Integrand size = 22, antiderivative size = 104 \[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\frac {2^n 3^{-1+2 n} x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )}{1+m}+\frac {2^{1+n} 9^{-1+n} a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},1-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )}{2+m} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {83, 126, 371} \[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\frac {2^n 3^{2 n-1} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},1-n,\frac {m+3}{2},\frac {4 a^2 x^2}{9}\right )}{m+1}+\frac {a 2^{n+1} 9^{n-1} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},1-n,\frac {m+4}{2},\frac {4 a^2 x^2}{9}\right )}{m+2} \]
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Rule 83
Rule 126
Rule 371
Rubi steps \begin{align*} \text {integral}& = 6 \int x^m (3-2 a x)^{-1+n} (6+4 a x)^{-1+n} \, dx+(4 a) \int x^{1+m} (3-2 a x)^{-1+n} (6+4 a x)^{-1+n} \, dx \\ & = 6 \int x^m \left (18-8 a^2 x^2\right )^{-1+n} \, dx+(4 a) \int x^{1+m} \left (18-8 a^2 x^2\right )^{-1+n} \, dx \\ & = \frac {2^n 3^{-1+2 n} x^{1+m} \, _2F_1\left (\frac {1+m}{2},1-n;\frac {3+m}{2};\frac {4 a^2 x^2}{9}\right )}{1+m}+\frac {2^{1+n} 9^{-1+n} a x^{2+m} \, _2F_1\left (\frac {2+m}{2},1-n;\frac {4+m}{2};\frac {4 a^2 x^2}{9}\right )}{2+m} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.15 \[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\frac {9^{-1+n} x^{1+m} \left (36-16 a^2 x^2\right )^n \left (18-8 a^2 x^2\right )^{-n} \left (2 a (1+m) x \operatorname {Hypergeometric2F1}\left (1+\frac {m}{2},1-n,2+\frac {m}{2},\frac {4 a^2 x^2}{9}\right )+3 (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},1-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )\right )}{(1+m) (2+m)} \]
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\[\int x^{m} \left (-2 a x +3\right )^{-1+n} \left (4 a x +6\right )^{n}d x\]
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\[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 1} x^{m} \,d x } \]
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\[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=2^{n} \int x^{m} \left (- 2 a x + 3\right )^{n - 1} \left (2 a x + 3\right )^{n}\, dx \]
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\[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 1} x^{m} \,d x } \]
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\[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 1} x^{m} \,d x } \]
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Timed out. \[ \int x^m (3-2 a x)^{-1+n} (6+4 a x)^n \, dx=\int x^m\,{\left (3-2\,a\,x\right )}^{n-1}\,{\left (4\,a\,x+6\right )}^n \,d x \]
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