\(\int x^3 (d+e x)^m (a+b x+c x^2) \, dx\) [117]

Optimal result

Integrand size = 21, antiderivative size = 202 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=-\frac {d^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^6 (1+m)}+\frac {d^2 \left (5 c d^2-e (4 b d-3 a e)\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {d \left (10 c d^2-3 e (2 b d-a e)\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {\left (10 c d^2-e (4 b d-a e)\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {(5 c d-b e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {c (d+e x)^{6+m}}{e^6 (6+m)} \] Output:

-d^3*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(1+m)/e^6/(1+m)+d^2*(5*c*d^2-e*(-3*a*e+4* 
b*d))*(e*x+d)^(2+m)/e^6/(2+m)-d*(10*c*d^2-3*e*(-a*e+2*b*d))*(e*x+d)^(3+m)/ 
e^6/(3+m)+(10*c*d^2-e*(-a*e+4*b*d))*(e*x+d)^(4+m)/e^6/(4+m)-(-b*e+5*c*d)*( 
e*x+d)^(5+m)/e^6/(5+m)+c*(e*x+d)^(6+m)/e^6/(6+m)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.85 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (-\frac {d^3 \left (c d^2+e (-b d+a e)\right )}{1+m}+\frac {d^2 \left (5 c d^2+e (-4 b d+3 a e)\right ) (d+e x)}{2+m}-\frac {d \left (10 c d^2+3 e (-2 b d+a e)\right ) (d+e x)^2}{3+m}+\frac {\left (10 c d^2+e (-4 b d+a e)\right ) (d+e x)^3}{4+m}-\frac {(5 c d-b e) (d+e x)^4}{5+m}+\frac {c (d+e x)^5}{6+m}\right )}{e^6} \] Input:

Integrate[x^3*(d + e*x)^m*(a + b*x + c*x^2),x]
 

Output:

((d + e*x)^(1 + m)*(-((d^3*(c*d^2 + e*(-(b*d) + a*e)))/(1 + m)) + (d^2*(5* 
c*d^2 + e*(-4*b*d + 3*a*e))*(d + e*x))/(2 + m) - (d*(10*c*d^2 + 3*e*(-2*b* 
d + a*e))*(d + e*x)^2)/(3 + m) + ((10*c*d^2 + e*(-4*b*d + a*e))*(d + e*x)^ 
3)/(4 + m) - ((5*c*d - b*e)*(d + e*x)^4)/(5 + m) + (c*(d + e*x)^5)/(6 + m) 
))/e^6
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 202, 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.

\(\displaystyle \int x^3 \left (a+b x+c x^2\right ) (d+e x)^m \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {d^2 (d+e x)^{m+1} \left (5 c d^2-e (4 b d-3 a e)\right )}{e^5}+\frac {d (d+e x)^{m+2} \left (3 e (2 b d-a e)-10 c d^2\right )}{e^5}+\frac {(d+e x)^{m+3} \left (10 c d^2-e (4 b d-a e)\right )}{e^5}-\frac {d^3 (d+e x)^m \left (a e^2-b d e+c d^2\right )}{e^5}+\frac {(b e-5 c d) (d+e x)^{m+4}}{e^5}+\frac {c (d+e x)^{m+5}}{e^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 (d+e x)^{m+2} \left (5 c d^2-e (4 b d-3 a e)\right )}{e^6 (m+2)}-\frac {d (d+e x)^{m+3} \left (10 c d^2-3 e (2 b d-a e)\right )}{e^6 (m+3)}+\frac {(d+e x)^{m+4} \left (10 c d^2-e (4 b d-a e)\right )}{e^6 (m+4)}-\frac {d^3 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^6 (m+1)}-\frac {(5 c d-b e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {c (d+e x)^{m+6}}{e^6 (m+6)}\)

Input:

Int[x^3*(d + e*x)^m*(a + b*x + c*x^2),x]
 

Output:

-((d^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (d^2*(5 
*c*d^2 - e*(4*b*d - 3*a*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (d*(10*c*d^ 
2 - 3*e*(2*b*d - a*e))*(d + e*x)^(3 + m))/(e^6*(3 + m)) + ((10*c*d^2 - e*( 
4*b*d - a*e))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - ((5*c*d - b*e)*(d + e*x)^ 
(5 + m))/(e^6*(5 + m)) + (c*(d + e*x)^(6 + m))/(e^6*(6 + 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. \(468\) vs. \(2(202)=404\).

