Integrand size = 29, antiderivative size = 69 \[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\frac {19^n (2+3 x)^{1+m+n} (3+4 x)^{-1-m-n} \operatorname {Hypergeometric2F1}\left (-1-m-n,-n,-m-n,\frac {12 (3+4 x)}{19 (2+3 x)}\right )}{1+m+n} \] Output:
19^n*(2+3*x)^(1+m+n)*(3+4*x)^(-1-m-n)*hypergeom([-n, -1-m-n],[-m-n],12*(3+ 4*x)/(38+57*x))/(1+m+n)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20 \[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\frac {\left (\frac {1}{4}+\frac {x}{3}\right )^{-1-m-n} (2+3 x)^{1+m} \left (-\frac {3+4 x}{2+9 x}\right )^{1+m+n} (2+9 x)^n \operatorname {Hypergeometric2F1}\left (1+m,2+m+n,2+m,\frac {38+57 x}{2+9 x}\right )}{1+m} \] Input:
Integrate[(2 + 3*x)^m*(3 + 4*x)^(-2 - m - n)*(2 + 9*x)^n,x]
Output:
((1/4 + x/3)^(-1 - m - n)*(2 + 3*x)^(1 + m)*(-((3 + 4*x)/(2 + 9*x)))^(1 + m + n)*(2 + 9*x)^n*Hypergeometric2F1[1 + m, 2 + m + n, 2 + m, (38 + 57*x)/ (2 + 9*x)])/(1 + m)
Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {142}
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.
\(\displaystyle \int (3 x+2)^m (9 x+2)^n (4 x+3)^{-m-n-2} \, dx\) |
\(\Big \downarrow \) 142 |
\(\displaystyle -\frac {12^{m+n+1} (3 x+2)^{m+1} (9 x+2)^{n+1} (4 x+3)^{-m-n-2} \left (-\frac {4 x+3}{9 x+2}\right )^{m+n+2} \operatorname {Hypergeometric2F1}\left (m+1,m+n+2,m+2,\frac {19 (3 x+2)}{9 x+2}\right )}{m+1}\) |
Input:
Int[(2 + 3*x)^m*(3 + 4*x)^(-2 - m - n)*(2 + 9*x)^n,x]
Output:
-((12^(1 + m + n)*(2 + 3*x)^(1 + m)*(3 + 4*x)^(-2 - m - n)*(-((3 + 4*x)/(2 + 9*x)))^(2 + m + n)*(2 + 9*x)^(1 + n)*Hypergeometric2F1[1 + m, 2 + m + n , 2 + m, (19*(2 + 3*x))/(2 + 9*x)])/(1 + m))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f *x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
\[\int \left (2+3 x \right )^{m} \left (3+4 x \right )^{-2-m -n} \left (2+9 x \right )^{n}d x\]
Input:
int((2+3*x)^m*(3+4*x)^(-2-m-n)*(2+9*x)^n,x)
Output:
int((2+3*x)^m*(3+4*x)^(-2-m-n)*(2+9*x)^n,x)
\[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\int { {\left (9 \, x + 2\right )}^{n} {\left (4 \, x + 3\right )}^{-m - n - 2} {\left (3 \, x + 2\right )}^{m} \,d x } \] Input:
integrate((2+3*x)^m*(3+4*x)^(-2-m-n)*(2+9*x)^n,x, algorithm="fricas")
Output:
integral((9*x + 2)^n*(4*x + 3)^(-m - n - 2)*(3*x + 2)^m, x)
\[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\int \left (3 x + 2\right )^{m} \left (4 x + 3\right )^{- m - n - 2} \left (9 x + 2\right )^{n}\, dx \] Input:
integrate((2+3*x)**m*(3+4*x)**(-2-m-n)*(2+9*x)**n,x)
Output:
Integral((3*x + 2)**m*(4*x + 3)**(-m - n - 2)*(9*x + 2)**n, x)
\[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\int { {\left (9 \, x + 2\right )}^{n} {\left (4 \, x + 3\right )}^{-m - n - 2} {\left (3 \, x + 2\right )}^{m} \,d x } \] Input:
integrate((2+3*x)^m*(3+4*x)^(-2-m-n)*(2+9*x)^n,x, algorithm="maxima")
Output:
integrate((9*x + 2)^n*(4*x + 3)^(-m - n - 2)*(3*x + 2)^m, x)
\[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\int { {\left (9 \, x + 2\right )}^{n} {\left (4 \, x + 3\right )}^{-m - n - 2} {\left (3 \, x + 2\right )}^{m} \,d x } \] Input:
integrate((2+3*x)^m*(3+4*x)^(-2-m-n)*(2+9*x)^n,x, algorithm="giac")
Output:
integrate((9*x + 2)^n*(4*x + 3)^(-m - n - 2)*(3*x + 2)^m, x)
Timed out. \[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\int \frac {{\left (3\,x+2\right )}^m\,{\left (9\,x+2\right )}^n}{{\left (4\,x+3\right )}^{m+n+2}} \,d x \] Input:
int(((3*x + 2)^m*(9*x + 2)^n)/(4*x + 3)^(m + n + 2),x)
Output:
int(((3*x + 2)^m*(9*x + 2)^n)/(4*x + 3)^(m + n + 2), x)
\[ \int (2+3 x)^m (3+4 x)^{-2-m-n} (2+9 x)^n \, dx=\int \frac {\left (3 x +2\right )^{m} \left (9 x +2\right )^{n}}{16 \left (4 x +3\right )^{m +n} x^{2}+24 \left (4 x +3\right )^{m +n} x +9 \left (4 x +3\right )^{m +n}}d x \] Input:
int((2+3*x)^m*(3+4*x)^(-2-m-n)*(2+9*x)^n,x)
Output:
int(((3*x + 2)**m*(9*x + 2)**n)/(16*(4*x + 3)**(m + n)*x**2 + 24*(4*x + 3) **(m + n)*x + 9*(4*x + 3)**(m + n)),x)