Integrand size = 19, antiderivative size = 71 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=\frac {a^2 x^{1+2 q} (c x)^m}{1+m+2 q}+\frac {2 a b x^{1+q+r} (c x)^m}{1+m+q+r}+\frac {b^2 x^{1+2 r} (c x)^m}{1+m+2 r} \] Output:
a^2*x^(1+2*q)*(c*x)^m/(1+m+2*q)+2*a*b*x^(1+q+r)*(c*x)^m/(1+m+q+r)+b^2*x^(1 +2*r)*(c*x)^m/(1+m+2*r)
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=x (c x)^m \left (\frac {a^2 x^{2 q}}{1+m+2 q}+\frac {b^2 x^{2 r}}{1+m+2 r}+\frac {2 a b x^{q+r}}{1+m+q+r}\right ) \] Input:
Integrate[(c*x)^m*(a*x^q + b*x^r)^2,x]
Output:
x*(c*x)^m*((a^2*x^(2*q))/(1 + m + 2*q) + (b^2*x^(2*r))/(1 + m + 2*r) + (2* a*b*x^(q + r))/(1 + m + q + r))
Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1939, 10, 802, 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.
\(\displaystyle \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx\) |
\(\Big \downarrow \) 1939 |
\(\displaystyle x^{-m} (c x)^m \int x^m \left (a x^q+b x^r\right )^2dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle x^{-m} (c x)^m \int x^{m+2 r} \left (a x^{q-r}+b\right )^2dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle x^{-m} (c x)^m \int \left (a^2 x^{m+2 q}+2 a b x^{m+q+r}+b^2 x^{m+2 r}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^{-m} (c x)^m \left (\frac {a^2 x^{m+2 q+1}}{m+2 q+1}+\frac {2 a b x^{m+q+r+1}}{m+q+r+1}+\frac {b^2 x^{m+2 r+1}}{m+2 r+1}\right )\) |
Input:
Int[(c*x)^m*(a*x^q + b*x^r)^2,x]
Output:
((c*x)^m*((a^2*x^(1 + m + 2*q))/(1 + m + 2*q) + (2*a*b*x^(1 + m + q + r))/ (1 + m + q + r) + (b^2*x^(1 + m + 2*r))/(1 + m + 2*r)))/x^m
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Int[(u_)^(m_.)*((a_.)*(v_)^(j_.) + (b_.)*(v_)^(n_.))^(p_.), x_Symbol] :> Si mp[u^m/(Coefficient[v, x, 1]*v^m) Subst[Int[x^m*(a*x^j + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, j, m, n, p}, x] && LinearPairQ[u, v, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.78 (sec) , antiderivative size = 367, normalized size of antiderivative = 5.17
method | result | size |
risch | \(\frac {x \left (8 a b q r \,x^{q} x^{r}+a^{2} m q \,x^{2 q}+3 a^{2} m r \,x^{2 q}+2 a^{2} q r \,x^{2 q}+3 b^{2} m q \,x^{2 r}+b^{2} m r \,x^{2 r}+2 b^{2} q r \,x^{2 r}+2 a b \,m^{2} x^{q} x^{r}+4 a b m r \,x^{q} x^{r}+4 a b m q \,x^{q} x^{r}+4 m a b \,x^{q} x^{r}+2 b a \,x^{r} x^{q}+4 x^{q} x^{r} a b q +4 x^{q} x^{r} a b r +b^{2} x^{2 r}+a^{2} x^{2 q}+a^{2} m^{2} x^{2 q}+2 m \,b^{2} x^{2 r}+2 b^{2} q^{2} x^{2 r}+2 m \,a^{2} x^{2 q}+2 a^{2} r^{2} x^{2 q}+b^{2} m^{2} x^{2 r}+x^{2 q} a^{2} q +3 x^{2 q} a^{2} r +3 x^{2 r} b^{2} q +x^{2 r} b^{2} r \right ) x^{m} c^{m} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c x \right ) m \left (\operatorname {csgn}\left (i c x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i c x \right )+\operatorname {csgn}\left (i c \right )\right )}{2}}}{\left (1+m +2 q \right ) \left (1+m +q +r \right ) \left (1+m +2 r \right )}\) | \(367\) |
