\(\int (d x)^m (a x^n+b x^{1+n}+c x^{2+n})^2 \, dx\) [83]

Optimal result

Integrand size = 28, antiderivative size = 126 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx=\frac {2 a b x^{2 (1+n)} (d x)^m}{2+m+2 n}+\frac {2 b c x^{2 (2+n)} (d x)^m}{4+m+2 n}+\frac {a^2 x^{1+2 n} (d x)^m}{1+m+2 n}+\frac {\left (b^2+2 a c\right ) x^{3+2 n} (d x)^m}{3+m+2 n}+\frac {c^2 x^{5+2 n} (d x)^m}{5+m+2 n} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx=\frac {x^{1+2 n} (d x)^m \left (\frac {2 x \left (-\frac {6 a b}{2+m+2 n}+\frac {\left (b^2 (1+m+2 n)-4 a c (4+m+2 n)\right ) x}{3+m+2 n}\right )}{(4+m+2 n) (5+m+2 n)}-\frac {2 x (b (7+m+2 n)+2 c (4+m+2 n) x) (a+x (b+c x))}{(4+m+2 n) (5+m+2 n)}+(a+x (b+c x))^2\right )}{1+m+2 n} \] Input:

Integrate[(d*x)^m*(a*x^n + b*x^(1 + n) + c*x^(2 + n))^2,x]
 

Output:

(x^(1 + 2*n)*(d*x)^m*((2*x*((-6*a*b)/(2 + m + 2*n) + ((b^2*(1 + m + 2*n) - 
 4*a*c*(4 + m + 2*n))*x)/(3 + m + 2*n)))/((4 + m + 2*n)*(5 + m + 2*n)) - ( 
2*x*(b*(7 + m + 2*n) + 2*c*(4 + m + 2*n)*x)*(a + x*(b + c*x)))/((4 + m + 2 
*n)*(5 + m + 2*n)) + (a + x*(b + c*x))^2))/(1 + m + 2*n)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2028, 30, 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.

\(\displaystyle \int (d x)^m \left (a x^n+b x^{n+1}+c x^{n+2}\right )^2 \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int x^{2 n} (d x)^m \left (a+b x+c x^2\right )^2dx\)

\(\Big \downarrow \) 30

\(\displaystyle x^{-m} (d x)^m \int x^{m+2 n} \left (c x^2+b x+a\right )^2dx\)

\(\Big \downarrow \) 1140

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d*x)^m*(a*x^n + b*x^(1 + n) + c*x^(2 + n))^2,x]
 

Output:

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

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

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]
 

rule 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(943\) vs. \(2(126)=252\).

Time = 3.18 (sec) , antiderivative size = 944, normalized size of antiderivative = 7.49