Time = 0.76 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.32

method	result	size
norman	\(\frac {c \,x^{6} {\mathrm e}^{m \ln \left (e x +d \right )}}{6+m}+\frac {\left (b e m +c d m +6 b e \right ) x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+11 m +30\right )}+\frac {\left (a \,e^{2} m^{2}+b d e \,m^{2}+11 a \,e^{2} m +6 b d e m -5 c \,d^{2} m +30 a \,e^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+15 m^{2}+74 m +120\right )}+\frac {m d \left (a \,e^{2} m^{2}+11 a \,e^{2} m -4 b d e m +30 a \,e^{2}-24 b d e +20 c \,d^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {6 d^{4} \left (a \,e^{2} m^{2}+11 a \,e^{2} m -4 b d e m +30 a \,e^{2}-24 b d e +20 c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{6} \left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right )}+\frac {6 m \,d^{3} \left (a \,e^{2} m^{2}+11 a \,e^{2} m -4 b d e m +30 a \,e^{2}-24 b d e +20 c \,d^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right )}-\frac {3 \left (a \,e^{2} m^{2}+11 a \,e^{2} m -4 b d e m +30 a \,e^{2}-24 b d e +20 c \,d^{2}\right ) d^{2} m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+20 m^{4}+155 m^{3}+580 m^{2}+1044 m +720\right )}\)	\(469\)
gosper	\(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{5} m^{5} x^{5}-b \,e^{5} m^{5} x^{4}-15 c \,e^{5} m^{4} x^{5}-a \,e^{5} m^{5} x^{3}-16 b \,e^{5} m^{4} x^{4}+5 c d \,e^{4} m^{4} x^{4}-85 c \,e^{5} m^{3} x^{5}-17 a \,e^{5} m^{4} x^{3}+4 b d \,e^{4} m^{4} x^{3}-95 b \,e^{5} m^{3} x^{4}+50 c d \,e^{4} m^{3} x^{4}-225 c \,e^{5} m^{2} x^{5}+3 a d \,e^{4} m^{4} x^{2}-107 a \,e^{5} m^{3} x^{3}+48 b d \,e^{4} m^{3} x^{3}-260 b \,e^{5} m^{2} x^{4}-20 c \,d^{2} e^{3} m^{3} x^{3}+175 c d \,e^{4} m^{2} x^{4}-274 c \,e^{5} m \,x^{5}+42 a d \,e^{4} m^{3} x^{2}-307 a \,e^{5} m^{2} x^{3}-12 b \,d^{2} e^{3} m^{3} x^{2}+188 b d \,e^{4} m^{2} x^{3}-324 b \,e^{5} m \,x^{4}-120 c \,d^{2} e^{3} m^{2} x^{3}+250 c d \,e^{4} m \,x^{4}-120 c \,x^{5} e^{5}-6 a \,d^{2} e^{3} m^{3} x +195 a d \,e^{4} m^{2} x^{2}-396 a \,e^{5} m \,x^{3}-108 b \,d^{2} e^{3} m^{2} x^{2}+288 b d \,e^{4} m \,x^{3}-144 b \,e^{5} x^{4}+60 c \,d^{3} e^{2} m^{2} x^{2}-220 c \,d^{2} e^{3} m \,x^{3}+120 x^{4} c d \,e^{4}-72 a \,d^{2} e^{3} m^{2} x +336 a d \,e^{4} m \,x^{2}-180 x^{3} a \,e^{5}+24 b \,d^{3} e^{2} m^{2} x -240 b \,d^{2} e^{3} m \,x^{2}+144 x^{3} b d \,e^{4}+180 c \,d^{3} e^{2} m \,x^{2}-120 x^{3} c \,d^{2} e^{3}+6 a \,d^{3} e^{2} m^{2}-246 a \,d^{2} e^{3} m x +180 x^{2} a d \,e^{4}+168 b \,d^{3} e^{2} m x -144 x^{2} b \,d^{2} e^{3}-120 c \,d^{4} e m x +120 x^{2} c \,d^{3} e^{2}+66 a \,d^{3} e^{2} m -180 x a \,d^{2} e^{3}-24 b \,d^{4} e m +144 x b \,d^{3} e^{2}-120 x c \,d^{4} e +180 a \,e^{2} d^{3}-144 b \,d^{4} e +120 c \,d^{5}\right )}{e^{6} \left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right )}\)	\(727\)
