\(\int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx\) [352]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\frac {2 x^{1+m} (3-2 a x)^{-1+n} (6+4 a x)^{-1+n}}{1-m-2 n}-\frac {2^{1+n} 9^{-1+n} (m+n) x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )}{(1+m) (1-m-2 n)}+\frac {2^{2+n} 3^{-3+2 n} a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )}{2+m} \] Output:

2*x^(1+m)*(-2*a*x+3)^(-1+n)*(4*a*x+6)^(-1+n)/(1-m-2*n)-2^(1+n)*9^(-1+n)*(m 
+n)*x^(1+m)*hypergeom([2-n, 1/2+1/2*m],[3/2+1/2*m],4/9*a^2*x^2)/(1+m)/(1-m 
-2*n)+2^(2+n)*3^(-3+2*n)*a*x^(2+m)*hypergeom([2-n, 1+1/2*m],[2+1/2*m],4/9* 
a^2*x^2)/(2+m)
 

Mathematica [A] (warning: unable to verify)

Time = 0.19 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.10 \[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\frac {9^{-2+n} x^{1+m} \left (36-16 a^2 x^2\right )^n \left (18-8 a^2 x^2\right )^{-n} \left (9 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )+4 a (1+m) x \left (3 (3+m) \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )+a (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},2-n,\frac {5+m}{2},\frac {4 a^2 x^2}{9}\right )\right )\right )}{(1+m) (2+m) (3+m)} \] Input:

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

Output:

(9^(-2 + n)*x^(1 + m)*(36 - 16*a^2*x^2)^n*(9*(6 + 5*m + m^2)*Hypergeometri 
c2F1[(1 + m)/2, 2 - n, (3 + m)/2, (4*a^2*x^2)/9] + 4*a*(1 + m)*x*(3*(3 + m 
)*Hypergeometric2F1[(2 + m)/2, 2 - n, (4 + m)/2, (4*a^2*x^2)/9] + a*(2 + m 
)*x*Hypergeometric2F1[(3 + m)/2, 2 - n, (5 + m)/2, (4*a^2*x^2)/9])))/((1 + 
 m)*(2 + m)*(3 + m)*(18 - 8*a^2*x^2)^n)
 

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {147, 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.

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

\(\Big \downarrow \) 147

\(\displaystyle \int \left (16 a^2 x^{m+2} (3-2 a x)^{n-2} (4 a x+6)^{n-2}+48 a x^{m+1} (3-2 a x)^{n-2} (4 a x+6)^{n-2}+36 x^m (3-2 a x)^{n-2} (4 a x+6)^{n-2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2^n 9^{n-1} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},2-n,\frac {m+3}{2},\frac {4 a^2 x^2}{9}\right )}{m+1}+\frac {a 2^{n+2} 3^{2 n-3} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},2-n,\frac {m+4}{2},\frac {4 a^2 x^2}{9}\right )}{m+2}+\frac {a^2 2^{n+2} 9^{n-2} x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {m+3}{2},2-n,\frac {m+5}{2},\frac {4 a^2 x^2}{9}\right )}{m+3}\)

Input:

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

Output:

(2^n*9^(-1 + n)*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, 2 - n, (3 + m)/2, ( 
4*a^2*x^2)/9])/(1 + m) + (2^(2 + n)*3^(-3 + 2*n)*a*x^(2 + m)*Hypergeometri 
c2F1[(2 + m)/2, 2 - n, (4 + m)/2, (4*a^2*x^2)/9])/(2 + m) + (2^(2 + n)*9^( 
-2 + n)*a^2*x^(3 + m)*Hypergeometric2F1[(3 + m)/2, 2 - n, (5 + m)/2, (4*a^ 
2*x^2)/9])/(3 + m)
 

Defintions of rubi rules used

rule 147

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

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\frac {-x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n}+8 \left (\int \frac {x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n} x}{8 a^{3} x^{3}-12 a^{2} x^{2}-18 a x +27}d x \right ) a^{3} m x +16 \left (\int \frac {x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n} x}{8 a^{3} x^{3}-12 a^{2} x^{2}-18 a x +27}d x \right ) a^{3} n x -12 \left (\int \frac {x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n} x}{8 a^{3} x^{3}-12 a^{2} x^{2}-18 a x +27}d x \right ) a^{2} m -24 \left (\int \frac {x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n} x}{8 a^{3} x^{3}-12 a^{2} x^{2}-18 a x +27}d x \right ) a^{2} n -18 \left (\int \frac {x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n}}{8 a^{3} x^{4}-12 a^{2} x^{3}-18 a \,x^{2}+27 x}d x \right ) a m x +27 \left (\int \frac {x^{m} \left (4 a x +6\right )^{n} \left (-2 a x +3\right )^{n}}{8 a^{3} x^{4}-12 a^{2} x^{3}-18 a \,x^{2}+27 x}d x \right ) m}{2 a \left (2 a x -3\right )} \] Input:

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

Output:

( - x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n + 8*int((x**m*(4*a*x + 6)**n*( - 
 2*a*x + 3)**n*x)/(8*a**3*x**3 - 12*a**2*x**2 - 18*a*x + 27),x)*a**3*m*x + 
 16*int((x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*x)/(8*a**3*x**3 - 12*a**2*x 
**2 - 18*a*x + 27),x)*a**3*n*x - 12*int((x**m*(4*a*x + 6)**n*( - 2*a*x + 3 
)**n*x)/(8*a**3*x**3 - 12*a**2*x**2 - 18*a*x + 27),x)*a**2*m - 24*int((x** 
m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*x)/(8*a**3*x**3 - 12*a**2*x**2 - 18*a*x 
 + 27),x)*a**2*n - 18*int((x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n)/(8*a**3* 
x**4 - 12*a**2*x**3 - 18*a*x**2 + 27*x),x)*a*m*x + 27*int((x**m*(4*a*x + 6 
)**n*( - 2*a*x + 3)**n)/(8*a**3*x**4 - 12*a**2*x**3 - 18*a*x**2 + 27*x),x) 
*m)/(2*a*(2*a*x - 3))