\(\int (d+e x)^m (b x+c x^2)^2 \, dx\) [239]

Optimal result

Integrand size = 19, antiderivative size = 159 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {d^2 (c d-b e)^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (\frac {d^2 (c d-b e)^2}{1+m}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)}{2+m}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {2 c (2 c d-b e) (d+e x)^3}{4+m}+\frac {c^2 (d+e x)^4}{5+m}\right )}{e^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 159, 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 = {1140, 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.

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

\(\Big \downarrow \) 1140

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, 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. \(418\) vs. \(2(159)=318\).

Time = 0.74 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.64

method	result	size
norman	\(\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {\left (b^{2} e^{2} m^{2}+2 b c d e \,m^{2}+9 b^{2} e^{2} m +10 b c d e m -4 c^{2} d^{2} m +20 b^{2} e^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {\left (2 b e m +c d m +10 b e \right ) c \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}+\frac {\left (b^{2} e^{2} m^{2}+9 b^{2} e^{2} m -6 b c d e m +20 b^{2} e^{2}-30 b c d e +12 c^{2} d^{2}\right ) d m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}+\frac {2 d^{3} \left (b^{2} e^{2} m^{2}+9 b^{2} e^{2} m -6 b c d e m +20 b^{2} e^{2}-30 b c d e +12 c^{2} d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}-\frac {2 m \,d^{2} \left (b^{2} e^{2} m^{2}+9 b^{2} e^{2} m -6 b c d e m +20 b^{2} e^{2}-30 b c d e +12 c^{2} d^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(419\)
gosper	\(\frac {\left (e x +d \right )^{1+m} \left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 x^{3} b c \,e^{4}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 d \,c^{2} x^{3} e^{3}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 x^{2} b^{2} e^{4}+72 b c \,d^{2} e^{2} m x -60 x^{2} b c d \,e^{3}-24 c^{2} d^{3} e m x +24 x^{2} c^{2} d^{2} e^{2}+18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 b c \,d^{2} e^{2} x -24 x \,c^{2} d^{3} e +40 d^{2} e^{2} b^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\)	\(547\)
orering	\(\frac {\left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 x^{3} b c \,e^{4}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 d \,c^{2} x^{3} e^{3}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 x^{2} b^{2} e^{4}+72 b c \,d^{2} e^{2} m x -60 x^{2} b c d \,e^{3}-24 c^{2} d^{3} e m x +24 x^{2} c^{2} d^{2} e^{2}+18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 b c \,d^{2} e^{2} x -24 x \,c^{2} d^{3} e +40 d^{2} e^{2} b^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right ) \left (e x +d \right ) \left (e x +d \right )^{m} \left (c \,x^{2}+b x \right )^{2}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) \left (c x +b \right )^{2} x^{2}}\)	\(571\)
risch	\(\frac {\left (c^{2} e^{5} m^{4} x^{5}+2 b c \,e^{5} m^{4} x^{4}+c^{2} d \,e^{4} m^{4} x^{4}+10 c^{2} e^{5} m^{3} x^{5}+b^{2} e^{5} m^{4} x^{3}+2 b c d \,e^{4} m^{4} x^{3}+22 b c \,e^{5} m^{3} x^{4}+6 c^{2} d \,e^{4} m^{3} x^{4}+35 c^{2} e^{5} m^{2} x^{5}+b^{2} d \,e^{4} m^{4} x^{2}+12 b^{2} e^{5} m^{3} x^{3}+16 b c d \,e^{4} m^{3} x^{3}+82 b c \,e^{5} m^{2} x^{4}-4 c^{2} d^{2} e^{3} m^{3} x^{3}+11 c^{2} d \,e^{4} m^{2} x^{4}+50 c^{2} e^{5} m \,x^{5}+10 b^{2} d \,e^{4} m^{3} x^{2}+49 b^{2} e^{5} m^{2} x^{3}-6 b c \,d^{2} e^{3} m^{3} x^{2}+34 b c d \,e^{4} m^{2} x^{3}+122 b c \,e^{5} m \,x^{4}-12 c^{2} d^{2} e^{3} m^{2} x^{3}+6 c^{2} d \,e^{4} m \,x^{4}+24 c^{2} e^{5} x^{5}-2 b^{2} d^{2} e^{3} m^{3} x +29 b^{2} d \,e^{4} m^{2} x^{2}+78 b^{2} e^{5} m \,x^{3}-36 b c \,d^{2} e^{3} m^{2} x^{2}+20 b c d \,e^{4} m \,x^{3}+60 b c \,e^{5} x^{4}+12 c^{2} d^{3} e^{2} m^{2} x^{2}-8 c^{2} d^{2} e^{3} m \,x^{3}-18 b^{2} d^{2} e^{3} m^{2} x +20 b^{2} d \,e^{4} m \,x^{2}+40 b^{2} e^{5} x^{3}+12 b c \,d^{3} e^{2} m^{2} x -30 b c \,d^{2} e^{3} m \,x^{2}+12 c^{2} d^{3} e^{2} m \,x^{2}+2 b^{2} d^{3} e^{2} m^{2}-40 b^{2} d^{2} e^{3} m x +60 b c \,d^{3} e^{2} m x -24 c^{2} d^{4} e m x +18 b^{2} d^{3} e^{2} m -12 b c \,d^{4} e m +40 b^{2} d^{3} e^{2}-60 b c \,d^{4} e +24 c^{2} d^{5}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{5}}\)	\(663\)
parallelrisch	\(\text {Expression too large to display}\)	\(1027\)

