\(\int x^2 (d+e x)^m (a+b x+c x^2) \, dx\) [118]

Optimal result

Integrand size = 21, antiderivative size = 161 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {d^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^5 (1+m)}-\frac {d \left (4 c d^2-e (3 b d-2 a e)\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c d^2-e (3 b d-a e)\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {(4 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c (d+e x)^{5+m}}{e^5 (5+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.85 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (\frac {d^2 \left (c d^2+e (-b d+a e)\right )}{1+m}-\frac {d \left (4 c d^2+e (-3 b d+2 a e)\right ) (d+e x)}{2+m}+\frac {\left (6 c d^2+e (-3 b d+a e)\right ) (d+e x)^2}{3+m}-\frac {(4 c d-b e) (d+e x)^3}{4+m}+\frac {c (d+e x)^4}{5+m}\right )}{e^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

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

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {d (d+e x)^{m+1} \left (e (3 b d-2 a e)-4 c d^2\right )}{e^4}+\frac {(d+e x)^{m+2} \left (6 c d^2-e (3 b d-a e)\right )}{e^4}+\frac {d^2 (d+e x)^m \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {(b e-4 c d) (d+e x)^{m+3}}{e^4}+\frac {c (d+e x)^{m+4}}{e^4}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1195

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]
 

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. \(373\) vs. \(2(161)=322\).