method	result	size
orering	\(\frac {\left (c^{2} m^{4} x^{4}+8 c^{2} m^{3} n \,x^{4}+24 c^{2} m^{2} n^{2} x^{4}+32 c^{2} m \,n^{3} x^{4}+16 c^{2} n^{4} x^{4}+2 b c \,m^{4} x^{3}+16 b c \,m^{3} n \,x^{3}+48 b c \,m^{2} n^{2} x^{3}+64 b c m \,n^{3} x^{3}+32 b c \,n^{4} x^{3}+10 c^{2} m^{3} x^{4}+60 c^{2} m^{2} n \,x^{4}+120 c^{2} m \,n^{2} x^{4}+80 c^{2} n^{3} x^{4}+2 a c \,m^{4} x^{2}+16 a c \,m^{3} n \,x^{2}+48 a c \,m^{2} n^{2} x^{2}+64 a c m \,n^{3} x^{2}+32 a c \,n^{4} x^{2}+b^{2} m^{4} x^{2}+8 b^{2} m^{3} n \,x^{2}+24 b^{2} m^{2} n^{2} x^{2}+32 b^{2} m \,n^{3} x^{2}+16 b^{2} n^{4} x^{2}+22 b c \,m^{3} x^{3}+132 b c \,m^{2} n \,x^{3}+264 b c m \,n^{2} x^{3}+176 b c \,n^{3} x^{3}+35 c^{2} m^{2} x^{4}+140 c^{2} m n \,x^{4}+140 c^{2} n^{2} x^{4}+2 a b \,m^{4} x +16 a b \,m^{3} n x +48 a b \,m^{2} n^{2} x +64 a b m \,n^{3} x +32 a b \,n^{4} x +24 a c \,m^{3} x^{2}+144 a c \,m^{2} n \,x^{2}+288 a c m \,n^{2} x^{2}+192 a c \,n^{3} x^{2}+12 b^{2} m^{3} x^{2}+72 b^{2} m^{2} n \,x^{2}+144 b^{2} m \,n^{2} x^{2}+96 b^{2} n^{3} x^{2}+82 b c \,m^{2} x^{3}+328 b c m n \,x^{3}+328 b c \,n^{2} x^{3}+50 c^{2} x^{4} m +100 c^{2} x^{4} n +a^{2} m^{4}+8 a^{2} m^{3} n +24 a^{2} m^{2} n^{2}+32 a^{2} m \,n^{3}+16 a^{2} n^{4}+26 a b \,m^{3} x +156 a b \,m^{2} n x +312 a b m \,n^{2} x +208 a b \,n^{3} x +98 a c \,m^{2} x^{2}+392 a c m n \,x^{2}+392 a c \,n^{2} x^{2}+49 b^{2} m^{2} x^{2}+196 b^{2} m n \,x^{2}+196 b^{2} n^{2} x^{2}+122 b c \,x^{3} m +244 b c \,x^{3} n +24 c^{2} x^{4}+14 a^{2} m^{3}+84 a^{2} m^{2} n +168 a^{2} m \,n^{2}+112 a^{2} n^{3}+118 a b \,m^{2} x +472 a b m n x +472 a b \,n^{2} x +156 a c \,x^{2} m +312 a c \,x^{2} n +78 b^{2} x^{2} m +156 b^{2} x^{2} n +60 b c \,x^{3}+71 a^{2} m^{2}+284 a^{2} m n +284 a^{2} n^{2}+214 a b x m +428 a b x n +80 a c \,x^{2}+40 b^{2} x^{2}+154 a^{2} m +308 a^{2} n +120 a b x +120 a^{2}\right ) x \left (d x \right )^{m} \left (x^{n} a +b \,x^{1+n}+c \,x^{2+n}\right )^{2}}{\left (1+m +2 n \right ) \left (2+m +2 n \right ) \left (3+m +2 n \right ) \left (4+m +2 n \right ) \left (5+m +2 n \right ) \left (c \,x^{2}+b x +a \right )^{2}}\)	\(944\)