orering | \(\frac {x \left (3 m^{2}+6 m q +6 m r +2 q^{2}+8 q r +2 r^{2}+3 m +3 q +3 r +1\right ) \left (c x \right )^{m} \left (a \,x^{q}+b \,x^{r}\right )^{2}}{\left (m^{2}+2 m q +2 m r +4 q r +2 m +2 q +2 r +1\right ) \left (1+m +q +r \right )}-\frac {3 x^{2} \left (m +q +r \right ) \left (\frac {\left (c x \right )^{m} m \left (a \,x^{q}+b \,x^{r}\right )^{2}}{x}+2 \left (c x \right )^{m} \left (a \,x^{q}+b \,x^{r}\right ) \left (\frac {a \,x^{q} q}{x}+\frac {b \,x^{r} r}{x}\right )\right )}{\left (m^{2}+2 m q +2 m r +4 q r +2 m +2 q +2 r +1\right ) \left (1+m +q +r \right )}+\frac {x^{3} \left (\frac {\left (c x \right )^{m} m^{2} \left (a \,x^{q}+b \,x^{r}\right )^{2}}{x^{2}}-\frac {\left (c x \right )^{m} m \left (a \,x^{q}+b \,x^{r}\right )^{2}}{x^{2}}+\frac {4 \left (c x \right )^{m} m \left (a \,x^{q}+b \,x^{r}\right ) \left (\frac {a \,x^{q} q}{x}+\frac {b \,x^{r} r}{x}\right )}{x}+2 \left (c x \right )^{m} \left (\frac {a \,x^{q} q}{x}+\frac {b \,x^{r} r}{x}\right )^{2}+2 \left (c x \right )^{m} \left (a \,x^{q}+b \,x^{r}\right ) \left (\frac {a \,x^{q} q^{2}}{x^{2}}-\frac {a \,x^{q} q}{x^{2}}+\frac {b \,x^{r} r^{2}}{x^{2}}-\frac {b \,x^{r} r}{x^{2}}\right )\right )}{m^{3}+3 m^{2} q +3 m^{2} r +2 m \,q^{2}+8 m q r +2 m \,r^{2}+4 q^{2} r +4 q \,r^{2}+3 m^{2}+6 m q +6 m r +2 q^{2}+8 q r +2 r^{2}+3 m +3 q +3 r +1}\) | \(468\) |
parallelrisch | \(\frac {4 x \,x^{q} x^{r} \left (c x \right )^{m} a b r +2 x \,x^{q} x^{r} \left (c x \right )^{m} a b \,m^{2}+4 x \,x^{q} x^{r} \left (c x \right )^{m} a b m +4 x \,x^{q} x^{r} \left (c x \right )^{m} a b q +4 x \,x^{q} x^{r} \left (c x \right )^{m} a b m q +4 x \,x^{q} x^{r} \left (c x \right )^{m} a b m r +8 x \,x^{q} x^{r} \left (c x \right )^{m} a b q r +2 x \,x^{q} x^{r} \left (c x \right )^{m} a b +x \,x^{2 q} \left (c x \right )^{m} a^{2} m q +3 x \,x^{2 q} \left (c x \right )^{m} a^{2} m r +2 x \,x^{2 q} \left (c x \right )^{m} a^{2} q r +3 x \,x^{2 r} \left (c x \right )^{m} b^{2} m q +x \,x^{2 r} \left (c x \right )^{m} b^{2} m r +2 x \,x^{2 r} \left (c x \right )^{m} b^{2} q r +x \,x^{2 q} \left (c x \right )^{m} a^{2} m^{2}+2 x \,x^{2 q} \left (c x \right )^{m} a^{2} r^{2}+x \,x^{2 r} \left (c x \right )^{m} b^{2} m^{2}+2 x \,x^{2 r} \left (c x \right )^{m} b^{2} q^{2}+2 x \,x^{2 q} \left (c x \right )^{m} a^{2} m +x \,x^{2 q} \left (c x \right )^{m} a^{2} q +3 x \,x^{2 q} \left (c x \right )^{m} a^{2} r +2 x \,x^{2 r} \left (c x \right )^{m} b^{2} m +3 x \,x^{2 r} \left (c x \right )^{m} b^{2} q +x \,x^{2 r} \left (c x \right )^{m} b^{2} r +x \,x^{2 q} \left (c x \right )^{m} a^{2}+x \,x^{2 r} \left (c x \right )^{m} b^{2}}{\left (1+m +2 q \right ) \left (1+m +q +r \right ) \left (1+m +2 r \right )}\) | \(476\) |
