} m^{2} + 11 \, c e^{4} m + 6 \, c e^{4}\right )} x^{4} - {\left (8 \, b e^{4} + {\left (c d e^{3} + b e^{4}\right )} m^{3} + {\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} m^{2} + 2 \, {\left (c d e^{3} + 7 \, b e^{4}\right )} m\right )} x^{3} - {\left (12 \, a e^{4} + {\left (b d e^{3} + a e^{4}\right )} m^{3} - {\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} m^{2} - {\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} m\right )} x^{2} - {\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} m - {\left (a d e^{3} m^{3} - {\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} m^{2} + 2 \, {\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \] Input:
integrate(x*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")
Output:
-(a*d^2*e^2*m^2 + 6*c*d^4 - 8*b*d^3*e + 12*a*d^2*e^2 - (c*e^4*m^3 + 6*c*e^ 4*m^2 + 11*c*e^4*m + 6*c*e^4)*x^4 - (8*b*e^4 + (c*d*e^3 + b*e^4)*m^3 + (3* c*d*e^3 + 7*b*e^4)*m^2 + 2*(c*d*e^3 + 7*b*e^4)*m)*x^3 - (12*a*e^4 + (b*d*e ^3 + a*e^4)*m^3 - (3*c*d^2*e^2 - 5*b*d*e^3 - 8*a*e^4)*m^2 - (3*c*d^2*e^2 - 4*b*d*e^3 - 19*a*e^4)*m)*x^2 - (2*b*d^3*e - 7*a*d^2*e^2)*m - (a*d*e^3*m^3 - (2*b*d^2*e^2 - 7*a*d*e^3)*m^2 + 2*(3*c*d^3*e - 4*b*d^2*e^2 + 6*a*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. 3267 vs. \(2 (104) = 208\).
Time = 0.98 (sec) , antiderivative size = 3267, normalized size of antiderivative = 27.00 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(x*(e*x+d)**m*(c*x**2+b*x+a),x)
Output:
Piecewise((d**m*(a*x**2/2 + b*x**3/3 + c*x**4/4), Eq(e, 0)), (-a*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*a*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*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) - 6*b*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*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*c *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*c*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*c*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*c*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*c*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*c*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*c*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*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*a*e**3* x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*d**2*e*log(d/e + x)/(2*d* *2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*b*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2* e**6*x**2) + 4*b*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*e **3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*d*...
Time = 0.05 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.78 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \] Input:
integrate(x*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="maxima")
Output:
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a/((m^2 + 3*m + 2)*e^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/((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*c/((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. 604 vs. \(2 (121) = 242\).