Input:

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

Output:

c^2/(5+m)*x^5*exp(m*ln(e*x+d))+(b^2*e^2*m^2+2*b*c*d*e*m^2+9*b^2*e^2*m+10*b 
*c*d*e*m-4*c^2*d^2*m+20*b^2*e^2)/e^2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x 
+d))+(2*b*e*m+c*d*m+10*b*e)/e*c/(m^2+9*m+20)*x^4*exp(m*ln(e*x+d))+(b^2*e^2 
*m^2+9*b^2*e^2*m-6*b*c*d*e*m+20*b^2*e^2-30*b*c*d*e+12*c^2*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*(b^2*e^2*m^2+9*b^2*e 
^2*m-6*b*c*d*e*m+20*b^2*e^2-30*b*c*d*e+12*c^2*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*(b^2*e^2*m^2+9*b^2*e^2*m-6 
*b*c*d*e*m+20*b^2*e^2-30*b*c*d*e+12*c^2*d^2)/(m^5+15*m^4+85*m^3+225*m^2+27 
4*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. 584 vs. \(2 (159) = 318\).

Time = 0.10 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.67 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {{\left (2 \, b^{2} d^{3} e^{2} m^{2} + 24 \, c^{2} d^{5} - 60 \, b c d^{4} e + 40 \, b^{2} d^{3} e^{2} + {\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} + {\left (60 \, b c e^{5} + {\left (c^{2} d e^{4} + 2 \, b c e^{5}\right )} m^{4} + 2 \, {\left (3 \, c^{2} d e^{4} + 11 \, b c e^{5}\right )} m^{3} + {\left (11 \, c^{2} d e^{4} + 82 \, b c e^{5}\right )} m^{2} + 2 \, {\left (3 \, c^{2} d e^{4} + 61 \, b c e^{5}\right )} m\right )} x^{4} + {\left (40 \, b^{2} e^{5} + {\left (2 \, b c d e^{4} + b^{2} e^{5}\right )} m^{4} - 4 \, {\left (c^{2} d^{2} e^{3} - 4 \, b c d e^{4} - 3 \, b^{2} e^{5}\right )} m^{3} - {\left (12 \, c^{2} d^{2} e^{3} - 34 \, b c d e^{4} - 49 \, b^{2} e^{5}\right )} m^{2} - 2 \, {\left (4 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} - 39 \, b^{2} e^{5}\right )} m\right )} x^{3} + {\left (b^{2} d e^{4} m^{4} - 2 \, {\left (3 \, b c d^{2} e^{3} - 5 \, b^{2} d e^{4}\right )} m^{3} + {\left (12 \, c^{2} d^{3} e^{2} - 36 \, b c d^{2} e^{3} + 29 \, b^{2} d e^{4}\right )} m^{2} + 2 \, {\left (6 \, c^{2} d^{3} e^{2} - 15 \, b c d^{2} e^{3} + 10 \, b^{2} d e^{4}\right )} m\right )} x^{2} - 6 \, {\left (2 \, b c d^{4} e - 3 \, b^{2} d^{3} e^{2}\right )} m - 2 \, {\left (b^{2} d^{2} e^{3} m^{3} - 3 \, {\left (2 \, b c d^{3} e^{2} - 3 \, b^{2} d^{2} e^{3}\right )} m^{2} + 2 \, {\left (6 \, c^{2} d^{4} e - 15 \, b c d^{3} e^{2} + 10 \, b^{2} d^{2} e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \] Input:

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

Output:

(2*b^2*d^3*e^2*m^2 + 24*c^2*d^5 - 60*b*c*d^4*e + 40*b^2*d^3*e^2 + (c^2*e^5 
*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5*m^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^5 + 
(60*b*c*e^5 + (c^2*d*e^4 + 2*b*c*e^5)*m^4 + 2*(3*c^2*d*e^4 + 11*b*c*e^5)*m 
^3 + (11*c^2*d*e^4 + 82*b*c*e^5)*m^2 + 2*(3*c^2*d*e^4 + 61*b*c*e^5)*m)*x^4 
 + (40*b^2*e^5 + (2*b*c*d*e^4 + b^2*e^5)*m^4 - 4*(c^2*d^2*e^3 - 4*b*c*d*e^ 
4 - 3*b^2*e^5)*m^3 - (12*c^2*d^2*e^3 - 34*b*c*d*e^4 - 49*b^2*e^5)*m^2 - 2* 
(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*b^2*e^5)*m)*x^3 + (b^2*d*e^4*m^4 - 2*(3 
*b*c*d^2*e^3 - 5*b^2*d*e^4)*m^3 + (12*c^2*d^3*e^2 - 36*b*c*d^2*e^3 + 29*b^ 
2*d*e^4)*m^2 + 2*(6*c^2*d^3*e^2 - 15*b*c*d^2*e^3 + 10*b^2*d*e^4)*m)*x^2 - 
6*(2*b*c*d^4*e - 3*b^2*d^3*e^2)*m - 2*(b^2*d^2*e^3*m^3 - 3*(2*b*c*d^3*e^2 
- 3*b^2*d^2*e^3)*m^2 + 2*(6*c^2*d^4*e - 15*b*c*d^3*e^2 + 10*b^2*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. 6418 vs. \(2 (144) = 288\).

Time = 1.39 (sec) , antiderivative size = 6418, normalized size of antiderivative = 40.36 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((d**m*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (-b** 
2*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*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 7 
2*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b**2*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) - 6*b*c*d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 4 
8*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*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) - 36*b*c*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) - 24*b*c*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**2*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**2*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**2*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**2*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**2*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**2*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**2*d*e**3*x**3*log(d/...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.00 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x)^2,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*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 2*((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*c/((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^2/((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. 1001 vs. \(2 (159) = 318\).

Time = 0.13 (sec) , antiderivative size = 1001, normalized size of antiderivative = 6.30 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