Input:
int((c*x)^m*(a*x^q+b*x^r)^2,x,method=_RETURNVERBOSE)
Output:
x*(8*a*b*q*r*x^q*x^r+b^2*(x^r)^2+2*a*b*m^2*x^q*x^r+4*a*b*m*r*x^q*x^r+4*a*b *m*q*x^q*x^r+a^2*(x^q)^2+4*m*a*b*x^q*x^r+a^2*m^2*(x^q)^2+2*a^2*r^2*(x^q)^2 +b^2*m^2*(x^r)^2+2*b*a*x^r*x^q+a^2*m*q*(x^q)^2+3*a^2*m*r*(x^q)^2+2*a^2*q*r *(x^q)^2+3*b^2*m*q*(x^r)^2+b^2*m*r*(x^r)^2+2*b^2*q*r*(x^r)^2+(x^q)^2*a^2*q +3*(x^q)^2*a^2*r+3*(x^r)^2*b^2*q+(x^r)^2*b^2*r+4*x^q*x^r*a*b*q+4*x^q*x^r*a *b*r+2*m*b^2*(x^r)^2+2*b^2*q^2*(x^r)^2+2*m*a^2*(x^q)^2)/(1+m+2*q)/(1+m+q+r )/(1+m+2*r)*x^m*c^m*exp(1/2*I*Pi*csgn(I*c*x)*m*(csgn(I*c*x)-csgn(I*x))*(-c sgn(I*c*x)+csgn(I*c)))
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (71) = 142\).
Time = 0.09 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.87 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=\frac {2 \, {\left (a b m^{2} + 2 \, a b m + a b + 2 \, {\left (a b m + a b\right )} q + 2 \, {\left (a b m + 2 \, a b q + a b\right )} r\right )} x x^{q} x^{r} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + {\left (a^{2} m^{2} + 2 \, a^{2} r^{2} + 2 \, a^{2} m + a^{2} + {\left (a^{2} m + a^{2}\right )} q + {\left (3 \, a^{2} m + 2 \, a^{2} q + 3 \, a^{2}\right )} r\right )} x x^{2 \, q} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + {\left (b^{2} m^{2} + 2 \, b^{2} q^{2} + 2 \, b^{2} m + b^{2} + 3 \, {\left (b^{2} m + b^{2}\right )} q + {\left (b^{2} m + 2 \, b^{2} q + b^{2}\right )} r\right )} x x^{2 \, r} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \, {\left (m + 1\right )} q^{2} + 2 \, {\left (m + 2 \, q + 1\right )} r^{2} + 3 \, m^{2} + 3 \, {\left (m^{2} + 2 \, m + 1\right )} q + {\left (3 \, m^{2} + 8 \, {\left (m + 1\right )} q + 4 \, q^{2} + 6 \, m + 3\right )} r + 3 \, m + 1} \] Input:
integrate((c*x)^m*(a*x^q+b*x^r)^2,x, algorithm="fricas")
Output:
(2*(a*b*m^2 + 2*a*b*m + a*b + 2*(a*b*m + a*b)*q + 2*(a*b*m + 2*a*b*q + a*b )*r)*x*x^q*x^r*e^(m*log(c) + m*log(x)) + (a^2*m^2 + 2*a^2*r^2 + 2*a^2*m + a^2 + (a^2*m + a^2)*q + (3*a^2*m + 2*a^2*q + 3*a^2)*r)*x*x^(2*q)*e^(m*log( c) + m*log(x)) + (b^2*m^2 + 2*b^2*q^2 + 2*b^2*m + b^2 + 3*(b^2*m + b^2)*q + (b^2*m + 2*b^2*q + b^2)*r)*x*x^(2*r)*e^(m*log(c) + m*log(x)))/(m^3 + 2*( m + 1)*q^2 + 2*(m + 2*q + 1)*r^2 + 3*m^2 + 3*(m^2 + 2*m + 1)*q + (3*m^2 + 8*(m + 1)*q + 4*q^2 + 6*m + 3)*r + 3*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 3398 vs. \(2 (66) = 132\).