Time = 0.23 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.99 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c e^{4} m^{3} x^{4} + {\left (e x + d\right )}^{m} c d e^{3} m^{3} x^{3} + {\left (e x + d\right )}^{m} b e^{4} m^{3} x^{3} + 6 \, {\left (e x + d\right )}^{m} c e^{4} m^{2} x^{4} + {\left (e x + d\right )}^{m} b d e^{3} m^{3} x^{2} + {\left (e x + d\right )}^{m} a e^{4} m^{3} x^{2} + 3 \, {\left (e x + d\right )}^{m} c d e^{3} m^{2} x^{3} + 7 \, {\left (e x + d\right )}^{m} b e^{4} m^{2} x^{3} + 11 \, {\left (e x + d\right )}^{m} c e^{4} m x^{4} + {\left (e x + d\right )}^{m} a d e^{3} m^{3} x - 3 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} m^{2} x^{2} + 5 \, {\left (e x + d\right )}^{m} b d e^{3} m^{2} x^{2} + 8 \, {\left (e x + d\right )}^{m} a e^{4} m^{2} x^{2} + 2 \, {\left (e x + d\right )}^{m} c d e^{3} m x^{3} + 14 \, {\left (e x + d\right )}^{m} b e^{4} m x^{3} + 6 \, {\left (e x + d\right )}^{m} c e^{4} x^{4} - 2 \, {\left (e x + d\right )}^{m} b d^{2} e^{2} m^{2} x + 7 \, {\left (e x + d\right )}^{m} a d e^{3} m^{2} x - 3 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} m x^{2} + 4 \, {\left (e x + d\right )}^{m} b d e^{3} m x^{2} + 19 \, {\left (e x + d\right )}^{m} a e^{4} m x^{2} + 8 \, {\left (e x + d\right )}^{m} b e^{4} x^{3} - {\left (e x + d\right )}^{m} a d^{2} e^{2} m^{2} + 6 \, {\left (e x + d\right )}^{m} c d^{3} e m x - 8 \, {\left (e x + d\right )}^{m} b d^{2} e^{2} m x + 12 \, {\left (e x + d\right )}^{m} a d e^{3} m x + 12 \, {\left (e x + d\right )}^{m} a e^{4} x^{2} + 2 \, {\left (e x + d\right )}^{m} b d^{3} e m - 7 \, {\left (e x + d\right )}^{m} a d^{2} e^{2} m - 6 \, {\left (e x + d\right )}^{m} c d^{4} + 8 \, {\left (e x + d\right )}^{m} b d^{3} e - 12 \, {\left (e x + d\right )}^{m} a d^{2} e^{2}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \] Input:
integrate(x*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="giac")
Output:
((e*x + d)^m*c*e^4*m^3*x^4 + (e*x + d)^m*c*d*e^3*m^3*x^3 + (e*x + d)^m*b*e ^4*m^3*x^3 + 6*(e*x + d)^m*c*e^4*m^2*x^4 + (e*x + d)^m*b*d*e^3*m^3*x^2 + ( e*x + d)^m*a*e^4*m^3*x^2 + 3*(e*x + d)^m*c*d*e^3*m^2*x^3 + 7*(e*x + d)^m*b *e^4*m^2*x^3 + 11*(e*x + d)^m*c*e^4*m*x^4 + (e*x + d)^m*a*d*e^3*m^3*x - 3* (e*x + d)^m*c*d^2*e^2*m^2*x^2 + 5*(e*x + d)^m*b*d*e^3*m^2*x^2 + 8*(e*x + d )^m*a*e^4*m^2*x^2 + 2*(e*x + d)^m*c*d*e^3*m*x^3 + 14*(e*x + d)^m*b*e^4*m*x ^3 + 6*(e*x + d)^m*c*e^4*x^4 - 2*(e*x + d)^m*b*d^2*e^2*m^2*x + 7*(e*x + d) ^m*a*d*e^3*m^2*x - 3*(e*x + d)^m*c*d^2*e^2*m*x^2 + 4*(e*x + d)^m*b*d*e^3*m *x^2 + 19*(e*x + d)^m*a*e^4*m*x^2 + 8*(e*x + d)^m*b*e^4*x^3 - (e*x + d)^m* a*d^2*e^2*m^2 + 6*(e*x + d)^m*c*d^3*e*m*x - 8*(e*x + d)^m*b*d^2*e^2*m*x + 12*(e*x + d)^m*a*d*e^3*m*x + 12*(e*x + d)^m*a*e^4*x^2 + 2*(e*x + d)^m*b*d^ 3*e*m - 7*(e*x + d)^m*a*d^2*e^2*m - 6*(e*x + d)^m*c*d^4 + 8*(e*x + d)^m*b* d^3*e - 12*(e*x + d)^m*a*d^2*e^2)/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50* e^4*m + 24*e^4)