Time = 0.70 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.32

method	result	size
norman	\(\frac {c \,x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {\left (b e m +c d m +5 b e \right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}+\frac {\left (a \,e^{2} m^{2}+b d e \,m^{2}+9 a \,e^{2} m +5 b d e m -4 c \,d^{2} m +20 a \,e^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {\left (a \,e^{2} m^{2}+9 a \,e^{2} m -3 b d e m +20 a \,e^{2}-15 b d e +12 c \,d^{2}\right ) d m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}+\frac {2 d^{3} \left (a \,e^{2} m^{2}+9 a \,e^{2} m -3 b d e m +20 a \,e^{2}-15 b d e +12 c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}-\frac {2 m \,d^{2} \left (a \,e^{2} m^{2}+9 a \,e^{2} m -3 b d e m +20 a \,e^{2}-15 b d e +12 c \,d^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(374\)
gosper	\(\frac {\left (e x +d \right )^{1+m} \left (c \,e^{4} m^{4} x^{4}+b \,e^{4} m^{4} x^{3}+10 c \,e^{4} m^{3} x^{4}+a \,e^{4} m^{4} x^{2}+11 b \,e^{4} m^{3} x^{3}-4 c d \,e^{3} m^{3} x^{3}+35 c \,e^{4} m^{2} x^{4}+12 a \,e^{4} m^{3} x^{2}-3 b d \,e^{3} m^{3} x^{2}+41 b \,e^{4} m^{2} x^{3}-24 c d \,e^{3} m^{2} x^{3}+50 c \,e^{4} m \,x^{4}-2 a d \,e^{3} m^{3} x +49 a \,e^{4} m^{2} x^{2}-24 b d \,e^{3} m^{2} x^{2}+61 b \,e^{4} m \,x^{3}+12 c \,d^{2} e^{2} m^{2} x^{2}-44 c d \,e^{3} m \,x^{3}+24 c \,x^{4} e^{4}-20 a d \,e^{3} m^{2} x +78 a \,e^{4} m \,x^{2}+6 b \,d^{2} e^{2} m^{2} x -51 b d \,e^{3} m \,x^{2}+30 b \,e^{4} x^{3}+36 c \,d^{2} e^{2} m \,x^{2}-24 c d \,e^{3} x^{3}+2 a \,d^{2} e^{2} m^{2}-58 a d \,e^{3} m x +40 a \,e^{4} x^{2}+36 b \,d^{2} e^{2} m x -30 b d \,e^{3} x^{2}-24 c \,d^{3} e m x +24 c \,d^{2} e^{2} x^{2}+18 a \,d^{2} e^{2} m -40 a d \,e^{3} x -6 b \,d^{3} e m +30 b \,d^{2} e^{2} x -24 c \,d^{3} e x +40 a \,d^{2} e^{2}-30 b \,d^{3} e +24 c \,d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(478\)
orering	\(\frac {\left (e x +d \right ) \left (c \,e^{4} m^{4} x^{4}+b \,e^{4} m^{4} x^{3}+10 c \,e^{4} m^{3} x^{4}+a \,e^{4} m^{4} x^{2}+11 b \,e^{4} m^{3} x^{3}-4 c d \,e^{3} m^{3} x^{3}+35 c \,e^{4} m^{2} x^{4}+12 a \,e^{4} m^{3} x^{2}-3 b d \,e^{3} m^{3} x^{2}+41 b \,e^{4} m^{2} x^{3}-24 c d \,e^{3} m^{2} x^{3}+50 c \,e^{4} m \,x^{4}-2 a d \,e^{3} m^{3} x +49 a \,e^{4} m^{2} x^{2}-24 b d \,e^{3} m^{2} x^{2}+61 b \,e^{4} m \,x^{3}+12 c \,d^{2} e^{2} m^{2} x^{2}-44 c d \,e^{3} m \,x^{3}+24 c \,x^{4} e^{4}-20 a d \,e^{3} m^{2} x +78 a \,e^{4} m \,x^{2}+6 b \,d^{2} e^{2} m^{2} x -51 b d \,e^{3} m \,x^{2}+30 b \,e^{4} x^{3}+36 c \,d^{2} e^{2} m \,x^{2}-24 c d \,e^{3} x^{3}+2 a \,d^{2} e^{2} m^{2}-58 a d \,e^{3} m x +40 a \,e^{4} x^{2}+36 b \,d^{2} e^{2} m x -30 b d \,e^{3} x^{2}-24 c \,d^{3} e m x +24 c \,d^{2} e^{2} x^{2}+18 a \,d^{2} e^{2} m -40 a d \,e^{3} x -6 b \,d^{3} e m +30 b \,d^{2} e^{2} x -24 c \,d^{3} e x +40 a \,d^{2} e^{2}-30 b \,d^{3} e +24 c \,d^{4}\right ) \left (e x +d \right )^{m}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(481\)