Time = 7.17 (sec) , antiderivative size = 3398, normalized size of antiderivative = 47.86 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((c*x)**m*(a*x**q+b*x**r)**2,x)
Output:
Piecewise((a**2*x*x**(2*r)*(c*x)**(-2*r - 1)*log(x) + 2*a*b*x*x**(2*r)*(c* x)**(-2*r - 1)*log(x) + b**2*x*x**(2*r)*(c*x)**(-2*r - 1)*log(x), Eq(q, r) & Eq(m, -2*r - 1)), (a**2*x*x**(2*q)*(c*x)**(-2*q - 1)*log(x) + 2*a*b*Pie cewise((x*x**q*x**r*(c*x)**(-2*q - 1)/(-q + r), Ne(q - r, 0)), (x*x**q*x** r*(c*x)**(-2*q - 1)*log(x), True)) + b**2*Piecewise((x*x**(2*r)*(c*x)**(-2 *q - 1)/(-2*q + 2*r), Ne(2*q - 2*r, 0)), (x*x**(2*r)*(c*x)**(-2*q - 1)*log (x), True)), Eq(m, -2*q - 1)), (a**2*Piecewise((x*x**(2*q)*(c*x)**(-2*r - 1)/(2*q - 2*r), Ne(2*q - 2*r, 0)), (x*x**(2*q)*(c*x)**(-2*r - 1)*log(x), T rue)) + 2*a*b*Piecewise((x*x**q*x**r*(c*x)**(-2*r - 1)/(q - r), Ne(q - r, 0)), (x*x**q*x**r*(c*x)**(-2*r - 1)*log(x), True)) + b**2*x*x**(2*r)*(c*x) **(-2*r - 1)*log(x), Eq(m, -2*r - 1)), (a**2*Piecewise((x*x**(2*q)*(c*x)** (-q - r - 1)/(q - r), Ne(q - r, 0)), (x*x**(2*q)*(c*x)**(-q - r - 1)*log(x ), True)) + 2*a*b*x*x**q*x**r*(c*x)**(-q - r - 1)*log(x) + b**2*Piecewise( (x*x**(2*r)*(c*x)**(-q - r - 1)/(-q + r), Ne(q - r, 0)), (x*x**(2*r)*(c*x) **(-q - r - 1)*log(x), True)), Eq(m, -q - r - 1)), (a**2*m**2*x*x**(2*q)*( c*x)**m/(m**3 + 3*m**2*q + 3*m**2*r + 3*m**2 + 2*m*q**2 + 8*m*q*r + 6*m*q + 2*m*r**2 + 6*m*r + 3*m + 4*q**2*r + 2*q**2 + 4*q*r**2 + 8*q*r + 3*q + 2* r**2 + 3*r + 1) + a**2*m*q*x*x**(2*q)*(c*x)**m/(m**3 + 3*m**2*q + 3*m**2*r + 3*m**2 + 2*m*q**2 + 8*m*q*r + 6*m*q + 2*m*r**2 + 6*m*r + 3*m + 4*q**2*r + 2*q**2 + 4*q*r**2 + 8*q*r + 3*q + 2*r**2 + 3*r + 1) + 3*a**2*m*r*x*x...
Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.18 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=\frac {a^{2} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, q \log \left (x\right )\right )}}{m + 2 \, q + 1} + \frac {2 \, a b c^{m} x e^{\left (m \log \left (x\right ) + q \log \left (x\right ) + r \log \left (x\right )\right )}}{m + q + r + 1} + \frac {b^{2} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right )\right )}}{m + 2 \, r + 1} \] Input:
integrate((c*x)^m*(a*x^q+b*x^r)^2,x, algorithm="maxima")
Output:
a^2*c^m*x*e^(m*log(x) + 2*q*log(x))/(m + 2*q + 1) + 2*a*b*c^m*x*e^(m*log(x ) + q*log(x) + r*log(x))/(m + q + r + 1) + b^2*c^m*x*e^(m*log(x) + 2*r*log (x))/(m + 2*r + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (71) = 142\).
Time = 0.23 (sec) , antiderivative size = 1159, normalized size of antiderivative = 16.32 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((c*x)^m*(a*x^q+b*x^r)^2,x, algorithm="giac")
Output:
(2*a*b*m^2*x*x^q*x^r*e^(m*log(c) + m*log(x)) + 4*a*b*m*q*x*x^q*x^r*e^(m*lo g(c) + m*log(x)) + 4*a*b*m*r*x*x^q*x^r*e^(m*log(c) + m*log(x)) + 8*a*b*q*r *x*x^q*x^r*e^(m*log(c) + m*log(x)) + a^2*m^2*x*x^(2*q)*e^(m*log(c) + m*log (x)) + a^2*m*q*x*x^(2*q)*e^(m*log(c) + m*log(x)) + 3*a^2*m*r*x*x^(2*q)*e^( m*log(c) + m*log(x)) + 2*a^2*q*r*x*x^(2*q)*e^(m*log(c) + m*log(x)) + 2*a^2 *r^2*x*x^(2*q)*e^(m*log(c) + m*log(x)) + 2*a*b*m^2*x*x^q*e^(m*log(c) + m*l og(x)) + 4*a*b*m*q*x*x^q*e^(m*log(c) + m*log(x)) + 4*a*b*m*r*x*x^q*e^(m*lo g(c) + m*log(x)) + 8*a*b*q*r*x*x^q*e^(m*log(c) + m*log(x)) + b^2*m^2*x*x^( 2*r)*e^(m*log(c) + m*log(x)) + 3*b^2*m*q*x*x^(2*r)*e^(m*log(c) + m*log(x)) + 2*b^2*q^2*x*x^(2*r)*e^(m*log(c) + m*log(x)) + b^2*m*r*x*x^(2*r)*e^(m*lo g(c) + m*log(x)) + 2*b^2*q*r*x*x^(2*r)*e^(m*log(c) + m*log(x)) + b^2*m^2*x *x^r*e^(m*log(c) + m*log(x)) + 3*b^2*m*q*x*x^r*e^(m*log(c) + m*log(x)) + 2 *b^2*q^2*x*x^r*e^(m*log(c) + m*log(x)) + b^2*m*r*x*x^r*e^(m*log(c) + m*log (x)) + 2*b^2*q*r*x*x^r*e^(m*log(c) + m*log(x)) + 4*a*b*m*x*x^q*x^r*e^(m*lo g(c) + m*log(x)) + 4*a*b*q*x*x^q*x^r*e^(m*log(c) + m*log(x)) + 4*a*b*r*x*x ^q*x^r*e^(m*log(c) + m*log(x)) + b^2*m^2*x*e^(m*log(c) + m*log(x)) + 3*b^2 *m*q*x*e^(m*log(c) + m*log(x)) + 2*b^2*q^2*x*e^(m*log(c) + m*log(x)) + b^2 *m*r*x*e^(m*log(c) + m*log(x)) + 2*b^2*q*r*x*e^(m*log(c) + m*log(x)) + 2*a ^2*m*x*x^(2*q)*e^(m*log(c) + m*log(x)) + a^2*q*x*x^(2*q)*e^(m*log(c) + m*l og(x)) + 3*a^2*r*x*x^(2*q)*e^(m*log(c) + m*log(x)) + 4*a*b*m*x*x^q*e^(m...