Time = 10.76 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.48 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {d^2\,\left (6\,c\,d^2-2\,b\,d\,e\,m-8\,b\,d\,e+a\,e^2\,m^2+7\,a\,e^2\,m+12\,a\,e^2\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^3\,\left (4\,b\,e+b\,e\,m+c\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,\left (-3\,c\,d^2\,m+b\,d\,e\,m^2+4\,b\,d\,e\,m+a\,e^2\,m^2+7\,a\,e^2\,m+12\,a\,e^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {d\,m\,x\,\left (6\,c\,d^2-2\,b\,d\,e\,m-8\,b\,d\,e+a\,e^2\,m^2+7\,a\,e^2\,m+12\,a\,e^2\right )}{e^3\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}\right ) \] Input:
int(x*(d + e*x)^m*(a + b*x + c*x^2),x)
Output:
(d + e*x)^m*((c*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^ 4 + 24) - (d^2*(12*a*e^2 + 6*c*d^2 + a*e^2*m^2 - 8*b*d*e + 7*a*e^2*m - 2*b *d*e*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^3*(4*b*e + b*e*m + c*d*m)*(3*m + m^2 + 2))/(e*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^2*(m + 1)*(12*a*e^2 + a*e^2*m^2 + 7*a*e^2*m - 3*c*d^2*m + b*d*e*m^2 + 4*b*d*e* m))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (d*m*x*(12*a*e^2 + 6*c*d^2 + a*e^2*m^2 - 8*b*d*e + 7*a*e^2*m - 2*b*d*e*m))/(e^3*(50*m + 35*m^2 + 10* m^3 + m^4 + 24)))
Time = 0.26 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.08 \[ \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {\left (e x +d \right )^{m} \left (c \,e^{4} m^{3} x^{4}+b \,e^{4} m^{3} x^{3}+c d \,e^{3} m^{3} x^{3}+6 c \,e^{4} m^{2} x^{4}+a \,e^{4} m^{3} x^{2}+b d \,e^{3} m^{3} x^{2}+7 b \,e^{4} m^{2} x^{3}+3 c d \,e^{3} m^{2} x^{3}+11 c \,e^{4} m \,x^{4}+a d \,e^{3} m^{3} x +8 a \,e^{4} m^{2} x^{2}+5 b d \,e^{3} m^{2} x^{2}+14 b \,e^{4} m \,x^{3}-3 c \,d^{2} e^{2} m^{2} x^{2}+2 c d \,e^{3} m \,x^{3}+6 c \,e^{4} x^{4}+7 a d \,e^{3} m^{2} x +19 a \,e^{4} m \,x^{2}-2 b \,d^{2} e^{2} m^{2} x +4 b d \,e^{3} m \,x^{2}+8 b \,e^{4} x^{3}-3 c \,d^{2} e^{2} m \,x^{2}-a \,d^{2} e^{2} m^{2}+12 a d \,e^{3} m x +12 a \,e^{4} x^{2}-8 b \,d^{2} e^{2} m x +6 c \,d^{3} e m x -7 a \,d^{2} e^{2} m +2 b \,d^{3} e m -12 a \,d^{2} e^{2}+8 b \,d^{3} e -6 c \,d^{4}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \] Input:
int(x*(e*x+d)^m*(c*x^2+b*x+a),x)
Output:
((d + e*x)**m*( - a*d**2*e**2*m**2 - 7*a*d**2*e**2*m - 12*a*d**2*e**2 + a* d*e**3*m**3*x + 7*a*d*e**3*m**2*x + 12*a*d*e**3*m*x + a*e**4*m**3*x**2 + 8 *a*e**4*m**2*x**2 + 19*a*e**4*m*x**2 + 12*a*e**4*x**2 + 2*b*d**3*e*m + 8*b *d**3*e - 2*b*d**2*e**2*m**2*x - 8*b*d**2*e**2*m*x + b*d*e**3*m**3*x**2 + 5*b*d*e**3*m**2*x**2 + 4*b*d*e**3*m*x**2 + b*e**4*m**3*x**3 + 7*b*e**4*m** 2*x**3 + 14*b*e**4*m*x**3 + 8*b*e**4*x**3 - 6*c*d**4 + 6*c*d**3*e*m*x - 3* c*d**2*e**2*m**2*x**2 - 3*c*d**2*e**2*m*x**2 + c*d*e**3*m**3*x**3 + 3*c*d* e**3*m**2*x**3 + 2*c*d*e**3*m*x**3 + c*e**4*m**3*x**4 + 6*c*e**4*m**2*x**4 + 11*c*e**4*m*x**4 + 6*c*e**4*x**4))/(e**4*(m**4 + 10*m**3 + 35*m**2 + 50 *m + 24))