((e*x + d)^m*c^2*e^5*m^4*x^5 + (e*x + d)^m*c^2*d*e^4*m^4*x^4 + 2*(e*x + d) 
^m*b*c*e^5*m^4*x^4 + 10*(e*x + d)^m*c^2*e^5*m^3*x^5 + 2*(e*x + d)^m*b*c*d* 
e^4*m^4*x^3 + (e*x + d)^m*b^2*e^5*m^4*x^3 + 6*(e*x + d)^m*c^2*d*e^4*m^3*x^ 
4 + 22*(e*x + d)^m*b*c*e^5*m^3*x^4 + 35*(e*x + d)^m*c^2*e^5*m^2*x^5 + (e*x 
 + d)^m*b^2*d*e^4*m^4*x^2 - 4*(e*x + d)^m*c^2*d^2*e^3*m^3*x^3 + 16*(e*x + 
d)^m*b*c*d*e^4*m^3*x^3 + 12*(e*x + d)^m*b^2*e^5*m^3*x^3 + 11*(e*x + d)^m*c 
^2*d*e^4*m^2*x^4 + 82*(e*x + d)^m*b*c*e^5*m^2*x^4 + 50*(e*x + d)^m*c^2*e^5 
*m*x^5 - 6*(e*x + d)^m*b*c*d^2*e^3*m^3*x^2 + 10*(e*x + d)^m*b^2*d*e^4*m^3* 
x^2 - 12*(e*x + d)^m*c^2*d^2*e^3*m^2*x^3 + 34*(e*x + d)^m*b*c*d*e^4*m^2*x^ 
3 + 49*(e*x + d)^m*b^2*e^5*m^2*x^3 + 6*(e*x + d)^m*c^2*d*e^4*m*x^4 + 122*( 
e*x + d)^m*b*c*e^5*m*x^4 + 24*(e*x + d)^m*c^2*e^5*x^5 - 2*(e*x + d)^m*b^2* 
d^2*e^3*m^3*x + 12*(e*x + d)^m*c^2*d^3*e^2*m^2*x^2 - 36*(e*x + d)^m*b*c*d^ 
2*e^3*m^2*x^2 + 29*(e*x + d)^m*b^2*d*e^4*m^2*x^2 - 8*(e*x + d)^m*c^2*d^2*e 
^3*m*x^3 + 20*(e*x + d)^m*b*c*d*e^4*m*x^3 + 78*(e*x + d)^m*b^2*e^5*m*x^3 + 
 60*(e*x + d)^m*b*c*e^5*x^4 + 12*(e*x + d)^m*b*c*d^3*e^2*m^2*x - 18*(e*x + 
 d)^m*b^2*d^2*e^3*m^2*x + 12*(e*x + d)^m*c^2*d^3*e^2*m*x^2 - 30*(e*x + d)^ 
m*b*c*d^2*e^3*m*x^2 + 20*(e*x + d)^m*b^2*d*e^4*m*x^2 + 40*(e*x + d)^m*b^2* 
e^5*x^3 + 2*(e*x + d)^m*b^2*d^3*e^2*m^2 - 24*(e*x + d)^m*c^2*d^4*e*m*x + 6 
0*(e*x + d)^m*b*c*d^3*e^2*m*x - 40*(e*x + d)^m*b^2*d^2*e^3*m*x - 12*(e*x + 
 d)^m*b*c*d^4*e*m + 18*(e*x + d)^m*b^2*d^3*e^2*m + 24*(e*x + d)^m*c^2*d...
 

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.92 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,d^3\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^3\,\left (m^2+3\,m+2\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2+2\,b\,c\,d\,e\,m^2+10\,b\,c\,d\,e\,m-4\,c^2\,d^2\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,x^4\,\left (10\,b\,e+2\,b\,e\,m+c\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}-\frac {2\,d^2\,m\,x\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^4\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {d\,m\,x^2\,\left (m+1\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \] Input:

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

Output:

(d + e*x)^m*((c^2*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*b^2*e^2 + 12*c^2*d^2 + 9*b^2 
*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*c*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*b^2*e^2 + 9*b^2*e 
^2*m - 4*c^2*d^2*m + b^2*e^2*m^2 + 10*b*c*d*e*m + 2*b*c*d*e*m^2))/(e^2*(27 
4*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c*x^4*(10*b*e + 2*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*b^2*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2* 
m^2 - 30*b*c*d*e - 6*b*c*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*b^2*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b 
^2*e^2*m^2 - 30*b*c*d*e - 6*b*c*d*e*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 1 
5*m^4 + m^5 + 120)))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 662, normalized size of antiderivative = 4.16 \[ \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx=\frac {\left (e x +d \right )^{m} \left (c^{2} e^{5} m^{4} x^{5}+2 b c \,e^{5} m^{4} x^{4}+c^{2} d \,e^{4} m^{4} x^{4}+10 c^{2} e^{5} m^{3} x^{5}+b^{2} e^{5} m^{4} x^{3}+2 b c d \,e^{4} m^{4} x^{3}+22 b c \,e^{5} m^{3} x^{4}+6 c^{2} d \,e^{4} m^{3} x^{4}+35 c^{2} e^{5} m^{2} x^{5}+b^{2} d \,e^{4} m^{4} x^{2}+12 b^{2} e^{5} m^{3} x^{3}+16 b c d \,e^{4} m^{3} x^{3}+82 b c \,e^{5} m^{2} x^{4}-4 c^{2} d^{2} e^{3} m^{3} x^{3}+11 c^{2} d \,e^{4} m^{2} x^{4}+50 c^{2} e^{5} m \,x^{5}+10 b^{2} d \,e^{4} m^{3} x^{2}+49 b^{2} e^{5} m^{2} x^{3}-6 b c \,d^{2} e^{3} m^{3} x^{2}+34 b c d \,e^{4} m^{2} x^{3}+122 b c \,e^{5} m \,x^{4}-12 c^{2} d^{2} e^{3} m^{2} x^{3}+6 c^{2} d \,e^{4} m \,x^{4}+24 c^{2} e^{5} x^{5}-2 b^{2} d^{2} e^{3} m^{3} x +29 b^{2} d \,e^{4} m^{2} x^{2}+78 b^{2} e^{5} m \,x^{3}-36 b c \,d^{2} e^{3} m^{2} x^{2}+20 b c d \,e^{4} m \,x^{3}+60 b c \,e^{5} x^{4}+12 c^{2} d^{3} e^{2} m^{2} x^{2}-8 c^{2} d^{2} e^{3} m \,x^{3}-18 b^{2} d^{2} e^{3} m^{2} x +20 b^{2} d \,e^{4} m \,x^{2}+40 b^{2} e^{5} x^{3}+12 b c \,d^{3} e^{2} m^{2} x -30 b c \,d^{2} e^{3} m \,x^{2}+12 c^{2} d^{3} e^{2} m \,x^{2}+2 b^{2} d^{3} e^{2} m^{2}-40 b^{2} d^{2} e^{3} m x +60 b c \,d^{3} e^{2} m x -24 c^{2} d^{4} e m x +18 b^{2} d^{3} e^{2} m -12 b c \,d^{4} e m +40 b^{2} d^{3} e^{2}-60 b c \,d^{4} e +24 c^{2} d^{5}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )} \] Input:

