\(\int (g x)^m (A+B x) (a^2+2 a b x+b^2 x^2) \, dx\) [467]

Optimal result

Integrand size = 27, antiderivative size = 91 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {a^2 A (g x)^{1+m}}{g (1+m)}+\frac {a (2 A b+a B) (g x)^{2+m}}{g^2 (2+m)}+\frac {b (A b+2 a B) (g x)^{3+m}}{g^3 (3+m)}+\frac {b^2 B (g x)^{4+m}}{g^4 (4+m)} \] Output:

a^2*A*(g*x)^(1+m)/g/(1+m)+a*(2*A*b+B*a)*(g*x)^(2+m)/g^2/(2+m)+b*(A*b+2*B*a 
)*(g*x)^(3+m)/g^3/(3+m)+b^2*B*(g*x)^(4+m)/g^4/(4+m)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(g x)^m \left (B x (a+b x)^3+(-a B (1+m)+A b (4+m)) x \left (\frac {a^2}{1+m}+\frac {2 a b x}{2+m}+\frac {b^2 x^2}{3+m}\right )\right )}{b (4+m)} \] Input:

Integrate[(g*x)^m*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2),x]
 

Output:

((g*x)^m*(B*x*(a + b*x)^3 + (-(a*B*(1 + m)) + A*b*(4 + m))*x*(a^2/(1 + m) 
+ (2*a*b*x)/(2 + m) + (b^2*x^2)/(3 + m))))/(b*(4 + m))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 85, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right ) (A+B x) (g x)^m \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int b^2 (g x)^m (a+b x)^2 (A+B x)dx}{b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \int (a+b x)^2 (A+B x) (g x)^mdx\)

\(\Big \downarrow \) 85

\(\displaystyle \int \left (a^2 A (g x)^m+\frac {b (g x)^{m+2} (2 a B+A b)}{g^2}+\frac {a (g x)^{m+1} (a B+2 A b)}{g}+\frac {b^2 B (g x)^{m+3}}{g^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 A (g x)^{m+1}}{g (m+1)}+\frac {b (g x)^{m+3} (2 a B+A b)}{g^3 (m+3)}+\frac {a (g x)^{m+2} (a B+2 A b)}{g^2 (m+2)}+\frac {b^2 B (g x)^{m+4}}{g^4 (m+4)}\)

Input:

Int[(g*x)^m*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2),x]
 

Output:

(a^2*A*(g*x)^(1 + m))/(g*(1 + m)) + (a*(2*A*b + a*B)*(g*x)^(2 + m))/(g^2*( 
2 + m)) + (b*(A*b + 2*a*B)*(g*x)^(3 + m))/(g^3*(3 + m)) + (b^2*B*(g*x)^(4 
+ m))/(g^4*(4 + m))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 85

