\(\int (g x)^m (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\) [465]

Optimal result

Integrand size = 29, antiderivative size = 219 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {a^6 A (g x)^{1+m}}{g (1+m)}+\frac {a^5 (6 A b+a B) (g x)^{2+m}}{g^2 (2+m)}+\frac {3 a^4 b (5 A b+2 a B) (g x)^{3+m}}{g^3 (3+m)}+\frac {5 a^3 b^2 (4 A b+3 a B) (g x)^{4+m}}{g^4 (4+m)}+\frac {5 a^2 b^3 (3 A b+4 a B) (g x)^{5+m}}{g^5 (5+m)}+\frac {3 a b^4 (2 A b+5 a B) (g x)^{6+m}}{g^6 (6+m)}+\frac {b^5 (A b+6 a B) (g x)^{7+m}}{g^7 (7+m)}+\frac {b^6 B (g x)^{8+m}}{g^8 (8+m)} \] Output:

a^6*A*(g*x)^(1+m)/g/(1+m)+a^5*(6*A*b+B*a)*(g*x)^(2+m)/g^2/(2+m)+3*a^4*b*(5 
*A*b+2*B*a)*(g*x)^(3+m)/g^3/(3+m)+5*a^3*b^2*(4*A*b+3*B*a)*(g*x)^(4+m)/g^4/ 
(4+m)+5*a^2*b^3*(3*A*b+4*B*a)*(g*x)^(5+m)/g^5/(5+m)+3*a*b^4*(2*A*b+5*B*a)* 
(g*x)^(6+m)/g^6/(6+m)+b^5*(A*b+6*B*a)*(g*x)^(7+m)/g^7/(7+m)+b^6*B*(g*x)^(8 
+m)/g^8/(8+m)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.63 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(g x)^m \left (B x (a+b x)^7+(-a B (1+m)+A b (8+m)) x \left (\frac {a^6}{1+m}+\frac {6 a^5 b x}{2+m}+\frac {15 a^4 b^2 x^2}{3+m}+\frac {20 a^3 b^3 x^3}{4+m}+\frac {15 a^2 b^4 x^4}{5+m}+\frac {6 a b^5 x^5}{6+m}+\frac {b^6 x^6}{7+m}\right )\right )}{b (8+m)} \] Input:

Integrate[(g*x)^m*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

((g*x)^m*(B*x*(a + b*x)^7 + (-(a*B*(1 + m)) + A*b*(8 + m))*x*(a^6/(1 + m) 
+ (6*a^5*b*x)/(2 + m) + (15*a^4*b^2*x^2)/(3 + m) + (20*a^3*b^3*x^3)/(4 + m 
) + (15*a^2*b^4*x^4)/(5 + m) + (6*a*b^5*x^5)/(6 + m) + (b^6*x^6)/(7 + m))) 
)/(b*(8 + m))
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x) (g x)^m \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int b^6 (g x)^m (a+b x)^6 (A+B x)dx}{b^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \int (a+b x)^6 (A+B x) (g x)^mdx\)

\(\Big \downarrow \) 85

\(\displaystyle \int \left (a^6 A (g x)^m+\frac {a^5 (g x)^{m+1} (a B+6 A b)}{g}+\frac {3 a^4 b (g x)^{m+2} (2 a B+5 A b)}{g^2}+\frac {5 a^3 b^2 (g x)^{m+3} (3 a B+4 A b)}{g^3}+\frac {5 a^2 b^3 (g x)^{m+4} (4 a B+3 A b)}{g^4}+\frac {b^5 (g x)^{m+6} (6 a B+A b)}{g^6}+\frac {3 a b^4 (g x)^{m+5} (5 a B+2 A b)}{g^5}+\frac {b^6 B (g x)^{m+7}}{g^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^6 A (g x)^{m+1}}{g (m+1)}+\frac {a^5 (g x)^{m+2} (a B+6 A b)}{g^2 (m+2)}+\frac {3 a^4 b (g x)^{m+3} (2 a B+5 A b)}{g^3 (m+3)}+\frac {5 a^3 b^2 (g x)^{m+4} (3 a B+4 A b)}{g^4 (m+4)}+\frac {5 a^2 b^3 (g x)^{m+5} (4 a B+3 A b)}{g^5 (m+5)}+\frac {b^5 (g x)^{m+7} (6 a B+A b)}{g^7 (m+7)}+\frac {3 a b^4 (g x)^{m+6} (5 a B+2 A b)}{g^6 (m+6)}+\frac {b^6 B (g x)^{m+8}}{g^8 (m+8)}\)

