Integrand size = 22, antiderivative size = 147 \[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=-\frac {x^{1+m} (3-2 a x)^{1+n} (6+4 a x)^{1+n}}{2 (3+m+2 n)}+\frac {18^{1+n} (2+m+n) x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )}{(1+m) (3+m+2 n)}-\frac {2^{2+n} 3^{1+2 n} a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )}{2+m} \] Output:
-1/2*x^(1+m)*(-2*a*x+3)^(1+n)*(4*a*x+6)^(1+n)/(3+m+2*n)+18^(1+n)*(2+m+n)*x ^(1+m)*hypergeom([-n, 1/2+1/2*m],[3/2+1/2*m],4/9*a^2*x^2)/(1+m)/(3+m+2*n)- 2^(2+n)*3^(1+2*n)*a*x^(2+m)*hypergeom([-n, 1+1/2*m],[2+1/2*m],4/9*a^2*x^2) /(2+m)
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.13 \[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=\frac {x^{1+m} \left (9-4 a^2 x^2\right )^n \left (\frac {1}{2}-\frac {2 a^2 x^2}{9}\right )^{-n} \left (9 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )-4 a (1+m) x \left (3 (3+m) \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )-a (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-n,\frac {5+m}{2},\frac {4 a^2 x^2}{9}\right )\right )\right )}{(1+m) (2+m) (3+m)} \] Input:
Integrate[x^m*(3 - 2*a*x)^(2 + n)*(6 + 4*a*x)^n,x]
Output:
(x^(1 + m)*(9 - 4*a^2*x^2)^n*(9*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/ 2, -n, (3 + m)/2, (4*a^2*x^2)/9] - 4*a*(1 + m)*x*(3*(3 + m)*Hypergeometric 2F1[(2 + m)/2, -n, (4 + m)/2, (4*a^2*x^2)/9] - a*(2 + m)*x*Hypergeometric2 F1[(3 + m)/2, -n, (5 + m)/2, (4*a^2*x^2)/9])))/((1 + m)*(2 + m)*(3 + m)*(1 /2 - (2*a^2*x^2)/9)^n)
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03, 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 |
| \(\displaystyle \int \left (4 a^2 x^{m+2} (3-2 a x)^n (4 a x+6)^n-12 a x^{m+1} (3-2 a x)^n (4 a x+6)^n+9 x^m (3-2 a x)^n (4 a x+6)^n\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2^n 9^{n+1} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-n,\frac {m+3}{2},\frac {4 a^2 x^2}{9}\right )}{m+1}-\frac {a 2^{n+2} 3^{2 n+1} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-n,\frac {m+4}{2},\frac {4 a^2 x^2}{9}\right )}{m+2}+\frac {a^2 2^{n+2} 9^n x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {m+3}{2},-n,\frac {m+5}{2},\frac {4 a^2 x^2}{9}\right )}{m+3}\) |
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, -n, (3 + m)/2, (4*a^ 2*x^2)/9])/(1 + m) - (2^(2 + n)*3^(1 + 2*n)*a*x^(2 + m)*Hypergeometric2F1[ (2 + m)/2, -n, (4 + m)/2, (4*a^2*x^2)/9])/(2 + m) + (2^(2 + n)*9^n*a^2*x^( 3 + m)*Hypergeometric2F1[(3 + m)/2, -n, (5 + m)/2, (4*a^2*x^2)/9])/(3 + m)
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]
\[\int x^{m} \left (-2 a x +3\right )^{2+n} \left (4 a x +6\right )^{n}d x\]
Input:
int(x^m*(-2*a*x+3)^(2+n)*(4*a*x+6)^n,x)
Output:
int(x^m*(-2*a*x+3)^(2+n)*(4*a*x+6)^n,x)
\[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n + 2} x^{m} \,d x } \] 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)
\[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=2^{n} \int x^{m} \left (- 2 a x + 3\right )^{n + 2} \left (2 a x + 3\right )^{n}\, dx \] 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)
\[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n + 2} x^{m} \,d x } \] 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)
\[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n + 2} x^{m} \,d x } \] 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)
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)
\[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx=\text {too large to display} \] Input:
int(x^m*(-2*a*x+3)^(2+n)*(4*a*x+6)^n,x)
Output:
(4*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**3*m**3*x**3 + 24*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**3*m**2*n*x**3 + 12*x**m*(4*a*x + 6)**n*( - 2*a *x + 3)**n*a**3*m**2*x**3 + 48*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**3* m*n**2*x**3 + 48*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**3*m*n*x**3 + 8*x **m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**3*m*x**3 + 32*x**m*(4*a*x + 6)**n* ( - 2*a*x + 3)**n*a**3*n**3*x**3 + 48*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)** n*a**3*n**2*x**3 + 16*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**3*n*x**3 - 12*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*m**3*x**2 - 72*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*m**2*n*x**2 - 48*x**m*(4*a*x + 6)**n*( - 2*a *x + 3)**n*a**2*m**2*x**2 - 144*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2 *m*n**2*x**2 - 192*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*m*n*x**2 - 3 6*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*m*x**2 - 96*x**m*(4*a*x + 6)* *n*( - 2*a*x + 3)**n*a**2*n**3*x**2 - 192*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*n**2*x**2 - 72*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a**2*n*x** 2 + 9*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a*m**3*x + 36*x**m*(4*a*x + 6) **n*( - 2*a*x + 3)**n*a*m**2*n*x + 45*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)** n*a*m**2*x + 36*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a*m*n**2*x + 144*x** m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a*m*n*x + 54*x**m*(4*a*x + 6)**n*( - 2* a*x + 3)**n*a*m*x + 108*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a*n**2*x + 1 08*x**m*(4*a*x + 6)**n*( - 2*a*x + 3)**n*a*n*x + 54*x**m*(4*a*x + 6)**n...