Optimal. Leaf size=42 \[ \frac {18^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {4 a^2 x^2}{9}\right )}{m+1} \]
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Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {125, 364} \[ \frac {18^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {4 a^2 x^2}{9}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 125
Rule 364
Rubi steps
\begin {align*} \int x^m (3-2 a x)^n (6+4 a x)^n \, dx &=\int x^m \left (18-8 a^2 x^2\right )^n \, dx\\ &=\frac {18^n x^{1+m} \, _2F_1\left (\frac {1+m}{2},-n;\frac {3+m}{2};\frac {4 a^2 x^2}{9}\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.64 \[ \frac {x^{m+1} (54-36 a x)^n (4 a x+6)^n \left (18-8 a^2 x^2\right )^{-n} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {4 a^2 x^2}{9}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-2 a x +3\right )^{n} \left (4 a x +6\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (3-2\,a\,x\right )}^n\,{\left (4\,a\,x+6\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 38.53, size = 238, normalized size = 5.67 \[ - \frac {3 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2}, 1 & \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - n, - \frac {m}{2} - n + \frac {1}{2} \\- \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - n, - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - n + \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2} & 0 \end {matrix} \middle | {\frac {9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{8 \pi a \Gamma \left (- n\right )} + \frac {3 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2}, 1 & \\- \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} & - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - n - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{8 \pi a \Gamma \left (- n\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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