risch	\(\frac {x \left (c^{2} m^{4} x^{4}+8 c^{2} m^{3} n \,x^{4}+24 c^{2} m^{2} n^{2} x^{4}+32 c^{2} m \,n^{3} x^{4}+16 c^{2} n^{4} x^{4}+2 b c \,m^{4} x^{3}+16 b c \,m^{3} n \,x^{3}+48 b c \,m^{2} n^{2} x^{3}+64 b c m \,n^{3} x^{3}+32 b c \,n^{4} x^{3}+10 c^{2} m^{3} x^{4}+60 c^{2} m^{2} n \,x^{4}+120 c^{2} m \,n^{2} x^{4}+80 c^{2} n^{3} x^{4}+2 a c \,m^{4} x^{2}+16 a c \,m^{3} n \,x^{2}+48 a c \,m^{2} n^{2} x^{2}+64 a c m \,n^{3} x^{2}+32 a c \,n^{4} x^{2}+b^{2} m^{4} x^{2}+8 b^{2} m^{3} n \,x^{2}+24 b^{2} m^{2} n^{2} x^{2}+32 b^{2} m \,n^{3} x^{2}+16 b^{2} n^{4} x^{2}+22 b c \,m^{3} x^{3}+132 b c \,m^{2} n \,x^{3}+264 b c m \,n^{2} x^{3}+176 b c \,n^{3} x^{3}+35 c^{2} m^{2} x^{4}+140 c^{2} m n \,x^{4}+140 c^{2} n^{2} x^{4}+2 a b \,m^{4} x +16 a b \,m^{3} n x +48 a b \,m^{2} n^{2} x +64 a b m \,n^{3} x +32 a b \,n^{4} x +24 a c \,m^{3} x^{2}+144 a c \,m^{2} n \,x^{2}+288 a c m \,n^{2} x^{2}+192 a c \,n^{3} x^{2}+12 b^{2} m^{3} x^{2}+72 b^{2} m^{2} n \,x^{2}+144 b^{2} m \,n^{2} x^{2}+96 b^{2} n^{3} x^{2}+82 b c \,m^{2} x^{3}+328 b c m n \,x^{3}+328 b c \,n^{2} x^{3}+50 c^{2} x^{4} m +100 c^{2} x^{4} n +a^{2} m^{4}+8 a^{2} m^{3} n +24 a^{2} m^{2} n^{2}+32 a^{2} m \,n^{3}+16 a^{2} n^{4}+26 a b \,m^{3} x +156 a b \,m^{2} n x +312 a b m \,n^{2} x +208 a b \,n^{3} x +98 a c \,m^{2} x^{2}+392 a c m n \,x^{2}+392 a c \,n^{2} x^{2}+49 b^{2} m^{2} x^{2}+196 b^{2} m n \,x^{2}+196 b^{2} n^{2} x^{2}+122 b c \,x^{3} m +244 b c \,x^{3} n +24 c^{2} x^{4}+14 a^{2} m^{3}+84 a^{2} m^{2} n +168 a^{2} m \,n^{2}+112 a^{2} n^{3}+118 a b \,m^{2} x +472 a b m n x +472 a b \,n^{2} x +156 a c \,x^{2} m +312 a c \,x^{2} n +78 b^{2} x^{2} m +156 b^{2} x^{2} n +60 b c \,x^{3}+71 a^{2} m^{2}+284 a^{2} m n +284 a^{2} n^{2}+214 a b x m +428 a b x n +80 a c \,x^{2}+40 b^{2} x^{2}+154 a^{2} m +308 a^{2} n +120 a b x +120 a^{2}\right ) x^{2 n} x^{m} d^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i d x \right ) \pi m \left (\operatorname {csgn}\left (i d x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right )}{2}}}{\left (1+m +2 n \right ) \left (2+m +2 n \right ) \left (3+m +2 n \right ) \left (4+m +2 n \right ) \left (5+m +2 n \right )}\)	\(956\)
parallelrisch	\(\text {Expression too large to display}\)	\(1924\)