Time = 9.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx={\left (c\,x\right )}^m\,\left (\frac {a^2\,x^{2\,q+1}}{m+2\,q+1}+\frac {b^2\,x^{2\,r+1}}{m+2\,r+1}+\frac {2\,a\,b\,x^{q+r+1}}{m+q+r+1}\right ) \] Input:
int((c*x)^m*(a*x^q + b*x^r)^2,x)
Output:
(c*x)^m*((a^2*x^(2*q + 1))/(m + 2*q + 1) + (b^2*x^(2*r + 1))/(m + 2*r + 1) + (2*a*b*x^(q + r + 1))/(m + q + r + 1))
Time = 0.21 (sec) , antiderivative size = 379, normalized size of antiderivative = 5.34 \[ \int (c x)^m \left (a x^q+b x^r\right )^2 \, dx=\frac {x^{m} c^{m} x \left (4 x^{q +r} a b m q +4 x^{q +r} a b m r +2 x^{2 q} a^{2} m +2 x^{2 q} a^{2} r^{2}+3 x^{2 q} a^{2} r +2 x^{q +r} a b +2 x^{2 r} b^{2} m +2 x^{2 r} b^{2} q^{2}+3 x^{2 r} b^{2} q +2 x^{2 q} a^{2} q r +2 x^{q +r} a b \,m^{2}+4 x^{q +r} a b m +4 x^{q +r} a b q +4 x^{q +r} a b r +3 x^{2 r} b^{2} m q +2 x^{2 r} b^{2} q r +3 x^{2 q} a^{2} m r +x^{2 q} a^{2} m^{2}+x^{2 q} a^{2} m q +x^{2 q} a^{2} q +x^{2 r} b^{2} m^{2}+x^{2 r} b^{2} m r +x^{2 r} b^{2} r +8 x^{q +r} a b q r +x^{2 q} a^{2}+x^{2 r} b^{2}\right )}{m^{3}+3 m^{2} q +3 m^{2} r +2 m \,q^{2}+8 m q r +2 m \,r^{2}+4 q^{2} r +4 q \,r^{2}+3 m^{2}+6 m q +6 m r +2 q^{2}+8 q r +2 r^{2}+3 m +3 q +3 r +1} \] Input:
int((c*x)^m*(a*x^q+b*x^r)^2,x)
Output:
(x**m*c**m*x*(x**(2*q)*a**2*m**2 + x**(2*q)*a**2*m*q + 3*x**(2*q)*a**2*m*r + 2*x**(2*q)*a**2*m + 2*x**(2*q)*a**2*q*r + x**(2*q)*a**2*q + 2*x**(2*q)* a**2*r**2 + 3*x**(2*q)*a**2*r + x**(2*q)*a**2 + 2*x**(q + r)*a*b*m**2 + 4* x**(q + r)*a*b*m*q + 4*x**(q + r)*a*b*m*r + 4*x**(q + r)*a*b*m + 8*x**(q + r)*a*b*q*r + 4*x**(q + r)*a*b*q + 4*x**(q + r)*a*b*r + 2*x**(q + r)*a*b + x**(2*r)*b**2*m**2 + 3*x**(2*r)*b**2*m*q + x**(2*r)*b**2*m*r + 2*x**(2*r) *b**2*m + 2*x**(2*r)*b**2*q**2 + 2*x**(2*r)*b**2*q*r + 3*x**(2*r)*b**2*q + x**(2*r)*b**2*r + x**(2*r)*b**2))/(m**3 + 3*m**2*q + 3*m**2*r + 3*m**2 + 2*m*q**2 + 8*m*q*r + 6*m*q + 2*m*r**2 + 6*m*r + 3*m + 4*q**2*r + 2*q**2 + 4*q*r**2 + 8*q*r + 3*q + 2*r**2 + 3*r + 1)