orering	\(-\frac {\left (e x +d \right )^{m} \left (-c \,e^{5} m^{5} x^{5}-b \,e^{5} m^{5} x^{4}-15 c \,e^{5} m^{4} x^{5}-a \,e^{5} m^{5} x^{3}-16 b \,e^{5} m^{4} x^{4}+5 c d \,e^{4} m^{4} x^{4}-85 c \,e^{5} m^{3} x^{5}-17 a \,e^{5} m^{4} x^{3}+4 b d \,e^{4} m^{4} x^{3}-95 b \,e^{5} m^{3} x^{4}+50 c d \,e^{4} m^{3} x^{4}-225 c \,e^{5} m^{2} x^{5}+3 a d \,e^{4} m^{4} x^{2}-107 a \,e^{5} m^{3} x^{3}+48 b d \,e^{4} m^{3} x^{3}-260 b \,e^{5} m^{2} x^{4}-20 c \,d^{2} e^{3} m^{3} x^{3}+175 c d \,e^{4} m^{2} x^{4}-274 c \,e^{5} m \,x^{5}+42 a d \,e^{4} m^{3} x^{2}-307 a \,e^{5} m^{2} x^{3}-12 b \,d^{2} e^{3} m^{3} x^{2}+188 b d \,e^{4} m^{2} x^{3}-324 b \,e^{5} m \,x^{4}-120 c \,d^{2} e^{3} m^{2} x^{3}+250 c d \,e^{4} m \,x^{4}-120 c \,x^{5} e^{5}-6 a \,d^{2} e^{3} m^{3} x +195 a d \,e^{4} m^{2} x^{2}-396 a \,e^{5} m \,x^{3}-108 b \,d^{2} e^{3} m^{2} x^{2}+288 b d \,e^{4} m \,x^{3}-144 b \,e^{5} x^{4}+60 c \,d^{3} e^{2} m^{2} x^{2}-220 c \,d^{2} e^{3} m \,x^{3}+120 x^{4} c d \,e^{4}-72 a \,d^{2} e^{3} m^{2} x +336 a d \,e^{4} m \,x^{2}-180 x^{3} a \,e^{5}+24 b \,d^{3} e^{2} m^{2} x -240 b \,d^{2} e^{3} m \,x^{2}+144 x^{3} b d \,e^{4}+180 c \,d^{3} e^{2} m \,x^{2}-120 x^{3} c \,d^{2} e^{3}+6 a \,d^{3} e^{2} m^{2}-246 a \,d^{2} e^{3} m x +180 x^{2} a d \,e^{4}+168 b \,d^{3} e^{2} m x -144 x^{2} b \,d^{2} e^{3}-120 c \,d^{4} e m x +120 x^{2} c \,d^{3} e^{2}+66 a \,d^{3} e^{2} m -180 x a \,d^{2} e^{3}-24 b \,d^{4} e m +144 x b \,d^{3} e^{2}-120 x c \,d^{4} e +180 a \,e^{2} d^{3}-144 b \,d^{4} e +120 c \,d^{5}\right ) \left (e x +d \right )}{e^{6} \left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right )}\)	\(730\)
risch	\(-\frac {\left (-c \,e^{6} m^{5} x^{6}-b \,e^{6} m^{5} x^{5}-c d \,e^{5} m^{5} x^{5}-15 c \,e^{6} m^{4} x^{6}-a \,e^{6} m^{5} x^{4}-b d \,e^{5} m^{5} x^{4}-16 b \,e^{6} m^{4} x^{5}-10 c d \,e^{5} m^{4} x^{5}-85 c \,e^{6} m^{3} x^{6}-a d \,e^{5} m^{5} x^{3}-17 a \,e^{6} m^{4} x^{4}-12 b d \,e^{5} m^{4} x^{4}-95 b \,e^{6} m^{3} x^{5}+5 c \,d^{2} e^{4} m^{4} x^{4}-35 c d \,e^{5} m^{3} x^{5}-225 c \,e^{6} m^{2} x^{6}-14 a d \,e^{5} m^{4} x^{3}-107 a \,e^{6} m^{3} x^{4}+4 b \,d^{2} e^{4} m^{4} x^{3}-47 b d \,e^{5} m^{3} x^{4}-260 b \,e^{6} m^{2} x^{5}+30 c \,d^{2} e^{4} m^{3} x^{4}-50 c d \,e^{5} m^{2} x^{5}-274 c \,e^{6} m \,x^{6}+3 a \,d^{2} e^{4} m^{4} x^{2}-65 a d \,e^{5} m^{3} x^{3}-307 a \,e^{6} m^{2} x^{4}+36 b \,d^{2} e^{4} m^{3} x^{3}-72 b d \,e^{5} m^{2} x^{4}-324 b \,e^{6} m \,x^{5}-20 c \,d^{3} e^{3} m^{3} x^{3}+55 c \,d^{2} e^{4} m^{2} x^{4}-24 c d \,e^{5} m \,x^{5}-120 c \,x^{6} e^{6}+36 a \,d^{2} e^{4} m^{3} x^{2}-112 a d \,e^{5} m^{2} x^{3}-396 a \,e^{6} m \,x^{4}-12 b \,d^{3} e^{3} m^{3} x^{2}+80 b \,d^{2} e^{4} m^{2} x^{3}-36 b d \,e^{5} m \,x^{4}-144 b \,e^{6} x^{5}-60 c \,d^{3} e^{3} m^{2} x^{3}+30 c \,d^{2} e^{4} m \,x^{4}-6 a \,d^{3} e^{3} m^{3} x +123 a \,d^{2} e^{4} m^{2} x^{2}-60 a d \,e^{5} m \,x^{3}-180 a \,e^{6} x^{4}-84 b \,d^{3} e^{3} m^{2} x^{2}+48 b \,d^{2} e^{4} m \,x^{3}+60 c \,d^{4} e^{2} m^{2} x^{2}-40 c \,d^{3} e^{3} m \,x^{3}-66 a \,d^{3} e^{3} m^{2} x +90 a \,d^{2} e^{4} m \,x^{2}+24 b \,d^{4} e^{2} m^{2} x -72 b \,d^{3} e^{3} m \,x^{2}+60 c \,d^{4} e^{2} m \,x^{2}+6 a \,d^{4} e^{2} m^{2}-180 a \,d^{3} e^{3} m x +144 b \,d^{4} e^{2} m x -120 c \,d^{5} e m x +66 a \,d^{4} e^{2} m -24 b \,d^{5} e m +180 a \,e^{2} d^{4}-144 b \,d^{5} e +120 c \,d^{6}\right ) \left (e x +d \right )^{m}}{\left (5+m \right ) \left (6+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{6}}\)	\(844\)
parallelrisch	\(\text {Expression too large to display}\)	\(1336\)