risch	\(\frac {\left (c \,e^{5} m^{4} x^{5}+b \,e^{5} m^{4} x^{4}+c d \,e^{4} m^{4} x^{4}+10 c \,e^{5} m^{3} x^{5}+a \,e^{5} m^{4} x^{3}+b d \,e^{4} m^{4} x^{3}+11 b \,e^{5} m^{3} x^{4}+6 c d \,e^{4} m^{3} x^{4}+35 c \,e^{5} m^{2} x^{5}+a d \,e^{4} m^{4} x^{2}+12 a \,e^{5} m^{3} x^{3}+8 b d \,e^{4} m^{3} x^{3}+41 b \,e^{5} m^{2} x^{4}-4 c \,d^{2} e^{3} m^{3} x^{3}+11 c d \,e^{4} m^{2} x^{4}+50 c \,e^{5} m \,x^{5}+10 a d \,e^{4} m^{3} x^{2}+49 a \,e^{5} m^{2} x^{3}-3 b \,d^{2} e^{3} m^{3} x^{2}+17 b d \,e^{4} m^{2} x^{3}+61 b \,e^{5} m \,x^{4}-12 c \,d^{2} e^{3} m^{2} x^{3}+6 c d \,e^{4} m \,x^{4}+24 c \,x^{5} e^{5}-2 a \,d^{2} e^{3} m^{3} x +29 a d \,e^{4} m^{2} x^{2}+78 a \,e^{5} m \,x^{3}-18 b \,d^{2} e^{3} m^{2} x^{2}+10 b d \,e^{4} m \,x^{3}+30 b \,e^{5} x^{4}+12 c \,d^{3} e^{2} m^{2} x^{2}-8 c \,d^{2} e^{3} m \,x^{3}-18 a \,d^{2} e^{3} m^{2} x +20 a d \,e^{4} m \,x^{2}+40 x^{3} a \,e^{5}+6 b \,d^{3} e^{2} m^{2} x -15 b \,d^{2} e^{3} m \,x^{2}+12 c \,d^{3} e^{2} m \,x^{2}+2 a \,d^{3} e^{2} m^{2}-40 a \,d^{2} e^{3} m x +30 b \,d^{3} e^{2} m x -24 c \,d^{4} e m x +18 a \,d^{3} e^{2} m -6 b \,d^{4} e m +40 a \,e^{2} d^{3}-30 b \,d^{4} e +24 c \,d^{5}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{5}}\)	\(583\)
parallelrisch	\(\frac {29 x^{2} \left (e x +d \right )^{m} a d \,e^{4} m^{2}-18 x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{3} m^{2}+12 x^{2} \left (e x +d \right )^{m} c \,d^{3} e^{2} m^{2}-2 x \left (e x +d \right )^{m} a \,d^{2} e^{3} m^{3}+20 x^{2} \left (e x +d \right )^{m} a d \,e^{4} m -15 x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{3} m +12 x^{2} \left (e x +d \right )^{m} c \,d^{3} e^{2} m -18 x \left (e x +d \right )^{m} a \,d^{2} e^{3} m^{2}+6 x \left (e x +d \right )^{m} b \,d^{3} e^{2} m^{2}-40 x \left (e x +d \right )^{m} a \,d^{2} e^{3} m -12 x^{3} \left (e x +d \right )^{m} c \,d^{2} e^{3} m^{2}+10 x^{2} \left (e x +d \right )^{m} a d \,e^{4} m^{3}-3 x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{3} m^{3}+50 x^{5} \left (e x +d \right )^{m} c \,e^{5} m +41 x^{4} \left (e x +d \right )^{m} b \,e^{5} m^{2}+12 x^{3} \left (e x +d \right )^{m} a \,e^{5} m^{3}+x^{5} \left (e x +d \right )^{m} c \,e^{5} m^{4}+10 x^{5} \left (e x +d \right )^{m} c \,e^{5} m^{3}+x^{4} \left (e x +d \right )^{m} b \,e^{5} m^{4}+35 x^{5} \left (e x +d \right )^{m} c \,e^{5} m^{2}+11 x^{4} \left (e x +d \right )^{m} b \,e^{5} m^{3}+x^{3} \left (e x +d \right )^{m} a \,e^{5} m^{4}+61 x^{4} \left (e x +d \right )^{m} b \,e^{5} m +49 x^{3} \left (e x +d \right )^{m} a \,e^{5} m^{2}+78 x^{3} \left (e x +d \right )^{m} a \,e^{5} m +30 x \left (e x +d \right )^{m} b \,d^{3} e^{2} m -24 x \left (e x +d \right )^{m} c \,d^{4} e m +10 x^{3} \left (e x +d \right )^{m} b d \,e^{4} m +2 \left (e x +d \right )^{m} a \,d^{3} e^{2} m^{2}+18 \left (e x +d \right )^{m} a \,d^{3} e^{2} m -6 \left (e x +d \right )^{m} b \,d^{4} e m +24 x^{5} \left (e x +d \right )^{m} c \,e^{5}+30 x^{4} \left (e x +d \right )^{m} b \,e^{5}+40 x^{3} \left (e x +d \right )^{m} a \,e^{5}+40 \left (e x +d \right )^{m} a \,d^{3} e^{2}-30 \left (e x +d \right )^{m} b \,d^{4} e -4 x^{3} \left (e x +d \right )^{m} c \,d^{2} e^{3} m^{3}+x^{2} \left (e x +d \right )^{m} a d \,e^{4} m^{4}+6 x^{4} \left (e x +d \right )^{m} c d \,e^{4} m +17 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^{4}+6 x^{4} \left (e x +d \right )^{m} c d \,e^{4} m^{3}+x^{3} \left (e x +d \right )^{m} b d \,e^{4} m^{4}+11 x^{4} \left (e x +d \right )^{m} c d \,e^{4} m^{2}+8 x^{3} \left (e x +d \right )^{m} b d \,e^{4} m^{3}-8 x^{3} \left (e x +d \right )^{m} c \,d^{2} e^{3} m +24 \left (e x +d \right )^{m} c \,d^{5}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(905\)

