Integrand size = 26, antiderivative size = 121 \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=-\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {2^{-1-m} \left (441-86 m+2 m^2\right ) (1+2 x)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (1+2 x))}{3 (1-m) m} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {92, 80, 71} \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {2^{-m-1} \left (2 m^2-86 m+441\right ) (2 x+1)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3 (2 x+1))}{3 (1-m) m}-\frac {1}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m}-\frac {7 (21-m) (3 x+2)^{m+1} (2 x+1)^{-m}}{3 m} \]
[In]
[Out]
Rule 71
Rule 80
Rule 92
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {1}{12} \int (1+2 x)^{-1-m} (2+3 x)^m (4 (82-5 m)-8 (65-2 m) x) \, dx \\ & = -\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {(24 (82-5 m)+8 (65-2 m) (-4 m+3 (1+m))) \int (1+2 x)^{-m} (2+3 x)^m \, dx}{24 m} \\ & = -\frac {7 (21-m) (1+2 x)^{-m} (2+3 x)^{1+m}}{3 m}-\frac {1}{3} (5-4 x) (1+2 x)^{-m} (2+3 x)^{1+m}+\frac {2^{-1-m} \left (441-86 m+2 m^2\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{3 (1-m) m} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79 \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\frac {1}{6} (1+2 x)^{-m} \left (\frac {14 (-21+m) (2+3 x)^{1+m}}{m}+2 (2+3 x)^{1+m} (-5+4 x)-\frac {2^{-m} \left (441-86 m+2 m^2\right ) (1+2 x) \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3-6 x)}{(-1+m) m}\right ) \]
[In]
[Out]
\[\int \left (5-4 x \right )^{2} \left (1+2 x \right )^{-1-m} \left (2+3 x \right )^{m}d x\]
[In]
[Out]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 1} {\left (4 \, x - 5\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int \left (2 x + 1\right )^{- m - 1} \left (3 x + 2\right )^{m} \left (4 x - 5\right )^{2}\, dx \]
[In]
[Out]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 1} {\left (4 \, x - 5\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int { {\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 1} {\left (4 \, x - 5\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^m \, dx=\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^2}{{\left (2\,x+1\right )}^{m+1}} \,d x \]
[In]
[Out]