Input:

int(x^3*(e*x+d)^m*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

c/(6+m)*x^6*exp(m*ln(e*x+d))+(b*e*m+c*d*m+6*b*e)/e/(m^2+11*m+30)*x^5*exp(m 
*ln(e*x+d))+(a*e^2*m^2+b*d*e*m^2+11*a*e^2*m+6*b*d*e*m-5*c*d^2*m+30*a*e^2)/ 
e^2/(m^3+15*m^2+74*m+120)*x^4*exp(m*ln(e*x+d))+m*d*(a*e^2*m^2+11*a*e^2*m-4 
*b*d*e*m+30*a*e^2-24*b*d*e+20*c*d^2)/e^3/(m^4+18*m^3+119*m^2+342*m+360)*x^ 
3*exp(m*ln(e*x+d))-6*d^4*(a*e^2*m^2+11*a*e^2*m-4*b*d*e*m+30*a*e^2-24*b*d*e 
+20*c*d^2)/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))+6/e^5*m*d^3*(a*e^2*m^2+11*a*e^2*m-4*b*d*e*m+30*a*e^2-24*b*d*e+20*c* 
d^2)/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)*x*exp(m*ln(e*x+d))-3 
*(a*e^2*m^2+11*a*e^2*m-4*b*d*e*m+30*a*e^2-24*b*d*e+20*c*d^2)*d^2/e^4*m/(m^ 
5+20*m^4+155*m^3+580*m^2+1044*m+720)*x^2*exp(m*ln(e*x+d))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (202) = 404\).