Input:

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

Output:

c/(5+m)*x^5*exp(m*ln(e*x+d))+(b*e*m+c*d*m+5*b*e)/e/(m^2+9*m+20)*x^4*exp(m* 
ln(e*x+d))+(a*e^2*m^2+b*d*e*m^2+9*a*e^2*m+5*b*d*e*m-4*c*d^2*m+20*a*e^2)/e^ 
2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x+d))+(a*e^2*m^2+9*a*e^2*m-3*b*d*e*m 
+20*a*e^2-15*b*d*e+12*c*d^2)*d/e^3*m/(m^4+14*m^3+71*m^2+154*m+120)*x^2*exp 
(m*ln(e*x+d))+2*d^3*(a*e^2*m^2+9*a*e^2*m-3*b*d*e*m+20*a*e^2-15*b*d*e+12*c* 
d^2)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*exp(m*ln(e*x+d))-2/e^4*m*d^ 
2*(a*e^2*m^2+9*a*e^2*m-3*b*d*e*m+20*a*e^2-15*b*d*e+12*c*d^2)/(m^5+15*m^4+8 
5*m^3+225*m^2+274*m+120)*x*exp(m*ln(e*x+d))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (161) = 322\).

Time = 0.08 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.09 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left (2 \, a d^{3} e^{2} m^{2} + 24 \, c d^{5} - 30 \, b d^{4} e + 40 \, a d^{3} e^{2} + {\left (c e^{5} m^{4} + 10 \, c e^{5} m^{3} + 35 \, c e^{5} m^{2} + 50 \, c e^{5} m + 24 \, c e^{5}\right )} x^{5} + {\left (30 \, b e^{5} + {\left (c d e^{4} + b e^{5}\right )} m^{4} + {\left (6 \, c d e^{4} + 11 \, b e^{5}\right )} m^{3} + {\left (11 \, c d e^{4} + 41 \, b e^{5}\right )} m^{2} + {\left (6 \, c d e^{4} + 61 \, b e^{5}\right )} m\right )} x^{4} + {\left (40 \, a e^{5} + {\left (b d e^{4} + a e^{5}\right )} m^{4} - 4 \, {\left (c d^{2} e^{3} - 2 \, b d e^{4} - 3 \, a e^{5}\right )} m^{3} - {\left (12 \, c d^{2} e^{3} - 17 \, b d e^{4} - 49 \, a e^{5}\right )} m^{2} - 2 \, {\left (4 \, c d^{2} e^{3} - 5 \, b d e^{4} - 39 \, a e^{5}\right )} m\right )} x^{3} + {\left (a d e^{4} m^{4} - {\left (3 \, b d^{2} e^{3} - 10 \, a d e^{4}\right )} m^{3} + {\left (12 \, c d^{3} e^{2} - 18 \, b d^{2} e^{3} + 29 \, a d e^{4}\right )} m^{2} + {\left (12 \, c d^{3} e^{2} - 15 \, b d^{2} e^{3} + 20 \, a d e^{4}\right )} m\right )} x^{2} - 6 \, {\left (b d^{4} e - 3 \, a d^{3} e^{2}\right )} m - 2 \, {\left (a d^{2} e^{3} m^{3} - 3 \, {\left (b d^{3} e^{2} - 3 \, a d^{2} e^{3}\right )} m^{2} + {\left (12 \, c d^{4} e - 15 \, b d^{3} e^{2} + 20 \, a d^{2} e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \] Input:

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

Output:

(2*a*d^3*e^2*m^2 + 24*c*d^5 - 30*b*d^4*e + 40*a*d^3*e^2 + (c*e^5*m^4 + 10* 
c*e^5*m^3 + 35*c*e^5*m^2 + 50*c*e^5*m + 24*c*e^5)*x^5 + (30*b*e^5 + (c*d*e 
^4 + b*e^5)*m^4 + (6*c*d*e^4 + 11*b*e^5)*m^3 + (11*c*d*e^4 + 41*b*e^5)*m^2 
 + (6*c*d*e^4 + 61*b*e^5)*m)*x^4 + (40*a*e^5 + (b*d*e^4 + a*e^5)*m^4 - 4*( 
c*d^2*e^3 - 2*b*d*e^4 - 3*a*e^5)*m^3 - (12*c*d^2*e^3 - 17*b*d*e^4 - 49*a*e 
^5)*m^2 - 2*(4*c*d^2*e^3 - 5*b*d*e^4 - 39*a*e^5)*m)*x^3 + (a*d*e^4*m^4 - ( 
3*b*d^2*e^3 - 10*a*d*e^4)*m^3 + (12*c*d^3*e^2 - 18*b*d^2*e^3 + 29*a*d*e^4) 
*m^2 + (12*c*d^3*e^2 - 15*b*d^2*e^3 + 20*a*d*e^4)*m)*x^2 - 6*(b*d^4*e - 3* 
a*d^3*e^2)*m - 2*(a*d^2*e^3*m^3 - 3*(b*d^3*e^2 - 3*a*d^2*e^3)*m^2 + (12*c* 
d^4*e - 15*b*d^3*e^2 + 20*a*d^2*e^3)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m 
^4 + 85*e^5*m^3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6195 vs. \(2 (141) = 282\).

Time = 1.52 (sec) , antiderivative size = 6195, normalized size of antiderivative = 38.48 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((d**m*(a*x**3/3 + b*x**4/4 + c*x**5/5), Eq(e, 0)), (-a*d**2*e**2 
/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12* 
e**9*x**4) - 4*a*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x* 
*2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*a*e**4*x**2/(12*d**4*e**5 + 48*d** 
3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 3*b*d**3*e 
/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12* 
e**9*x**4) - 12*b*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e** 
7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 18*b*d*e**3*x**2/(12*d**4*e**5 + 
 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12* 
b*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8 
*x**3 + 12*e**9*x**4) + 12*c*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e** 
6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 25*c*d**4/(12*d 
**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x 
**4) + 48*c*d**3*e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2 
*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 88*c*d**3*e*x/(12*d**4*e**5 
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 72 
*c*d**2*e**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e* 
*7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 108*c*d**2*e**2*x**2/(12*d**4*e 
**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) 
+ 48*c*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.94 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \] Input:

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

Output:

((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/((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/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^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*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274* 
m + 120)*e^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (161) = 322\).

Time = 0.21 (sec) , antiderivative size = 921, normalized size of antiderivative = 5.72 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

((e*x + d)^m*c*e^5*m^4*x^5 + (e*x + d)^m*c*d*e^4*m^4*x^4 + (e*x + d)^m*b*e 
^5*m^4*x^4 + 10*(e*x + d)^m*c*e^5*m^3*x^5 + (e*x + d)^m*b*d*e^4*m^4*x^3 + 
(e*x + d)^m*a*e^5*m^4*x^3 + 6*(e*x + d)^m*c*d*e^4*m^3*x^4 + 11*(e*x + d)^m 
*b*e^5*m^3*x^4 + 35*(e*x + d)^m*c*e^5*m^2*x^5 + (e*x + d)^m*a*d*e^4*m^4*x^ 
2 - 4*(e*x + d)^m*c*d^2*e^3*m^3*x^3 + 8*(e*x + d)^m*b*d*e^4*m^3*x^3 + 12*( 
e*x + d)^m*a*e^5*m^3*x^3 + 11*(e*x + d)^m*c*d*e^4*m^2*x^4 + 41*(e*x + d)^m 
*b*e^5*m^2*x^4 + 50*(e*x + d)^m*c*e^5*m*x^5 - 3*(e*x + d)^m*b*d^2*e^3*m^3* 
x^2 + 10*(e*x + d)^m*a*d*e^4*m^3*x^2 - 12*(e*x + d)^m*c*d^2*e^3*m^2*x^3 + 
17*(e*x + d)^m*b*d*e^4*m^2*x^3 + 49*(e*x + d)^m*a*e^5*m^2*x^3 + 6*(e*x + d 
)^m*c*d*e^4*m*x^4 + 61*(e*x + d)^m*b*e^5*m*x^4 + 24*(e*x + d)^m*c*e^5*x^5 
- 2*(e*x + d)^m*a*d^2*e^3*m^3*x + 12*(e*x + d)^m*c*d^3*e^2*m^2*x^2 - 18*(e 
*x + d)^m*b*d^2*e^3*m^2*x^2 + 29*(e*x + d)^m*a*d*e^4*m^2*x^2 - 8*(e*x + d) 
^m*c*d^2*e^3*m*x^3 + 10*(e*x + d)^m*b*d*e^4*m*x^3 + 78*(e*x + d)^m*a*e^5*m 
*x^3 + 30*(e*x + d)^m*b*e^5*x^4 + 6*(e*x + d)^m*b*d^3*e^2*m^2*x - 18*(e*x 
+ d)^m*a*d^2*e^3*m^2*x + 12*(e*x + d)^m*c*d^3*e^2*m*x^2 - 15*(e*x + d)^m*b 
*d^2*e^3*m*x^2 + 20*(e*x + d)^m*a*d*e^4*m*x^2 + 40*(e*x + d)^m*a*e^5*x^3 + 
 2*(e*x + d)^m*a*d^3*e^2*m^2 - 24*(e*x + d)^m*c*d^4*e*m*x + 30*(e*x + d)^m 
*b*d^3*e^2*m*x - 40*(e*x + d)^m*a*d^2*e^3*m*x - 6*(e*x + d)^m*b*d^4*e*m + 
18*(e*x + d)^m*a*d^3*e^2*m + 24*(e*x + d)^m*c*d^5 - 30*(e*x + d)^m*b*d^4*e 
 + 40*(e*x + d)^m*a*d^3*e^2)/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e...
 