Input:

Int[(g*x)^m*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

(a^6*A*(g*x)^(1 + m))/(g*(1 + m)) + (a^5*(6*A*b + a*B)*(g*x)^(2 + m))/(g^2 
*(2 + m)) + (3*a^4*b*(5*A*b + 2*a*B)*(g*x)^(3 + m))/(g^3*(3 + m)) + (5*a^3 
*b^2*(4*A*b + 3*a*B)*(g*x)^(4 + m))/(g^4*(4 + m)) + (5*a^2*b^3*(3*A*b + 4* 
a*B)*(g*x)^(5 + m))/(g^5*(5 + m)) + (3*a*b^4*(2*A*b + 5*a*B)*(g*x)^(6 + m) 
)/(g^6*(6 + m)) + (b^5*(A*b + 6*a*B)*(g*x)^(7 + m))/(g^7*(7 + m)) + (b^6*B 
*(g*x)^(8 + m))/(g^8*(8 + 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 = 1.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {A \,a^{6} x \,{\mathrm e}^{m \ln \left (g x \right )}}{1+m}+\frac {B \,b^{6} x^{8} {\mathrm e}^{m \ln \left (g x \right )}}{8+m}+\frac {a^{5} \left (6 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (g x \right )}}{2+m}+\frac {b^{5} \left (A b +6 B a \right ) x^{7} {\mathrm e}^{m \ln \left (g x \right )}}{7+m}+\frac {3 a \,b^{4} \left (2 A b +5 B a \right ) x^{6} {\mathrm e}^{m \ln \left (g x \right )}}{6+m}+\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right ) x^{5} {\mathrm e}^{m \ln \left (g x \right )}}{5+m}+\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (g x \right )}}{4+m}+\frac {3 a^{4} b \left (5 A b +2 B a \right ) x^{3} {\mathrm e}^{m \ln \left (g x \right )}}{3+m}\)	\(218\)
gosper	\(\text {Expression too large to display}\)	\(1439\)
risch	\(\text {Expression too large to display}\)	\(1439\)
orering	\(\text {Expression too large to display}\)	\(1464\)
parallelrisch	\(\text {Expression too large to display}\)	\(2033\)

Input:

int((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

A*a^6/(1+m)*x*exp(m*ln(g*x))+B*b^6/(8+m)*x^8*exp(m*ln(g*x))+a^5*(6*A*b+B*a 
)/(2+m)*x^2*exp(m*ln(g*x))+b^5*(A*b+6*B*a)/(7+m)*x^7*exp(m*ln(g*x))+3*a*b^ 
4*(2*A*b+5*B*a)/(6+m)*x^6*exp(m*ln(g*x))+5*a^2*b^3*(3*A*b+4*B*a)/(5+m)*x^5 
*exp(m*ln(g*x))+5*a^3*b^2*(4*A*b+3*B*a)/(4+m)*x^4*exp(m*ln(g*x))+3*a^4*b*( 
5*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. 1193 vs. \(2 (219) = 438\).

Time = 0.09 (sec) , antiderivative size = 1193, normalized size of antiderivative = 5.45 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