Time = 0.09 (sec) , antiderivative size = 709, normalized size of antiderivative = 3.51 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate(x^3*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

-(6*a*d^4*e^2*m^2 + 120*c*d^6 - 144*b*d^5*e + 180*a*d^4*e^2 - (c*e^6*m^5 + 
 15*c*e^6*m^4 + 85*c*e^6*m^3 + 225*c*e^6*m^2 + 274*c*e^6*m + 120*c*e^6)*x^ 
6 - (144*b*e^6 + (c*d*e^5 + b*e^6)*m^5 + 2*(5*c*d*e^5 + 8*b*e^6)*m^4 + 5*( 
7*c*d*e^5 + 19*b*e^6)*m^3 + 10*(5*c*d*e^5 + 26*b*e^6)*m^2 + 12*(2*c*d*e^5 
+ 27*b*e^6)*m)*x^5 - (180*a*e^6 + (b*d*e^5 + a*e^6)*m^5 - (5*c*d^2*e^4 - 1 
2*b*d*e^5 - 17*a*e^6)*m^4 - (30*c*d^2*e^4 - 47*b*d*e^5 - 107*a*e^6)*m^3 - 
(55*c*d^2*e^4 - 72*b*d*e^5 - 307*a*e^6)*m^2 - 6*(5*c*d^2*e^4 - 6*b*d*e^5 - 
 66*a*e^6)*m)*x^4 - (a*d*e^5*m^5 - 2*(2*b*d^2*e^4 - 7*a*d*e^5)*m^4 + (20*c 
*d^3*e^3 - 36*b*d^2*e^4 + 65*a*d*e^5)*m^3 + 4*(15*c*d^3*e^3 - 20*b*d^2*e^4 
 + 28*a*d*e^5)*m^2 + 4*(10*c*d^3*e^3 - 12*b*d^2*e^4 + 15*a*d*e^5)*m)*x^3 + 
 3*(a*d^2*e^4*m^4 - 4*(b*d^3*e^3 - 3*a*d^2*e^4)*m^3 + (20*c*d^4*e^2 - 28*b 
*d^3*e^3 + 41*a*d^2*e^4)*m^2 + 2*(10*c*d^4*e^2 - 12*b*d^3*e^3 + 15*a*d^2*e 
^4)*m)*x^2 - 6*(4*b*d^5*e - 11*a*d^4*e^2)*m - 6*(a*d^3*e^3*m^3 - (4*b*d^4* 
e^2 - 11*a*d^3*e^3)*m^2 + 2*(10*c*d^5*e - 12*b*d^4*e^2 + 15*a*d^3*e^3)*m)* 
x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^6*m^3 + 1624*e^ 
6*m^2 + 1764*e^6*m + 720*e^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10479 vs. \(2 (178) = 356\).

Time = 2.47 (sec) , antiderivative size = 10479, normalized size of antiderivative = 51.88 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate(x**3*(e*x+d)**m*(c*x**2+b*x+a),x)
 

Output:

Piecewise((d**m*(a*x**4/4 + b*x**5/5 + c*x**6/6), Eq(e, 0)), (-3*a*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) - 15*a*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) - 30*a*d*e**4*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 6 
00*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) 
 - 30*a*e**5*x**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 6 
00*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 12*b*d**4*e/(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*b*d**3*e**2*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*b*d**2*e**3*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 
0*b*d*e**4*x**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*b*e**5*x**4/(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*c*d**5*log(d/e + x)/(60*d**5*e**6 + 30 
0*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) + 137*c*d**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) + 30...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (202) = 404\).

Time = 0.05 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.11 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} a}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} b}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \] Input:

integrate(x^3*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

((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/((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*b/((m^5 + 1 
5*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^5 + 15*m^4 + 85*m^3 + 2 
25*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*m^2 + 2 
*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x 
 + d)^m*c/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^ 
6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1304 vs. \(2 (202) = 404\).

