\(\int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx\) [349]

Optimal result

Integrand size = 22, antiderivative size = 99 \[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\frac {2^n 3^{1+2 n} x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )}{1+m}-\frac {2^{1+n} 9^n a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )}{2+m} \] Output:

2^n*3^(1+2*n)*x^(1+m)*hypergeom([-n, 1/2+1/2*m],[3/2+1/2*m],4/9*a^2*x^2)/( 
1+m)-2^(1+n)*9^n*a*x^(2+m)*hypergeom([-n, 1+1/2*m],[2+1/2*m],4/9*a^2*x^2)/ 
(2+m)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16 \[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\frac {x^{1+m} \left (9-4 a^2 x^2\right )^n \left (\frac {1}{2}-\frac {2 a^2 x^2}{9}\right )^{-n} \left (3 (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )-2 a (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )\right )}{(1+m) (2+m)} \] Input:

Integrate[x^m*(3 - 2*a*x)^(1 + n)*(6 + 4*a*x)^n,x]
 

Output:

(x^(1 + m)*(9 - 4*a^2*x^2)^n*(3*(2 + m)*Hypergeometric2F1[(1 + m)/2, -n, ( 
3 + m)/2, (4*a^2*x^2)/9] - 2*a*(1 + m)*x*Hypergeometric2F1[(2 + m)/2, -n, 
(4 + m)/2, (4*a^2*x^2)/9]))/((1 + m)*(2 + m)*(1/2 - (2*a^2*x^2)/9)^n)
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {92, 135, 278}

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.

\(\displaystyle \int x^m (3-2 a x)^{n+1} (4 a x+6)^n \, dx\)

\(\Big \downarrow \) 92

\(\displaystyle 3 \int x^m (3-2 a x)^n (4 a x+6)^ndx-2 a \int x^{m+1} (3-2 a x)^n (4 a x+6)^ndx\)

\(\Big \downarrow \) 135

\(\displaystyle 3 \int x^m \left (18-8 a^2 x^2\right )^ndx-2 a \int x^{m+1} \left (18-8 a^2 x^2\right )^ndx\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {2^n 3^{2 n+1} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-n,\frac {m+3}{2},\frac {4 a^2 x^2}{9}\right )}{m+1}-\frac {a 2^{n+1} 9^n x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-n,\frac {m+4}{2},\frac {4 a^2 x^2}{9}\right )}{m+2}\)

Input:

Int[x^m*(3 - 2*a*x)^(1 + n)*(6 + 4*a*x)^n,x]
 

Output:

(2^n*3^(1 + 2*n)*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4* 
a^2*x^2)/9])/(1 + m) - (2^(1 + n)*9^n*a*x^(2 + m)*Hypergeometric2F1[(2 + m 
)/2, -n, (4 + m)/2, (4*a^2*x^2)/9])/(2 + m)
 

Defintions of rubi rules used

rule 92

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Simp[a   Int[(a + b*x)^n*(c + d*x)^n*(f*x)^p, x], x] + Simp[b/f   In 
t[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, f, m, 
 n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] &&  !RationalQ[p] && 
!IGtQ[m, 0] && NeQ[m + n + p + 2, 0]
 

rule 135

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, 
 x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 0]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

Maple [F]

\[\int x^{m} \left (-2 a x +3\right )^{1+n} \left (4 a x +6\right )^{n}d x\]

Input:

int(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x)
 

Output:

int(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x)
 

Fricas [F]

\[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n + 1} x^{m} \,d x } \] Input:

integrate(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x, algorithm="fricas")
 

Output:

integral((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m, x)
 

Sympy [F]

\[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=2^{n} \int x^{m} \left (- 2 a x + 3\right )^{n + 1} \left (2 a x + 3\right )^{n}\, dx \] Input:

integrate(x**m*(-2*a*x+3)**(1+n)*(4*a*x+6)**n,x)
 

Output:

2**n*Integral(x**m*(-2*a*x + 3)**(n + 1)*(2*a*x + 3)**n, x)
 

Maxima [F]

\[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n + 1} x^{m} \,d x } \] Input:

integrate(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x, algorithm="maxima")
 

Output:

integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m, x)
 

Giac [F]

\[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n + 1} x^{m} \,d x } \] Input:

integrate(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x, algorithm="giac")
 

Output:

integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\int x^m\,{\left (3-2\,a\,x\right )}^{n+1}\,{\left (4\,a\,x+6\right )}^n \,d x \] Input:

int(x^m*(3 - 2*a*x)^(n + 1)*(4*a*x + 6)^n,x)
 

Output:

int(x^m*(3 - 2*a*x)^(n + 1)*(4*a*x + 6)^n, x)
 

Reduce [F]

\[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx=\text {too large to display} \] Input:

int(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x)
 

Output:

( - 2*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*m**2*x**2 - 8*x**m*(4*a*x 
 + 6)**n*( - 2*a*x + 3)**n*a**2*m*n*x**2 - 2*x**m*(4*a*x + 6)**n*( - 2*a*x 
 + 3)**n*a**2*m*x**2 - 8*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*n**2*x 
**2 - 4*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*n*x**2 + 3*x**m*(4*a*x 
+ 6)**n*( - 2*a*x + 3)**n*a*m**2*x + 12*x**m*(4*a*x + 6)**n*( - 2*a*x + 3) 
**n*a*m*n*x + 6*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a*m*x + 12*x**m*(4*a 
*x + 6)**n*( - 2*a*x + 3)**n*a*n**2*x + 12*x**m*(4*a*x + 6)**n*( - 2*a*x + 
 3)**n*a*n*x + 9*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*m*n + 18*x**m*(4*a* 
x + 6)**n*( - 2*a*x + 3)**n*n**2 + 9*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n 
*n + 81*int((x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n)/(4*a**2*m**3*x**3 + 24 
*a**2*m**2*n*x**3 + 12*a**2*m**2*x**3 + 48*a**2*m*n**2*x**3 + 48*a**2*m*n* 
x**3 + 8*a**2*m*x**3 + 32*a**2*n**3*x**3 + 48*a**2*n**2*x**3 + 16*a**2*n*x 
**3 - 9*m**3*x - 54*m**2*n*x - 27*m**2*x - 108*m*n**2*x - 108*m*n*x - 18*m 
*x - 72*n**3*x - 108*n**2*x - 36*n*x),x)*m**5*n + 648*int((x**m*(4*a*x + 6 
)**n*( - 2*a*x + 3)**n)/(4*a**2*m**3*x**3 + 24*a**2*m**2*n*x**3 + 12*a**2* 
m**2*x**3 + 48*a**2*m*n**2*x**3 + 48*a**2*m*n*x**3 + 8*a**2*m*x**3 + 32*a* 
*2*n**3*x**3 + 48*a**2*n**2*x**3 + 16*a**2*n*x**3 - 9*m**3*x - 54*m**2*n*x 
 - 27*m**2*x - 108*m*n**2*x - 108*m*n*x - 18*m*x - 72*n**3*x - 108*n**2*x 
- 36*n*x),x)*m**4*n**2 + 324*int((x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n)/( 
4*a**2*m**3*x**3 + 24*a**2*m**2*n*x**3 + 12*a**2*m**2*x**3 + 48*a**2*m*...