((B*b^6*m^7 + 28*B*b^6*m^6 + 322*B*b^6*m^5 + 1960*B*b^6*m^4 + 6769*B*b^6*m 
^3 + 13132*B*b^6*m^2 + 13068*B*b^6*m + 5040*B*b^6)*x^8 + ((6*B*a*b^5 + A*b 
^6)*m^7 + 34560*B*a*b^5 + 5760*A*b^6 + 29*(6*B*a*b^5 + A*b^6)*m^6 + 343*(6 
*B*a*b^5 + A*b^6)*m^5 + 2135*(6*B*a*b^5 + A*b^6)*m^4 + 7504*(6*B*a*b^5 + A 
*b^6)*m^3 + 14756*(6*B*a*b^5 + A*b^6)*m^2 + 14832*(6*B*a*b^5 + A*b^6)*m)*x 
^7 + 3*((5*B*a^2*b^4 + 2*A*a*b^5)*m^7 + 33600*B*a^2*b^4 + 13440*A*a*b^5 + 
30*(5*B*a^2*b^4 + 2*A*a*b^5)*m^6 + 366*(5*B*a^2*b^4 + 2*A*a*b^5)*m^5 + 234 
0*(5*B*a^2*b^4 + 2*A*a*b^5)*m^4 + 8409*(5*B*a^2*b^4 + 2*A*a*b^5)*m^3 + 168 
30*(5*B*a^2*b^4 + 2*A*a*b^5)*m^2 + 17144*(5*B*a^2*b^4 + 2*A*a*b^5)*m)*x^6 
+ 5*((4*B*a^3*b^3 + 3*A*a^2*b^4)*m^7 + 32256*B*a^3*b^3 + 24192*A*a^2*b^4 + 
 31*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^6 + 391*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^5 
+ 2581*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^4 + 9544*(4*B*a^3*b^3 + 3*A*a^2*b^4)* 
m^3 + 19564*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^2 + 20304*(4*B*a^3*b^3 + 3*A*a^2 
*b^4)*m)*x^5 + 5*((3*B*a^4*b^2 + 4*A*a^3*b^3)*m^7 + 30240*B*a^4*b^2 + 4032 
0*A*a^3*b^3 + 32*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^6 + 418*(3*B*a^4*b^2 + 4*A* 
a^3*b^3)*m^5 + 2864*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^4 + 10993*(3*B*a^4*b^2 + 
 4*A*a^3*b^3)*m^3 + 23312*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^2 + 24876*(3*B*a^4 
*b^2 + 4*A*a^3*b^3)*m)*x^4 + 3*((2*B*a^5*b + 5*A*a^4*b^2)*m^7 + 26880*B*a^ 
5*b + 67200*A*a^4*b^2 + 33*(2*B*a^5*b + 5*A*a^4*b^2)*m^6 + 447*(2*B*a^5*b 
+ 5*A*a^4*b^2)*m^5 + 3195*(2*B*a^5*b + 5*A*a^4*b^2)*m^4 + 12864*(2*B*a^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7961 vs. \(2 (211) = 422\).

Time = 0.87 (sec) , antiderivative size = 7961, normalized size of antiderivative = 36.35 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((g*x)**m*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