Time = 0.24 (sec) , antiderivative size = 1304, normalized size of antiderivative = 6.46 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^3*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

((e*x + d)^m*c*e^6*m^5*x^6 + (e*x + d)^m*c*d*e^5*m^5*x^5 + (e*x + d)^m*b*e 
^6*m^5*x^5 + 15*(e*x + d)^m*c*e^6*m^4*x^6 + (e*x + d)^m*b*d*e^5*m^5*x^4 + 
(e*x + d)^m*a*e^6*m^5*x^4 + 10*(e*x + d)^m*c*d*e^5*m^4*x^5 + 16*(e*x + d)^ 
m*b*e^6*m^4*x^5 + 85*(e*x + d)^m*c*e^6*m^3*x^6 + (e*x + d)^m*a*d*e^5*m^5*x 
^3 - 5*(e*x + d)^m*c*d^2*e^4*m^4*x^4 + 12*(e*x + d)^m*b*d*e^5*m^4*x^4 + 17 
*(e*x + d)^m*a*e^6*m^4*x^4 + 35*(e*x + d)^m*c*d*e^5*m^3*x^5 + 95*(e*x + d) 
^m*b*e^6*m^3*x^5 + 225*(e*x + d)^m*c*e^6*m^2*x^6 - 4*(e*x + d)^m*b*d^2*e^4 
*m^4*x^3 + 14*(e*x + d)^m*a*d*e^5*m^4*x^3 - 30*(e*x + d)^m*c*d^2*e^4*m^3*x 
^4 + 47*(e*x + d)^m*b*d*e^5*m^3*x^4 + 107*(e*x + d)^m*a*e^6*m^3*x^4 + 50*( 
e*x + d)^m*c*d*e^5*m^2*x^5 + 260*(e*x + d)^m*b*e^6*m^2*x^5 + 274*(e*x + d) 
^m*c*e^6*m*x^6 - 3*(e*x + d)^m*a*d^2*e^4*m^4*x^2 + 20*(e*x + d)^m*c*d^3*e^ 
3*m^3*x^3 - 36*(e*x + d)^m*b*d^2*e^4*m^3*x^3 + 65*(e*x + d)^m*a*d*e^5*m^3* 
x^3 - 55*(e*x + d)^m*c*d^2*e^4*m^2*x^4 + 72*(e*x + d)^m*b*d*e^5*m^2*x^4 + 
307*(e*x + d)^m*a*e^6*m^2*x^4 + 24*(e*x + d)^m*c*d*e^5*m*x^5 + 324*(e*x + 
d)^m*b*e^6*m*x^5 + 120*(e*x + d)^m*c*e^6*x^6 + 12*(e*x + d)^m*b*d^3*e^3*m^ 
3*x^2 - 36*(e*x + d)^m*a*d^2*e^4*m^3*x^2 + 60*(e*x + d)^m*c*d^3*e^3*m^2*x^ 
3 - 80*(e*x + d)^m*b*d^2*e^4*m^2*x^3 + 112*(e*x + d)^m*a*d*e^5*m^2*x^3 - 3 
0*(e*x + d)^m*c*d^2*e^4*m*x^4 + 36*(e*x + d)^m*b*d*e^5*m*x^4 + 396*(e*x + 
d)^m*a*e^6*m*x^4 + 144*(e*x + d)^m*b*e^6*x^5 + 6*(e*x + d)^m*a*d^3*e^3*m^3 
*x - 60*(e*x + d)^m*c*d^4*e^2*m^2*x^2 + 84*(e*x + d)^m*b*d^3*e^3*m^2*x^...
 

Mupad [B] (verification not implemented)

Time = 10.99 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.74 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {6\,d^4\,\left (20\,c\,d^2-4\,b\,d\,e\,m-24\,b\,d\,e+a\,e^2\,m^2+11\,a\,e^2\,m+30\,a\,e^2\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^4\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (-5\,c\,d^2\,m+b\,d\,e\,m^2+6\,b\,d\,e\,m+a\,e^2\,m^2+11\,a\,e^2\,m+30\,a\,e^2\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^5\,\left (6\,b\,e+b\,e\,m+c\,d\,m\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {6\,d^3\,m\,x\,\left (20\,c\,d^2-4\,b\,d\,e\,m-24\,b\,d\,e+a\,e^2\,m^2+11\,a\,e^2\,m+30\,a\,e^2\right )}{e^5\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}-\frac {3\,d^2\,m\,x^2\,\left (m+1\right )\,\left (20\,c\,d^2-4\,b\,d\,e\,m-24\,b\,d\,e+a\,e^2\,m^2+11\,a\,e^2\,m+30\,a\,e^2\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {d\,m\,x^3\,\left (m^2+3\,m+2\right )\,\left (20\,c\,d^2-4\,b\,d\,e\,m-24\,b\,d\,e+a\,e^2\,m^2+11\,a\,e^2\,m+30\,a\,e^2\right )}{e^3\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}\right ) \] Input:

int(x^3*(d + e*x)^m*(a + b*x + c*x^2),x)
 