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : 
> Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, 
 d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* 
f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n 
 + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 
1])
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {B \,b^{2} x^{4} {\mathrm e}^{m \ln \left (g x \right )}}{4+m}+\frac {a \left (2 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (g x \right )}}{2+m}+\frac {a^{2} A x \,{\mathrm e}^{m \ln \left (g x \right )}}{1+m}+\frac {b \left (A b +2 B a \right ) x^{3} {\mathrm e}^{m \ln \left (g x \right )}}{3+m}\)	\(90\)
gosper	\(\frac {\left (g x \right )^{m} \left (B \,b^{2} m^{3} x^{3}+A \,b^{2} m^{3} x^{2}+2 B a b \,m^{3} x^{2}+6 B \,b^{2} m^{2} x^{3}+2 A a b \,m^{3} x +7 A \,b^{2} m^{2} x^{2}+B \,a^{2} m^{3} x +14 B a b \,m^{2} x^{2}+11 B \,b^{2} m \,x^{3}+A \,a^{2} m^{3}+16 A a b \,m^{2} x +14 A \,b^{2} m \,x^{2}+8 B \,a^{2} m^{2} x +28 B a b m \,x^{2}+6 x^{3} B \,b^{2}+9 A \,a^{2} m^{2}+38 A a b m x +8 x^{2} b^{2} A +19 B \,a^{2} m x +16 B a \,x^{2} b +26 a^{2} A m +24 a b A x +12 a^{2} B x +24 a^{2} A \right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(247\)
risch	\(\frac {\left (g x \right )^{m} \left (B \,b^{2} m^{3} x^{3}+A \,b^{2} m^{3} x^{2}+2 B a b \,m^{3} x^{2}+6 B \,b^{2} m^{2} x^{3}+2 A a b \,m^{3} x +7 A \,b^{2} m^{2} x^{2}+B \,a^{2} m^{3} x +14 B a b \,m^{2} x^{2}+11 B \,b^{2} m \,x^{3}+A \,a^{2} m^{3}+16 A a b \,m^{2} x +14 A \,b^{2} m \,x^{2}+8 B \,a^{2} m^{2} x +28 B a b m \,x^{2}+6 x^{3} B \,b^{2}+9 A \,a^{2} m^{2}+38 A a b m x +8 x^{2} b^{2} A +19 B \,a^{2} m x +16 B a \,x^{2} b +26 a^{2} A m +24 a b A x +12 a^{2} B x +24 a^{2} A \right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(247\)
orering	\(\frac {\left (B \,b^{2} m^{3} x^{3}+A \,b^{2} m^{3} x^{2}+2 B a b \,m^{3} x^{2}+6 B \,b^{2} m^{2} x^{3}+2 A a b \,m^{3} x +7 A \,b^{2} m^{2} x^{2}+B \,a^{2} m^{3} x +14 B a b \,m^{2} x^{2}+11 B \,b^{2} m \,x^{3}+A \,a^{2} m^{3}+16 A a b \,m^{2} x +14 A \,b^{2} m \,x^{2}+8 B \,a^{2} m^{2} x +28 B a b m \,x^{2}+6 x^{3} B \,b^{2}+9 A \,a^{2} m^{2}+38 A a b m x +8 x^{2} b^{2} A +19 B \,a^{2} m x +16 B a \,x^{2} b +26 a^{2} A m +24 a b A x +12 a^{2} B x +24 a^{2} A \right ) x \left (g x \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \left (b x +a \right )^{2}}\)	\(270\)
parallelrisch	\(\frac {26 A x \left (g x \right )^{m} a^{2} m +7 A \,x^{3} \left (g x \right )^{m} b^{2} m^{2}+11 B \,x^{4} \left (g x \right )^{m} b^{2} m +B \,x^{2} \left (g x \right )^{m} a^{2} m^{3}+14 A \,x^{3} \left (g x \right )^{m} b^{2} m +A x \left (g x \right )^{m} a^{2} m^{3}+9 A x \left (g x \right )^{m} a^{2} m^{2}+16 B \,x^{3} \left (g x \right )^{m} a b +19 B \,x^{2} \left (g x \right )^{m} a^{2} m +24 A \,x^{2} \left (g x \right )^{m} a b +B \,x^{4} \left (g x \right )^{m} b^{2} m^{3}+A \,x^{3} \left (g x \right )^{m} b^{2} m^{3}+6 B \,x^{4} \left (g x \right )^{m} b^{2} m^{2}+8 B \,x^{2} \left (g x \right )^{m} a^{2} m^{2}+12 B \,x^{2} \left (g x \right )^{m} a^{2}+24 A x \left (g x \right )^{m} a^{2}+2 B \,x^{3} \left (g x \right )^{m} a b \,m^{3}+2 A \,x^{2} \left (g x \right )^{m} a b \,m^{3}+14 B \,x^{3} \left (g x \right )^{m} a b \,m^{2}+16 A \,x^{2} \left (g x \right )^{m} a b \,m^{2}+28 B \,x^{3} \left (g x \right )^{m} a b m +38 A \,x^{2} \left (g x \right )^{m} a b m +6 B \,x^{4} \left (g x \right )^{m} b^{2}+8 A \,x^{3} \left (g x \right )^{m} b^{2}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(381\)

Input:

int((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)
 

Output:

B*b^2/(4+m)*x^4*exp(m*ln(g*x))+a*(2*A*b+B*a)/(2+m)*x^2*exp(m*ln(g*x))+a^2* 
A/(1+m)*x*exp(m*ln(g*x))+b*(A*b+2*B*a)/(3+m)*x^3*exp(m*ln(g*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (91) = 182\).

Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.38 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {{\left ({\left (B b^{2} m^{3} + 6 \, B b^{2} m^{2} + 11 \, B b^{2} m + 6 \, B b^{2}\right )} x^{4} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 16 \, B a b + 8 \, A b^{2} + 7 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 14 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 12 \, B a^{2} + 24 \, A a b + 8 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 19 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} + {\left (A a^{2} m^{3} + 9 \, A a^{2} m^{2} + 26 \, A a^{2} m + 24 \, A a^{2}\right )} x\right )} \left (g x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \] Input:

integrate((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")
 

Output:

((B*b^2*m^3 + 6*B*b^2*m^2 + 11*B*b^2*m + 6*B*b^2)*x^4 + ((2*B*a*b + A*b^2) 
*m^3 + 16*B*a*b + 8*A*b^2 + 7*(2*B*a*b + A*b^2)*m^2 + 14*(2*B*a*b + A*b^2) 
*m)*x^3 + ((B*a^2 + 2*A*a*b)*m^3 + 12*B*a^2 + 24*A*a*b + 8*(B*a^2 + 2*A*a* 
b)*m^2 + 19*(B*a^2 + 2*A*a*b)*m)*x^2 + (A*a^2*m^3 + 9*A*a^2*m^2 + 26*A*a^2 
*m + 24*A*a^2)*x)*(g*x)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (82) = 164\).