Piecewise(((-A*a**6/(7*x**7) - A*a**5*b/x**6 - 3*A*a**4*b**2/x**5 - 5*A*a* 
*3*b**3/x**4 - 5*A*a**2*b**4/x**3 - 3*A*a*b**5/x**2 - A*b**6/x - B*a**6/(6 
*x**6) - 6*B*a**5*b/(5*x**5) - 15*B*a**4*b**2/(4*x**4) - 20*B*a**3*b**3/(3 
*x**3) - 15*B*a**2*b**4/(2*x**2) - 6*B*a*b**5/x + B*b**6*log(x))/g**8, Eq( 
m, -8)), ((-A*a**6/(6*x**6) - 6*A*a**5*b/(5*x**5) - 15*A*a**4*b**2/(4*x**4 
) - 20*A*a**3*b**3/(3*x**3) - 15*A*a**2*b**4/(2*x**2) - 6*A*a*b**5/x + A*b 
**6*log(x) - B*a**6/(5*x**5) - 3*B*a**5*b/(2*x**4) - 5*B*a**4*b**2/x**3 - 
10*B*a**3*b**3/x**2 - 15*B*a**2*b**4/x + 6*B*a*b**5*log(x) + B*b**6*x)/g** 
7, Eq(m, -7)), ((-A*a**6/(5*x**5) - 3*A*a**5*b/(2*x**4) - 5*A*a**4*b**2/x* 
*3 - 10*A*a**3*b**3/x**2 - 15*A*a**2*b**4/x + 6*A*a*b**5*log(x) + A*b**6*x 
 - B*a**6/(4*x**4) - 2*B*a**5*b/x**3 - 15*B*a**4*b**2/(2*x**2) - 20*B*a**3 
*b**3/x + 15*B*a**2*b**4*log(x) + 6*B*a*b**5*x + B*b**6*x**2/2)/g**6, Eq(m 
, -6)), ((-A*a**6/(4*x**4) - 2*A*a**5*b/x**3 - 15*A*a**4*b**2/(2*x**2) - 2 
0*A*a**3*b**3/x + 15*A*a**2*b**4*log(x) + 6*A*a*b**5*x + A*b**6*x**2/2 - B 
*a**6/(3*x**3) - 3*B*a**5*b/x**2 - 15*B*a**4*b**2/x + 20*B*a**3*b**3*log(x 
) + 15*B*a**2*b**4*x + 3*B*a*b**5*x**2 + B*b**6*x**3/3)/g**5, Eq(m, -5)), 
((-A*a**6/(3*x**3) - 3*A*a**5*b/x**2 - 15*A*a**4*b**2/x + 20*A*a**3*b**3*l 
og(x) + 15*A*a**2*b**4*x + 3*A*a*b**5*x**2 + A*b**6*x**3/3 - B*a**6/(2*x** 
2) - 6*B*a**5*b/x + 15*B*a**4*b**2*log(x) + 20*B*a**3*b**3*x + 15*B*a**2*b 
**4*x**2/2 + 2*B*a*b**5*x**3 + B*b**6*x**4/4)/g**4, Eq(m, -4)), ((-A*a*...
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.37 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {B b^{6} g^{m} x^{8} x^{m}}{m + 8} + \frac {6 \, B a b^{5} g^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{6} g^{m} x^{7} x^{m}}{m + 7} + \frac {15 \, B a^{2} b^{4} g^{m} x^{6} x^{m}}{m + 6} + \frac {6 \, A a b^{5} g^{m} x^{6} x^{m}}{m + 6} + \frac {20 \, B a^{3} b^{3} g^{m} x^{5} x^{m}}{m + 5} + \frac {15 \, A a^{2} b^{4} g^{m} x^{5} x^{m}}{m + 5} + \frac {15 \, B a^{4} b^{2} g^{m} x^{4} x^{m}}{m + 4} + \frac {20 \, A a^{3} b^{3} g^{m} x^{4} x^{m}}{m + 4} + \frac {6 \, B a^{5} b g^{m} x^{3} x^{m}}{m + 3} + \frac {15 \, A a^{4} b^{2} g^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{6} g^{m} x^{2} x^{m}}{m + 2} + \frac {6 \, A a^{5} b g^{m} x^{2} x^{m}}{m + 2} + \frac {\left (g x\right )^{m + 1} A a^{6}}{g {\left (m + 1\right )}} \] Input:

integrate((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
 

Output:

B*b^6*g^m*x^8*x^m/(m + 8) + 6*B*a*b^5*g^m*x^7*x^m/(m + 7) + A*b^6*g^m*x^7* 
x^m/(m + 7) + 15*B*a^2*b^4*g^m*x^6*x^m/(m + 6) + 6*A*a*b^5*g^m*x^6*x^m/(m 
+ 6) + 20*B*a^3*b^3*g^m*x^5*x^m/(m + 5) + 15*A*a^2*b^4*g^m*x^5*x^m/(m + 5) 
 + 15*B*a^4*b^2*g^m*x^4*x^m/(m + 4) + 20*A*a^3*b^3*g^m*x^4*x^m/(m + 4) + 6 
*B*a^5*b*g^m*x^3*x^m/(m + 3) + 15*A*a^4*b^2*g^m*x^3*x^m/(m + 3) + B*a^6*g^ 
m*x^2*x^m/(m + 2) + 6*A*a^5*b*g^m*x^2*x^m/(m + 2) + (g*x)^(m + 1)*A*a^6/(g 
*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (219) = 438\).