Input:

int((d*x)^m*(x^n*a+b*x^(1+n)+c*x^(2+n))^2,x,method=_RETURNVERBOSE)
 

Output:

(c^2*m^4*x^4+8*c^2*m^3*n*x^4+24*c^2*m^2*n^2*x^4+32*c^2*m*n^3*x^4+16*c^2*n^ 
4*x^4+2*b*c*m^4*x^3+16*b*c*m^3*n*x^3+48*b*c*m^2*n^2*x^3+64*b*c*m*n^3*x^3+3 
2*b*c*n^4*x^3+10*c^2*m^3*x^4+60*c^2*m^2*n*x^4+120*c^2*m*n^2*x^4+80*c^2*n^3 
*x^4+2*a*c*m^4*x^2+16*a*c*m^3*n*x^2+48*a*c*m^2*n^2*x^2+64*a*c*m*n^3*x^2+32 
*a*c*n^4*x^2+b^2*m^4*x^2+8*b^2*m^3*n*x^2+24*b^2*m^2*n^2*x^2+32*b^2*m*n^3*x 
^2+16*b^2*n^4*x^2+22*b*c*m^3*x^3+132*b*c*m^2*n*x^3+264*b*c*m*n^2*x^3+176*b 
*c*n^3*x^3+35*c^2*m^2*x^4+140*c^2*m*n*x^4+140*c^2*n^2*x^4+2*a*b*m^4*x+16*a 
*b*m^3*n*x+48*a*b*m^2*n^2*x+64*a*b*m*n^3*x+32*a*b*n^4*x+24*a*c*m^3*x^2+144 
*a*c*m^2*n*x^2+288*a*c*m*n^2*x^2+192*a*c*n^3*x^2+12*b^2*m^3*x^2+72*b^2*m^2 
*n*x^2+144*b^2*m*n^2*x^2+96*b^2*n^3*x^2+82*b*c*m^2*x^3+328*b*c*m*n*x^3+328 
*b*c*n^2*x^3+50*c^2*m*x^4+100*c^2*n*x^4+a^2*m^4+8*a^2*m^3*n+24*a^2*m^2*n^2 
+32*a^2*m*n^3+16*a^2*n^4+26*a*b*m^3*x+156*a*b*m^2*n*x+312*a*b*m*n^2*x+208* 
a*b*n^3*x+98*a*c*m^2*x^2+392*a*c*m*n*x^2+392*a*c*n^2*x^2+49*b^2*m^2*x^2+19 
6*b^2*m*n*x^2+196*b^2*n^2*x^2+122*b*c*m*x^3+244*b*c*n*x^3+24*c^2*x^4+14*a^ 
2*m^3+84*a^2*m^2*n+168*a^2*m*n^2+112*a^2*n^3+118*a*b*m^2*x+472*a*b*m*n*x+4 
72*a*b*n^2*x+156*a*c*m*x^2+312*a*c*n*x^2+78*b^2*m*x^2+156*b^2*n*x^2+60*b*c 
*x^3+71*a^2*m^2+284*a^2*m*n+284*a^2*n^2+214*a*b*m*x+428*a*b*n*x+80*a*c*x^2 
+40*b^2*x^2+154*a^2*m+308*a^2*n+120*a*b*x+120*a^2)/(1+m+2*n)/(2+m+2*n)/(3+ 
m+2*n)/(4+m+2*n)/(5+m+2*n)/(c*x^2+b*x+a)^2*x*(d*x)^m*(x^n*a+b*x^(1+n)+c*x^ 
(2+n))^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (126) = 252\).

Time = 0.11 (sec) , antiderivative size = 753, normalized size of antiderivative = 5.98 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^2,x, algorithm="fricas")
 

Output:

(a^2*m^4 + 16*a^2*n^4 + 14*a^2*m^3 + (c^2*m^4 + 16*c^2*n^4 + 10*c^2*m^3 + 
35*c^2*m^2 + 16*(2*c^2*m + 5*c^2)*n^3 + 50*c^2*m + 4*(6*c^2*m^2 + 30*c^2*m 
 + 35*c^2)*n^2 + 24*c^2 + 4*(2*c^2*m^3 + 15*c^2*m^2 + 35*c^2*m + 25*c^2)*n 
)*x^4 + 71*a^2*m^2 + 16*(2*a^2*m + 7*a^2)*n^3 + 2*(b*c*m^4 + 16*b*c*n^4 + 
11*b*c*m^3 + 41*b*c*m^2 + 8*(4*b*c*m + 11*b*c)*n^3 + 61*b*c*m + 4*(6*b*c*m 
^2 + 33*b*c*m + 41*b*c)*n^2 + 30*b*c + 2*(4*b*c*m^3 + 33*b*c*m^2 + 82*b*c* 
m + 61*b*c)*n)*x^3 + 154*a^2*m + 4*(6*a^2*m^2 + 42*a^2*m + 71*a^2)*n^2 + ( 
(b^2 + 2*a*c)*m^4 + 16*(b^2 + 2*a*c)*n^4 + 12*(b^2 + 2*a*c)*m^3 + 32*(3*b^ 
2 + 6*a*c + (b^2 + 2*a*c)*m)*n^3 + 49*(b^2 + 2*a*c)*m^2 + 4*(6*(b^2 + 2*a* 
c)*m^2 + 49*b^2 + 98*a*c + 36*(b^2 + 2*a*c)*m)*n^2 + 40*b^2 + 80*a*c + 78* 
(b^2 + 2*a*c)*m + 4*(2*(b^2 + 2*a*c)*m^3 + 18*(b^2 + 2*a*c)*m^2 + 39*b^2 + 
 78*a*c + 49*(b^2 + 2*a*c)*m)*n)*x^2 + 120*a^2 + 4*(2*a^2*m^3 + 21*a^2*m^2 
 + 71*a^2*m + 77*a^2)*n + 2*(a*b*m^4 + 16*a*b*n^4 + 13*a*b*m^3 + 59*a*b*m^ 
2 + 8*(4*a*b*m + 13*a*b)*n^3 + 107*a*b*m + 4*(6*a*b*m^2 + 39*a*b*m + 59*a* 
b)*n^2 + 60*a*b + 2*(4*a*b*m^3 + 39*a*b*m^2 + 118*a*b*m + 107*a*b)*n)*x)*x 
^(2*n + 4)*e^(m*log(d) + m*log(x))/((m^5 + 80*(m + 3)*n^4 + 32*n^5 + 15*m^ 
4 + 40*(2*m^2 + 12*m + 17)*n^3 + 85*m^3 + 20*(2*m^3 + 18*m^2 + 51*m + 45)* 
n^2 + 225*m^2 + 2*(5*m^4 + 60*m^3 + 255*m^2 + 450*m + 274)*n + 274*m + 120 
)*x^3)
 