Mupad [B] (verification not implemented)

Time = 10.82 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.60 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,d^3\,\left (12\,c\,d^2-3\,b\,d\,e\,m-15\,b\,d\,e+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^3\,\left (m^2+3\,m+2\right )\,\left (-4\,c\,d^2\,m+b\,d\,e\,m^2+5\,b\,d\,e\,m+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^4\,\left (5\,b\,e+b\,e\,m+c\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}-\frac {2\,d^2\,m\,x\,\left (12\,c\,d^2-3\,b\,d\,e\,m-15\,b\,d\,e+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^4\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {d\,m\,x^2\,\left (m+1\right )\,\left (12\,c\,d^2-3\,b\,d\,e\,m-15\,b\,d\,e+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \] Input:

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

Output:

(d + e*x)^m*((c*x^5*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 
+ 85*m^3 + 15*m^4 + m^5 + 120) + (2*d^3*(20*a*e^2 + 12*c*d^2 + a*e^2*m^2 - 
 15*b*d*e + 9*a*e^2*m - 3*b*d*e*m))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^ 
4 + m^5 + 120)) + (x^3*(3*m + m^2 + 2)*(20*a*e^2 + a*e^2*m^2 + 9*a*e^2*m - 
 4*c*d^2*m + b*d*e*m^2 + 5*b*d*e*m))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m 
^4 + m^5 + 120)) + (x^4*(5*b*e + b*e*m + c*d*m)*(11*m + 6*m^2 + m^3 + 6))/ 
(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) - (2*d^2*m*x*(20*a*e^2 
 + 12*c*d^2 + a*e^2*m^2 - 15*b*d*e + 9*a*e^2*m - 3*b*d*e*m))/(e^4*(274*m + 
 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (d*m*x^2*(m + 1)*(20*a*e^2 + 12 
*c*d^2 + a*e^2*m^2 - 15*b*d*e + 9*a*e^2*m - 3*b*d*e*m))/(e^3*(274*m + 225* 
m^2 + 85*m^3 + 15*m^4 + m^5 + 120)))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 582, normalized size of antiderivative = 3.61 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {\left (e x +d \right )^{m} \left (c \,e^{5} m^{4} x^{5}+b \,e^{5} m^{4} x^{4}+c d \,e^{4} m^{4} x^{4}+10 c \,e^{5} m^{3} x^{5}+a \,e^{5} m^{4} x^{3}+b d \,e^{4} m^{4} x^{3}+11 b \,e^{5} m^{3} x^{4}+6 c d \,e^{4} m^{3} x^{4}+35 c \,e^{5} m^{2} x^{5}+a d \,e^{4} m^{4} x^{2}+12 a \,e^{5} m^{3} x^{3}+8 b d \,e^{4} m^{3} x^{3}+41 b \,e^{5} m^{2} x^{4}-4 c \,d^{2} e^{3} m^{3} x^{3}+11 c d \,e^{4} m^{2} x^{4}+50 c \,e^{5} m \,x^{5}+10 a d \,e^{4} m^{3} x^{2}+49 a \,e^{5} m^{2} x^{3}-3 b \,d^{2} e^{3} m^{3} x^{2}+17 b d \,e^{4} m^{2} x^{3}+61 b \,e^{5} m \,x^{4}-12 c \,d^{2} e^{3} m^{2} x^{3}+6 c d \,e^{4} m \,x^{4}+24 c \,e^{5} x^{5}-2 a \,d^{2} e^{3} m^{3} x +29 a d \,e^{4} m^{2} x^{2}+78 a \,e^{5} m \,x^{3}-18 b \,d^{2} e^{3} m^{2} x^{2}+10 b d \,e^{4} m \,x^{3}+30 b \,e^{5} x^{4}+12 c \,d^{3} e^{2} m^{2} x^{2}-8 c \,d^{2} e^{3} m \,x^{3}-18 a \,d^{2} e^{3} m^{2} x +20 a d \,e^{4} m \,x^{2}+40 a \,e^{5} x^{3}+6 b \,d^{3} e^{2} m^{2} x -15 b \,d^{2} e^{3} m \,x^{2}+12 c \,d^{3} e^{2} m \,x^{2}+2 a \,d^{3} e^{2} m^{2}-40 a \,d^{2} e^{3} m x +30 b \,d^{3} e^{2} m x -24 c \,d^{4} e m x +18 a \,d^{3} e^{2} m -6 b \,d^{4} e m +40 a \,d^{3} e^{2}-30 b \,d^{4} e +24 c \,d^{5}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )} \] Input:

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

Output:

((d + e*x)**m*(2*a*d**3*e**2*m**2 + 18*a*d**3*e**2*m + 40*a*d**3*e**2 - 2* 
a*d**2*e**3*m**3*x - 18*a*d**2*e**3*m**2*x - 40*a*d**2*e**3*m*x + a*d*e**4 
*m**4*x**2 + 10*a*d*e**4*m**3*x**2 + 29*a*d*e**4*m**2*x**2 + 20*a*d*e**4*m 
*x**2 + a*e**5*m**4*x**3 + 12*a*e**5*m**3*x**3 + 49*a*e**5*m**2*x**3 + 78* 
a*e**5*m*x**3 + 40*a*e**5*x**3 - 6*b*d**4*e*m - 30*b*d**4*e + 6*b*d**3*e** 
2*m**2*x + 30*b*d**3*e**2*m*x - 3*b*d**2*e**3*m**3*x**2 - 18*b*d**2*e**3*m 
**2*x**2 - 15*b*d**2*e**3*m*x**2 + b*d*e**4*m**4*x**3 + 8*b*d*e**4*m**3*x* 
*3 + 17*b*d*e**4*m**2*x**3 + 10*b*d*e**4*m*x**3 + b*e**5*m**4*x**4 + 11*b* 
e**5*m**3*x**4 + 41*b*e**5*m**2*x**4 + 61*b*e**5*m*x**4 + 30*b*e**5*x**4 + 
 24*c*d**5 - 24*c*d**4*e*m*x + 12*c*d**3*e**2*m**2*x**2 + 12*c*d**3*e**2*m 
*x**2 - 4*c*d**2*e**3*m**3*x**3 - 12*c*d**2*e**3*m**2*x**3 - 8*c*d**2*e**3 
*m*x**3 + c*d*e**4*m**4*x**4 + 6*c*d*e**4*m**3*x**4 + 11*c*d*e**4*m**2*x** 
4 + 6*c*d*e**4*m*x**4 + c*e**5*m**4*x**5 + 10*c*e**5*m**3*x**5 + 35*c*e**5 
*m**2*x**5 + 50*c*e**5*m*x**5 + 24*c*e**5*x**5))/(e**5*(m**5 + 15*m**4 + 8 
5*m**3 + 225*m**2 + 274*m + 120))