Time = 0.28 (sec) , antiderivative size = 2032, normalized size of antiderivative = 9.28 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

((g*x)^m*B*b^6*m^7*x^8 + 6*(g*x)^m*B*a*b^5*m^7*x^7 + (g*x)^m*A*b^6*m^7*x^7 
 + 28*(g*x)^m*B*b^6*m^6*x^8 + 15*(g*x)^m*B*a^2*b^4*m^7*x^6 + 6*(g*x)^m*A*a 
*b^5*m^7*x^6 + 174*(g*x)^m*B*a*b^5*m^6*x^7 + 29*(g*x)^m*A*b^6*m^6*x^7 + 32 
2*(g*x)^m*B*b^6*m^5*x^8 + 20*(g*x)^m*B*a^3*b^3*m^7*x^5 + 15*(g*x)^m*A*a^2* 
b^4*m^7*x^5 + 450*(g*x)^m*B*a^2*b^4*m^6*x^6 + 180*(g*x)^m*A*a*b^5*m^6*x^6 
+ 2058*(g*x)^m*B*a*b^5*m^5*x^7 + 343*(g*x)^m*A*b^6*m^5*x^7 + 1960*(g*x)^m* 
B*b^6*m^4*x^8 + 15*(g*x)^m*B*a^4*b^2*m^7*x^4 + 20*(g*x)^m*A*a^3*b^3*m^7*x^ 
4 + 620*(g*x)^m*B*a^3*b^3*m^6*x^5 + 465*(g*x)^m*A*a^2*b^4*m^6*x^5 + 5490*( 
g*x)^m*B*a^2*b^4*m^5*x^6 + 2196*(g*x)^m*A*a*b^5*m^5*x^6 + 12810*(g*x)^m*B* 
a*b^5*m^4*x^7 + 2135*(g*x)^m*A*b^6*m^4*x^7 + 6769*(g*x)^m*B*b^6*m^3*x^8 + 
6*(g*x)^m*B*a^5*b*m^7*x^3 + 15*(g*x)^m*A*a^4*b^2*m^7*x^3 + 480*(g*x)^m*B*a 
^4*b^2*m^6*x^4 + 640*(g*x)^m*A*a^3*b^3*m^6*x^4 + 7820*(g*x)^m*B*a^3*b^3*m^ 
5*x^5 + 5865*(g*x)^m*A*a^2*b^4*m^5*x^5 + 35100*(g*x)^m*B*a^2*b^4*m^4*x^6 + 
 14040*(g*x)^m*A*a*b^5*m^4*x^6 + 45024*(g*x)^m*B*a*b^5*m^3*x^7 + 7504*(g*x 
)^m*A*b^6*m^3*x^7 + 13132*(g*x)^m*B*b^6*m^2*x^8 + (g*x)^m*B*a^6*m^7*x^2 + 
6*(g*x)^m*A*a^5*b*m^7*x^2 + 198*(g*x)^m*B*a^5*b*m^6*x^3 + 495*(g*x)^m*A*a^ 
4*b^2*m^6*x^3 + 6270*(g*x)^m*B*a^4*b^2*m^5*x^4 + 8360*(g*x)^m*A*a^3*b^3*m^ 
5*x^4 + 51620*(g*x)^m*B*a^3*b^3*m^4*x^5 + 38715*(g*x)^m*A*a^2*b^4*m^4*x^5 
+ 126135*(g*x)^m*B*a^2*b^4*m^3*x^6 + 50454*(g*x)^m*A*a*b^5*m^3*x^6 + 88536 
*(g*x)^m*B*a*b^5*m^2*x^7 + 14756*(g*x)^m*A*b^6*m^2*x^7 + 13068*(g*x)^m*...
 

Mupad [B] (verification not implemented)

Time = 11.54 (sec) , antiderivative size = 745, normalized size of antiderivative = 3.40 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {A\,a^6\,x\,{\left (g\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,b^6\,x^8\,{\left (g\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^5\,x^2\,{\left (g\,x\right )}^m\,\left (6\,A\,b+B\,a\right )\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b^5\,x^7\,{\left (g\,x\right )}^m\,\left (A\,b+6\,B\,a\right )\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,a^2\,b^3\,x^5\,{\left (g\,x\right )}^m\,\left (3\,A\,b+4\,B\,a\right )\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,a^3\,b^2\,x^4\,{\left (g\,x\right )}^m\,\left (4\,A\,b+3\,B\,a\right )\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a\,b^4\,x^6\,{\left (g\,x\right )}^m\,\left (2\,A\,b+5\,B\,a\right )\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^4\,b\,x^3\,{\left (g\,x\right )}^m\,\left (5\,A\,b+2\,B\,a\right )\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \] Input:

int((g*x)^m*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
 

Output:

(A*a^6*x*(g*x)^m*(69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 3 
5*m^6 + m^7 + 40320))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 453 
6*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (B*b^6*x^8*(g*x)^m*(13068*m + 13 
132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))/(109584*m 
+ 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 
 40320) + (a^5*x^2*(g*x)^m*(6*A*b + B*a)*(44712*m + 36706*m^2 + 15289*m^3 
+ 3580*m^4 + 478*m^5 + 34*m^6 + m^7 + 20160))/(109584*m + 118124*m^2 + 672 
84*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (b^5*x^7 
*(g*x)^m*(A*b + 6*B*a)*(14832*m + 14756*m^2 + 7504*m^3 + 2135*m^4 + 343*m^ 
5 + 29*m^6 + m^7 + 5760))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 
 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (5*a^2*b^3*x^5*(g*x)^m*(3*A* 
b + 4*B*a)*(20304*m + 19564*m^2 + 9544*m^3 + 2581*m^4 + 391*m^5 + 31*m^6 + 
 m^7 + 8064))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 
546*m^6 + 36*m^7 + m^8 + 40320) + (5*a^3*b^2*x^4*(g*x)^m*(4*A*b + 3*B*a)*( 
24876*m + 23312*m^2 + 10993*m^3 + 2864*m^4 + 418*m^5 + 32*m^6 + m^7 + 1008 
0))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 
36*m^7 + m^8 + 40320) + (3*a*b^4*x^6*(g*x)^m*(2*A*b + 5*B*a)*(17144*m + 16 
830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m^6 + m^7 + 6720))/(109584*m 
+ 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 
 40320) + (3*a^4*b*x^3*(g*x)^m*(5*A*b + 2*B*a)*(32048*m + 28692*m^2 + 1...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 783, normalized size of antiderivative = 3.58 \[ \int (g x)^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x^{m} g^{m} x \left (b^{7} m^{7} x^{7}+7 a \,b^{6} m^{7} x^{6}+28 b^{7} m^{6} x^{7}+21 a^{2} b^{5} m^{7} x^{5}+203 a \,b^{6} m^{6} x^{6}+322 b^{7} m^{5} x^{7}+35 a^{3} b^{4} m^{7} x^{4}+630 a^{2} b^{5} m^{6} x^{5}+2401 a \,b^{6} m^{5} x^{6}+1960 b^{7} m^{4} x^{7}+35 a^{4} b^{3} m^{7} x^{3}+1085 a^{3} b^{4} m^{6} x^{4}+7686 a^{2} b^{5} m^{5} x^{5}+14945 a \,b^{6} m^{4} x^{6}+6769 b^{7} m^{3} x^{7}+21 a^{5} b^{2} m^{7} x^{2}+1120 a^{4} b^{3} m^{6} x^{3}+13685 a^{3} b^{4} m^{5} x^{4}+49140 a^{2} b^{5} m^{4} x^{5}+52528 a \,b^{6} m^{3} x^{6}+13132 b^{7} m^{2} x^{7}+7 a^{6} b \,m^{7} x +693 a^{5} b^{2} m^{6} x^{2}+14630 a^{4} b^{3} m^{5} x^{3}+90335 a^{3} b^{4} m^{4} x^{4}+176589 a^{2} b^{5} m^{3} x^{5}+103292 a \,b^{6} m^{2} x^{6}+13068 b^{7} m \,x^{7}+a^{7} m^{7}+238 a^{6} b \,m^{6} x +9387 a^{5} b^{2} m^{5} x^{2}+100240 a^{4} b^{3} m^{4} x^{3}+334040 a^{3} b^{4} m^{3} x^{4}+353430 a^{2} b^{5} m^{2} x^{5}+103824 a \,b^{6} m \,x^{6}+5040 b^{7} x^{7}+35 a^{7} m^{6}+3346 a^{6} b \,m^{5} x +67095 a^{5} b^{2} m^{4} x^{2}+384755 a^{4} b^{3} m^{3} x^{3}+684740 a^{3} b^{4} m^{2} x^{4}+360024 a^{2} b^{5} m \,x^{5}+40320 a \,b^{6} x^{6}+511 a^{7} m^{5}+25060 a^{6} b \,m^{4} x +270144 a^{5} b^{2} m^{3} x^{2}+815920 a^{4} b^{3} m^{2} x^{3}+710640 a^{3} b^{4} m \,x^{4}+141120 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+107023 a^{6} b \,m^{3} x +602532 a^{5} b^{2} m^{2} x^{2}+870660 a^{4} b^{3} m \,x^{3}+282240 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+256942 a^{6} b \,m^{2} x +673008 a^{5} b^{2} m \,x^{2}+352800 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+312984 a^{6} b m x +282240 a^{5} b^{2} x^{2}+69264 a^{7} m +141120 a^{6} b x +40320 a^{7}\right )}{m^{8}+36 m^{7}+546 m^{6}+4536 m^{5}+22449 m^{4}+67284 m^{3}+118124 m^{2}+109584 m +40320} \] Input:

int((g*x)^m*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(x**m*g**m*x*(a**7*m**7 + 35*a**7*m**6 + 511*a**7*m**5 + 4025*a**7*m**4 + 
18424*a**7*m**3 + 48860*a**7*m**2 + 69264*a**7*m + 40320*a**7 + 7*a**6*b*m 
**7*x + 238*a**6*b*m**6*x + 3346*a**6*b*m**5*x + 25060*a**6*b*m**4*x + 107 
023*a**6*b*m**3*x + 256942*a**6*b*m**2*x + 312984*a**6*b*m*x + 141120*a**6 
*b*x + 21*a**5*b**2*m**7*x**2 + 693*a**5*b**2*m**6*x**2 + 9387*a**5*b**2*m 
**5*x**2 + 67095*a**5*b**2*m**4*x**2 + 270144*a**5*b**2*m**3*x**2 + 602532 
*a**5*b**2*m**2*x**2 + 673008*a**5*b**2*m*x**2 + 282240*a**5*b**2*x**2 + 3 
5*a**4*b**3*m**7*x**3 + 1120*a**4*b**3*m**6*x**3 + 14630*a**4*b**3*m**5*x* 
*3 + 100240*a**4*b**3*m**4*x**3 + 384755*a**4*b**3*m**3*x**3 + 815920*a**4 
*b**3*m**2*x**3 + 870660*a**4*b**3*m*x**3 + 352800*a**4*b**3*x**3 + 35*a** 
3*b**4*m**7*x**4 + 1085*a**3*b**4*m**6*x**4 + 13685*a**3*b**4*m**5*x**4 + 
90335*a**3*b**4*m**4*x**4 + 334040*a**3*b**4*m**3*x**4 + 684740*a**3*b**4* 
m**2*x**4 + 710640*a**3*b**4*m*x**4 + 282240*a**3*b**4*x**4 + 21*a**2*b**5 
*m**7*x**5 + 630*a**2*b**5*m**6*x**5 + 7686*a**2*b**5*m**5*x**5 + 49140*a* 
*2*b**5*m**4*x**5 + 176589*a**2*b**5*m**3*x**5 + 353430*a**2*b**5*m**2*x** 
5 + 360024*a**2*b**5*m*x**5 + 141120*a**2*b**5*x**5 + 7*a*b**6*m**7*x**6 + 
 203*a*b**6*m**6*x**6 + 2401*a*b**6*m**5*x**6 + 14945*a*b**6*m**4*x**6 + 5 
2528*a*b**6*m**3*x**6 + 103292*a*b**6*m**2*x**6 + 103824*a*b**6*m*x**6 + 4 
0320*a*b**6*x**6 + b**7*m**7*x**7 + 28*b**7*m**6*x**7 + 322*b**7*m**5*x**7 
 + 1960*b**7*m**4*x**7 + 6769*b**7*m**3*x**7 + 13132*b**7*m**2*x**7 + 1...