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

Output:

((d + e*x)**m*(2*b**2*d**3*e**2*m**2 + 18*b**2*d**3*e**2*m + 40*b**2*d**3* 
e**2 - 2*b**2*d**2*e**3*m**3*x - 18*b**2*d**2*e**3*m**2*x - 40*b**2*d**2*e 
**3*m*x + b**2*d*e**4*m**4*x**2 + 10*b**2*d*e**4*m**3*x**2 + 29*b**2*d*e** 
4*m**2*x**2 + 20*b**2*d*e**4*m*x**2 + b**2*e**5*m**4*x**3 + 12*b**2*e**5*m 
**3*x**3 + 49*b**2*e**5*m**2*x**3 + 78*b**2*e**5*m*x**3 + 40*b**2*e**5*x** 
3 - 12*b*c*d**4*e*m - 60*b*c*d**4*e + 12*b*c*d**3*e**2*m**2*x + 60*b*c*d** 
3*e**2*m*x - 6*b*c*d**2*e**3*m**3*x**2 - 36*b*c*d**2*e**3*m**2*x**2 - 30*b 
*c*d**2*e**3*m*x**2 + 2*b*c*d*e**4*m**4*x**3 + 16*b*c*d*e**4*m**3*x**3 + 3 
4*b*c*d*e**4*m**2*x**3 + 20*b*c*d*e**4*m*x**3 + 2*b*c*e**5*m**4*x**4 + 22* 
b*c*e**5*m**3*x**4 + 82*b*c*e**5*m**2*x**4 + 122*b*c*e**5*m*x**4 + 60*b*c* 
e**5*x**4 + 24*c**2*d**5 - 24*c**2*d**4*e*m*x + 12*c**2*d**3*e**2*m**2*x** 
2 + 12*c**2*d**3*e**2*m*x**2 - 4*c**2*d**2*e**3*m**3*x**3 - 12*c**2*d**2*e 
**3*m**2*x**3 - 8*c**2*d**2*e**3*m*x**3 + c**2*d*e**4*m**4*x**4 + 6*c**2*d 
*e**4*m**3*x**4 + 11*c**2*d*e**4*m**2*x**4 + 6*c**2*d*e**4*m*x**4 + c**2*e 
**5*m**4*x**5 + 10*c**2*e**5*m**3*x**5 + 35*c**2*e**5*m**2*x**5 + 50*c**2* 
e**5*m*x**5 + 24*c**2*e**5*x**5))/(e**5*(m**5 + 15*m**4 + 85*m**3 + 225*m* 
*2 + 274*m + 120))