Output:

(d + e*x)^m*((c*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764 
*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (6*d^4*(30*a*e^2 
 + 20*c*d^2 + a*e^2*m^2 - 24*b*d*e + 11*a*e^2*m - 4*b*d*e*m))/(e^6*(1764*m 
 + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x^4*(11*m + 6*m^ 
2 + m^3 + 6)*(30*a*e^2 + a*e^2*m^2 + 11*a*e^2*m - 5*c*d^2*m + b*d*e*m^2 + 
6*b*d*e*m))/(e^2*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 7 
20)) + (x^5*(6*b*e + b*e*m + c*d*m)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/( 
e*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (6*d^3*m 
*x*(30*a*e^2 + 20*c*d^2 + a*e^2*m^2 - 24*b*d*e + 11*a*e^2*m - 4*b*d*e*m))/ 
(e^5*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) - (3*d^ 
2*m*x^2*(m + 1)*(30*a*e^2 + 20*c*d^2 + a*e^2*m^2 - 24*b*d*e + 11*a*e^2*m - 
 4*b*d*e*m))/(e^4*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 
720)) + (d*m*x^3*(3*m + m^2 + 2)*(30*a*e^2 + 20*c*d^2 + a*e^2*m^2 - 24*b*d 
*e + 11*a*e^2*m - 4*b*d*e*m))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 
+ 21*m^5 + m^6 + 720)))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 836, normalized size of antiderivative = 4.14 \[ \int x^3 (d+e x)^m \left (a+b x+c x^2\right ) \, dx=\frac {\left (e x +d \right )^{m} \left (c \,e^{6} m^{5} x^{6}+b \,e^{6} m^{5} x^{5}+c d \,e^{5} m^{5} x^{5}+15 c \,e^{6} m^{4} x^{6}+a \,e^{6} m^{5} x^{4}+b d \,e^{5} m^{5} x^{4}+16 b \,e^{6} m^{4} x^{5}+10 c d \,e^{5} m^{4} x^{5}+85 c \,e^{6} m^{3} x^{6}+a d \,e^{5} m^{5} x^{3}+17 a \,e^{6} m^{4} x^{4}+12 b d \,e^{5} m^{4} x^{4}+95 b \,e^{6} m^{3} x^{5}-5 c \,d^{2} e^{4} m^{4} x^{4}+35 c d \,e^{5} m^{3} x^{5}+225 c \,e^{6} m^{2} x^{6}+14 a d \,e^{5} m^{4} x^{3}+107 a \,e^{6} m^{3} x^{4}-4 b \,d^{2} e^{4} m^{4} x^{3}+47 b d \,e^{5} m^{3} x^{4}+260 b \,e^{6} m^{2} x^{5}-30 c \,d^{2} e^{4} m^{3} x^{4}+50 c d \,e^{5} m^{2} x^{5}+274 c \,e^{6} m \,x^{6}-3 a \,d^{2} e^{4} m^{4} x^{2}+65 a d \,e^{5} m^{3} x^{3}+307 a \,e^{6} m^{2} x^{4}-36 b \,d^{2} e^{4} m^{3} x^{3}+72 b d \,e^{5} m^{2} x^{4}+324 b \,e^{6} m \,x^{5}+20 c \,d^{3} e^{3} m^{3} x^{3}-55 c \,d^{2} e^{4} m^{2} x^{4}+24 c d \,e^{5} m \,x^{5}+120 c \,e^{6} x^{6}-36 a \,d^{2} e^{4} m^{3} x^{2}+112 a d \,e^{5} m^{2} x^{3}+396 a \,e^{6} m \,x^{4}+12 b \,d^{3} e^{3} m^{3} x^{2}-80 b \,d^{2} e^{4} m^{2} x^{3}+36 b d \,e^{5} m \,x^{4}+144 b \,e^{6} x^{5}+60 c \,d^{3} e^{3} m^{2} x^{3}-30 c \,d^{2} e^{4} m \,x^{4}+6 a \,d^{3} e^{3} m^{3} x -123 a \,d^{2} e^{4} m^{2} x^{2}+60 a d \,e^{5} m \,x^{3}+180 a \,e^{6} x^{4}+84 b \,d^{3} e^{3} m^{2} x^{2}-48 b \,d^{2} e^{4} m \,x^{3}-60 c \,d^{4} e^{2} m^{2} x^{2}+40 c \,d^{3} e^{3} m \,x^{3}+66 a \,d^{3} e^{3} m^{2} x -90 a \,d^{2} e^{4} m \,x^{2}-24 b \,d^{4} e^{2} m^{2} x +72 b \,d^{3} e^{3} m \,x^{2}-60 c \,d^{4} e^{2} m \,x^{2}-6 a \,d^{4} e^{2} m^{2}+180 a \,d^{3} e^{3} m x -144 b \,d^{4} e^{2} m x +120 c \,d^{5} e m x -66 a \,d^{4} e^{2} m +24 b \,d^{5} e m -180 a \,d^{4} e^{2}+144 b \,d^{5} e -120 c \,d^{6}\right )}{e^{6} \left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right )} \] Input:

int(x^3*(e*x+d)^m*(c*x^2+b*x+a),x)
 

Output:

((d + e*x)**m*( - 6*a*d**4*e**2*m**2 - 66*a*d**4*e**2*m - 180*a*d**4*e**2 
+ 6*a*d**3*e**3*m**3*x + 66*a*d**3*e**3*m**2*x + 180*a*d**3*e**3*m*x - 3*a 
*d**2*e**4*m**4*x**2 - 36*a*d**2*e**4*m**3*x**2 - 123*a*d**2*e**4*m**2*x** 
2 - 90*a*d**2*e**4*m*x**2 + a*d*e**5*m**5*x**3 + 14*a*d*e**5*m**4*x**3 + 6 
5*a*d*e**5*m**3*x**3 + 112*a*d*e**5*m**2*x**3 + 60*a*d*e**5*m*x**3 + a*e** 
6*m**5*x**4 + 17*a*e**6*m**4*x**4 + 107*a*e**6*m**3*x**4 + 307*a*e**6*m**2 
*x**4 + 396*a*e**6*m*x**4 + 180*a*e**6*x**4 + 24*b*d**5*e*m + 144*b*d**5*e 
 - 24*b*d**4*e**2*m**2*x - 144*b*d**4*e**2*m*x + 12*b*d**3*e**3*m**3*x**2 
+ 84*b*d**3*e**3*m**2*x**2 + 72*b*d**3*e**3*m*x**2 - 4*b*d**2*e**4*m**4*x* 
*3 - 36*b*d**2*e**4*m**3*x**3 - 80*b*d**2*e**4*m**2*x**3 - 48*b*d**2*e**4* 
m*x**3 + b*d*e**5*m**5*x**4 + 12*b*d*e**5*m**4*x**4 + 47*b*d*e**5*m**3*x** 
4 + 72*b*d*e**5*m**2*x**4 + 36*b*d*e**5*m*x**4 + b*e**6*m**5*x**5 + 16*b*e 
**6*m**4*x**5 + 95*b*e**6*m**3*x**5 + 260*b*e**6*m**2*x**5 + 324*b*e**6*m* 
x**5 + 144*b*e**6*x**5 - 120*c*d**6 + 120*c*d**5*e*m*x - 60*c*d**4*e**2*m* 
*2*x**2 - 60*c*d**4*e**2*m*x**2 + 20*c*d**3*e**3*m**3*x**3 + 60*c*d**3*e** 
3*m**2*x**3 + 40*c*d**3*e**3*m*x**3 - 5*c*d**2*e**4*m**4*x**4 - 30*c*d**2* 
e**4*m**3*x**4 - 55*c*d**2*e**4*m**2*x**4 - 30*c*d**2*e**4*m*x**4 + c*d*e* 
*5*m**5*x**5 + 10*c*d*e**5*m**4*x**5 + 35*c*d*e**5*m**3*x**5 + 50*c*d*e**5 
*m**2*x**5 + 24*c*d*e**5*m*x**5 + c*e**6*m**5*x**6 + 15*c*e**6*m**4*x**6 + 
 85*c*e**6*m**3*x**6 + 225*c*e**6*m**2*x**6 + 274*c*e**6*m*x**6 + 120*c...