Sympy [F(-1)]

Timed out. \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx=\text {Timed out} \] Input:

integrate((d*x)**m*(a*x**n+b*x**(1+n)+c*x**(2+n))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.37 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx=\frac {c^{2} d^{m} x^{5} e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 5} + \frac {2 \, b c d^{m} x^{4} e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 4} + \frac {b^{2} d^{m} x^{3} e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 3} + \frac {2 \, a c d^{m} x^{3} e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 3} + \frac {2 \, a b d^{m} x^{2} e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 2} + \frac {a^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} \] Input:

integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^2,x, algorithm="maxima")
 

Output:

c^2*d^m*x^5*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 5) + 2*b*c*d^m*x^4*e^(m*l 
og(x) + 2*n*log(x))/(m + 2*n + 4) + b^2*d^m*x^3*e^(m*log(x) + 2*n*log(x))/ 
(m + 2*n + 3) + 2*a*c*d^m*x^3*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 3) + 2* 
a*b*d^m*x^2*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 2) + a^2*d^m*x*e^(m*log(x 
) + 2*n*log(x))/(m + 2*n + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2370 vs. \(2 (126) = 252\).

Time = 0.20 (sec) , antiderivative size = 2370, normalized size of antiderivative = 18.81 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^2,x, algorithm="giac")
 

Output:

(c^2*m^4*x^5*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*c^2*m^3*n*x^5*x^(2*n)*e^( 
m*log(d) + m*log(x)) + 24*c^2*m^2*n^2*x^5*x^(2*n)*e^(m*log(d) + m*log(x)) 
+ 32*c^2*m*n^3*x^5*x^(2*n)*e^(m*log(d) + m*log(x)) + 16*c^2*n^4*x^5*x^(2*n 
)*e^(m*log(d) + m*log(x)) + 2*b*c*m^4*x^4*x^(2*n)*e^(m*log(d) + m*log(x)) 
+ 16*b*c*m^3*n*x^4*x^(2*n)*e^(m*log(d) + m*log(x)) + 48*b*c*m^2*n^2*x^4*x^ 
(2*n)*e^(m*log(d) + m*log(x)) + 64*b*c*m*n^3*x^4*x^(2*n)*e^(m*log(d) + m*l 
og(x)) + 32*b*c*n^4*x^4*x^(2*n)*e^(m*log(d) + m*log(x)) + 10*c^2*m^3*x^5*x 
^(2*n)*e^(m*log(d) + m*log(x)) + 60*c^2*m^2*n*x^5*x^(2*n)*e^(m*log(d) + m* 
log(x)) + 120*c^2*m*n^2*x^5*x^(2*n)*e^(m*log(d) + m*log(x)) + 80*c^2*n^3*x 
^5*x^(2*n)*e^(m*log(d) + m*log(x)) + b^2*m^4*x^3*x^(2*n)*e^(m*log(d) + m*l 
og(x)) + 2*a*c*m^4*x^3*x^(2*n)*e^(m*log(d) + m*log(x)) + 8*b^2*m^3*n*x^3*x 
^(2*n)*e^(m*log(d) + m*log(x)) + 16*a*c*m^3*n*x^3*x^(2*n)*e^(m*log(d) + m* 
log(x)) + 24*b^2*m^2*n^2*x^3*x^(2*n)*e^(m*log(d) + m*log(x)) + 48*a*c*m^2* 
n^2*x^3*x^(2*n)*e^(m*log(d) + m*log(x)) + 32*b^2*m*n^3*x^3*x^(2*n)*e^(m*lo 
g(d) + m*log(x)) + 64*a*c*m*n^3*x^3*x^(2*n)*e^(m*log(d) + m*log(x)) + 16*b 
^2*n^4*x^3*x^(2*n)*e^(m*log(d) + m*log(x)) + 32*a*c*n^4*x^3*x^(2*n)*e^(m*l 
og(d) + m*log(x)) + 22*b*c*m^3*x^4*x^(2*n)*e^(m*log(d) + m*log(x)) + 132*b 
*c*m^2*n*x^4*x^(2*n)*e^(m*log(d) + m*log(x)) + 264*b*c*m*n^2*x^4*x^(2*n)*e 
^(m*log(d) + m*log(x)) + 176*b*c*n^3*x^4*x^(2*n)*e^(m*log(d) + m*log(x)) + 
 35*c^2*m^2*x^5*x^(2*n)*e^(m*log(d) + m*log(x)) + 140*c^2*m*n*x^5*x^(2*...
 