Time = 0.33 (sec) , antiderivative size = 1073, normalized size of antiderivative = 11.79 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate((g*x)**m*(B*x+A)*(b**2*x**2+2*a*b*x+a**2),x)
 

Output:

Piecewise(((-A*a**2/(3*x**3) - A*a*b/x**2 - A*b**2/x - B*a**2/(2*x**2) - 2 
*B*a*b/x + B*b**2*log(x))/g**4, Eq(m, -4)), ((-A*a**2/(2*x**2) - 2*A*a*b/x 
 + A*b**2*log(x) - B*a**2/x + 2*B*a*b*log(x) + B*b**2*x)/g**3, Eq(m, -3)), 
 ((-A*a**2/x + 2*A*a*b*log(x) + A*b**2*x + B*a**2*log(x) + 2*B*a*b*x + B*b 
**2*x**2/2)/g**2, Eq(m, -2)), ((A*a**2*log(x) + 2*A*a*b*x + A*b**2*x**2/2 
+ B*a**2*x + B*a*b*x**2 + B*b**2*x**3/3)/g, Eq(m, -1)), (A*a**2*m**3*x*(g* 
x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 9*A*a**2*m**2*x*(g*x)**m/(m 
**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*A*a**2*m*x*(g*x)**m/(m**4 + 10*m 
**3 + 35*m**2 + 50*m + 24) + 24*A*a**2*x*(g*x)**m/(m**4 + 10*m**3 + 35*m** 
2 + 50*m + 24) + 2*A*a*b*m**3*x**2*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50 
*m + 24) + 16*A*a*b*m**2*x**2*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 
24) + 38*A*a*b*m*x**2*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 24 
*A*a*b*x**2*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + A*b**2*m**3* 
x**3*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 7*A*b**2*m**2*x**3* 
(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 14*A*b**2*m*x**3*(g*x)** 
m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*A*b**2*x**3*(g*x)**m/(m**4 + 
10*m**3 + 35*m**2 + 50*m + 24) + B*a**2*m**3*x**2*(g*x)**m/(m**4 + 10*m**3 
 + 35*m**2 + 50*m + 24) + 8*B*a**2*m**2*x**2*(g*x)**m/(m**4 + 10*m**3 + 35 
*m**2 + 50*m + 24) + 19*B*a**2*m*x**2*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 
 50*m + 24) + 12*B*a**2*x**2*(g*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m ...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {B b^{2} g^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, B a b g^{m} x^{3} x^{m}}{m + 3} + \frac {A b^{2} g^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{2} g^{m} x^{2} x^{m}}{m + 2} + \frac {2 \, A a b g^{m} x^{2} x^{m}}{m + 2} + \frac {\left (g x\right )^{m + 1} A a^{2}}{g {\left (m + 1\right )}} \] Input:

integrate((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")
 

Output:

B*b^2*g^m*x^4*x^m/(m + 4) + 2*B*a*b*g^m*x^3*x^m/(m + 3) + A*b^2*g^m*x^3*x^ 
m/(m + 3) + B*a^2*g^m*x^2*x^m/(m + 2) + 2*A*a*b*g^m*x^2*x^m/(m + 2) + (g*x 
)^(m + 1)*A*a^2/(g*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (91) = 182\).

Time = 0.26 (sec) , antiderivative size = 380, normalized size of antiderivative = 4.18 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {\left (g x\right )^{m} B b^{2} m^{3} x^{4} + 2 \, \left (g x\right )^{m} B a b m^{3} x^{3} + \left (g x\right )^{m} A b^{2} m^{3} x^{3} + 6 \, \left (g x\right )^{m} B b^{2} m^{2} x^{4} + \left (g x\right )^{m} B a^{2} m^{3} x^{2} + 2 \, \left (g x\right )^{m} A a b m^{3} x^{2} + 14 \, \left (g x\right )^{m} B a b m^{2} x^{3} + 7 \, \left (g x\right )^{m} A b^{2} m^{2} x^{3} + 11 \, \left (g x\right )^{m} B b^{2} m x^{4} + \left (g x\right )^{m} A a^{2} m^{3} x + 8 \, \left (g x\right )^{m} B a^{2} m^{2} x^{2} + 16 \, \left (g x\right )^{m} A a b m^{2} x^{2} + 28 \, \left (g x\right )^{m} B a b m x^{3} + 14 \, \left (g x\right )^{m} A b^{2} m x^{3} + 6 \, \left (g x\right )^{m} B b^{2} x^{4} + 9 \, \left (g x\right )^{m} A a^{2} m^{2} x + 19 \, \left (g x\right )^{m} B a^{2} m x^{2} + 38 \, \left (g x\right )^{m} A a b m x^{2} + 16 \, \left (g x\right )^{m} B a b x^{3} + 8 \, \left (g x\right )^{m} A b^{2} x^{3} + 26 \, \left (g x\right )^{m} A a^{2} m x + 12 \, \left (g x\right )^{m} B a^{2} x^{2} + 24 \, \left (g x\right )^{m} A a b x^{2} + 24 \, \left (g x\right )^{m} A a^{2} x}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \] Input:

integrate((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")
 

Output:

((g*x)^m*B*b^2*m^3*x^4 + 2*(g*x)^m*B*a*b*m^3*x^3 + (g*x)^m*A*b^2*m^3*x^3 + 
 6*(g*x)^m*B*b^2*m^2*x^4 + (g*x)^m*B*a^2*m^3*x^2 + 2*(g*x)^m*A*a*b*m^3*x^2 
 + 14*(g*x)^m*B*a*b*m^2*x^3 + 7*(g*x)^m*A*b^2*m^2*x^3 + 11*(g*x)^m*B*b^2*m 
*x^4 + (g*x)^m*A*a^2*m^3*x + 8*(g*x)^m*B*a^2*m^2*x^2 + 16*(g*x)^m*A*a*b*m^ 
2*x^2 + 28*(g*x)^m*B*a*b*m*x^3 + 14*(g*x)^m*A*b^2*m*x^3 + 6*(g*x)^m*B*b^2* 
x^4 + 9*(g*x)^m*A*a^2*m^2*x + 19*(g*x)^m*B*a^2*m*x^2 + 38*(g*x)^m*A*a*b*m* 
x^2 + 16*(g*x)^m*B*a*b*x^3 + 8*(g*x)^m*A*b^2*x^3 + 26*(g*x)^m*A*a^2*m*x + 
12*(g*x)^m*B*a^2*x^2 + 24*(g*x)^m*A*a*b*x^2 + 24*(g*x)^m*A*a^2*x)/(m^4 + 1 
0*m^3 + 35*m^2 + 50*m + 24)
 

Mupad [B] (verification not implemented)

Time = 11.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.97 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx={\left (g\,x\right )}^m\,\left (\frac {B\,b^2\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {A\,a^2\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,x^2\,\left (2\,A\,b+B\,a\right )\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b\,x^3\,\left (A\,b+2\,B\,a\right )\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \] Input:

int((g*x)^m*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x),x)
 

Output:

(g*x)^m*((B*b^2*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^ 
4 + 24) + (A*a^2*x*(26*m + 9*m^2 + m^3 + 24))/(50*m + 35*m^2 + 10*m^3 + m^ 
4 + 24) + (a*x^2*(2*A*b + B*a)*(19*m + 8*m^2 + m^3 + 12))/(50*m + 35*m^2 + 
 10*m^3 + m^4 + 24) + (b*x^3*(A*b + 2*B*a)*(14*m + 7*m^2 + m^3 + 8))/(50*m 
 + 35*m^2 + 10*m^3 + m^4 + 24))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.88 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {x^{m} g^{m} x \left (b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}+3 a^{2} b \,m^{3} x +21 a \,b^{2} m^{2} x^{2}+11 b^{3} m \,x^{3}+a^{3} m^{3}+24 a^{2} b \,m^{2} x +42 a \,b^{2} m \,x^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}+57 a^{2} b m x +24 a \,b^{2} x^{2}+26 a^{3} m +36 a^{2} b x +24 a^{3}\right )}{m^{4}+10 m^{3}+35 m^{2}+50 m +24} \] Input:

int((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x)
 

Output:

(x**m*g**m*x*(a**3*m**3 + 9*a**3*m**2 + 26*a**3*m + 24*a**3 + 3*a**2*b*m** 
3*x + 24*a**2*b*m**2*x + 57*a**2*b*m*x + 36*a**2*b*x + 3*a*b**2*m**3*x**2 
+ 21*a*b**2*m**2*x**2 + 42*a*b**2*m*x**2 + 24*a*b**2*x**2 + b**3*m**3*x**3 
 + 6*b**3*m**2*x**3 + 11*b**3*m*x**3 + 6*b**3*x**3))/(m**4 + 10*m**3 + 35* 
m**2 + 50*m + 24)