Mupad [B] (verification not implemented)

Time = 12.80 (sec) , antiderivative size = 1210, normalized size of antiderivative = 9.60 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx =\text {Too large to display} \] Input:

int((d*x)^m*(a*x^n + b*x^(n + 1) + c*x^(n + 2))^2,x)
 

Output:

(b^2*x*x^(2*n + 2)*(d*x)^m*(78*m + 156*n + 196*m*n + 144*m*n^2 + 72*m^2*n 
+ 32*m*n^3 + 8*m^3*n + 49*m^2 + 12*m^3 + m^4 + 196*n^2 + 96*n^3 + 16*n^4 + 
 24*m^2*n^2 + 40))/(274*m + 548*n + 900*m*n + 1020*m*n^2 + 510*m^2*n + 480 
*m*n^3 + 120*m^3*n + 80*m*n^4 + 10*m^4*n + 225*m^2 + 85*m^3 + 15*m^4 + m^5 
 + 900*n^2 + 680*n^3 + 240*n^4 + 32*n^5 + 360*m^2*n^2 + 80*m^2*n^3 + 40*m^ 
3*n^2 + 120) + (c^2*x*x^(2*n + 4)*(d*x)^m*(50*m + 100*n + 140*m*n + 120*m* 
n^2 + 60*m^2*n + 32*m*n^3 + 8*m^3*n + 35*m^2 + 10*m^3 + m^4 + 140*n^2 + 80 
*n^3 + 16*n^4 + 24*m^2*n^2 + 24))/(274*m + 548*n + 900*m*n + 1020*m*n^2 + 
510*m^2*n + 480*m*n^3 + 120*m^3*n + 80*m*n^4 + 10*m^4*n + 225*m^2 + 85*m^3 
 + 15*m^4 + m^5 + 900*n^2 + 680*n^3 + 240*n^4 + 32*n^5 + 360*m^2*n^2 + 80* 
m^2*n^3 + 40*m^3*n^2 + 120) + (a^2*x*x^(2*n)*(d*x)^m*(154*m + 308*n + 284* 
m*n + 168*m*n^2 + 84*m^2*n + 32*m*n^3 + 8*m^3*n + 71*m^2 + 14*m^3 + m^4 + 
284*n^2 + 112*n^3 + 16*n^4 + 24*m^2*n^2 + 120))/(274*m + 548*n + 900*m*n + 
 1020*m*n^2 + 510*m^2*n + 480*m*n^3 + 120*m^3*n + 80*m*n^4 + 10*m^4*n + 22 
5*m^2 + 85*m^3 + 15*m^4 + m^5 + 900*n^2 + 680*n^3 + 240*n^4 + 32*n^5 + 360 
*m^2*n^2 + 80*m^2*n^3 + 40*m^3*n^2 + 120) + (2*b*c*x*x^(n + 1)*x^(n + 2)*( 
d*x)^m*(61*m + 122*n + 164*m*n + 132*m*n^2 + 66*m^2*n + 32*m*n^3 + 8*m^3*n 
 + 41*m^2 + 11*m^3 + m^4 + 164*n^2 + 88*n^3 + 16*n^4 + 24*m^2*n^2 + 30))/( 
274*m + 548*n + 900*m*n + 1020*m*n^2 + 510*m^2*n + 480*m*n^3 + 120*m^3*n + 
 80*m*n^4 + 10*m^4*n + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 900*n^2 + 680*...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 986, normalized size of antiderivative = 7.83 \[ \int (d x)^m \left (a x^n+b x^{1+n}+c x^{2+n}\right )^2 \, dx =\text {Too large to display} \] Input:

int((d*x)^m*(a*x^n+b*x^(1+n)+c*x^(2+n))^2,x)
 

Output:

(x**(m + 2*n)*d**m*x*(a**2*m**4 + 8*a**2*m**3*n + 14*a**2*m**3 + 24*a**2*m 
**2*n**2 + 84*a**2*m**2*n + 71*a**2*m**2 + 32*a**2*m*n**3 + 168*a**2*m*n** 
2 + 284*a**2*m*n + 154*a**2*m + 16*a**2*n**4 + 112*a**2*n**3 + 284*a**2*n* 
*2 + 308*a**2*n + 120*a**2 + 2*a*b*m**4*x + 16*a*b*m**3*n*x + 26*a*b*m**3* 
x + 48*a*b*m**2*n**2*x + 156*a*b*m**2*n*x + 118*a*b*m**2*x + 64*a*b*m*n**3 
*x + 312*a*b*m*n**2*x + 472*a*b*m*n*x + 214*a*b*m*x + 32*a*b*n**4*x + 208* 
a*b*n**3*x + 472*a*b*n**2*x + 428*a*b*n*x + 120*a*b*x + 2*a*c*m**4*x**2 + 
16*a*c*m**3*n*x**2 + 24*a*c*m**3*x**2 + 48*a*c*m**2*n**2*x**2 + 144*a*c*m* 
*2*n*x**2 + 98*a*c*m**2*x**2 + 64*a*c*m*n**3*x**2 + 288*a*c*m*n**2*x**2 + 
392*a*c*m*n*x**2 + 156*a*c*m*x**2 + 32*a*c*n**4*x**2 + 192*a*c*n**3*x**2 + 
 392*a*c*n**2*x**2 + 312*a*c*n*x**2 + 80*a*c*x**2 + b**2*m**4*x**2 + 8*b** 
2*m**3*n*x**2 + 12*b**2*m**3*x**2 + 24*b**2*m**2*n**2*x**2 + 72*b**2*m**2* 
n*x**2 + 49*b**2*m**2*x**2 + 32*b**2*m*n**3*x**2 + 144*b**2*m*n**2*x**2 + 
196*b**2*m*n*x**2 + 78*b**2*m*x**2 + 16*b**2*n**4*x**2 + 96*b**2*n**3*x**2 
 + 196*b**2*n**2*x**2 + 156*b**2*n*x**2 + 40*b**2*x**2 + 2*b*c*m**4*x**3 + 
 16*b*c*m**3*n*x**3 + 22*b*c*m**3*x**3 + 48*b*c*m**2*n**2*x**3 + 132*b*c*m 
**2*n*x**3 + 82*b*c*m**2*x**3 + 64*b*c*m*n**3*x**3 + 264*b*c*m*n**2*x**3 + 
 328*b*c*m*n*x**3 + 122*b*c*m*x**3 + 32*b*c*n**4*x**3 + 176*b*c*n**3*x**3 
+ 328*b*c*n**2*x**3 + 244*b*c*n*x**3 + 60*b*c*x**3 + c**2*m**4*x**4 + 8*c* 
*2*m**3*n*x**4 + 10*c**2*m**3*x**4 + 24*c**2*m**2*n**2*x**4 + 60*c**2*m...