Integrand size = 19, antiderivative size = 57 \[ \int (a+b x)^m (a c-b c x)^m \, dx=x (a+b x)^m (a c-b c x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},\frac {b^2 x^2}{a^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {42, 252, 251} \[ \int (a+b x)^m (a c-b c x)^m \, dx=x (a+b x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m} (a c-b c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},\frac {b^2 x^2}{a^2}\right ) \]
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Rule 42
Rule 251
Rule 252
Rubi steps \begin{align*} \text {integral}& = \left ((a+b x)^m (a c-b c x)^m \left (a^2 c-b^2 c x^2\right )^{-m}\right ) \int \left (a^2 c-b^2 c x^2\right )^m \, dx \\ & = \left ((a+b x)^m (a c-b c x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m}\right ) \int \left (1-\frac {b^2 x^2}{a^2}\right )^m \, dx \\ & = x (a+b x)^m (a c-b c x)^m \left (1-\frac {b^2 x^2}{a^2}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {b^2 x^2}{a^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.26 \[ \int (a+b x)^m (a c-b c x)^m \, dx=-\frac {2^m (a-b x) (c (a-b x))^m (a+b x)^m \left (\frac {a+b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,1+m,2+m,\frac {a-b x}{2 a}\right )}{b (1+m)} \]
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\[\int \left (b x +a \right )^{m} \left (-b c x +a c \right )^{m}d x\]
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\[ \int (a+b x)^m (a c-b c x)^m \, dx=\int { {\left (-b c x + a c\right )}^{m} {\left (b x + a\right )}^{m} \,d x } \]
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Result contains complex when optimal does not.
Time = 4.78 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.56 \[ \int (a+b x)^m (a c-b c x)^m \, dx=\frac {a^{2 m + 1} c^{m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, 1 & \frac {1}{2}, - m, \frac {1}{2} - m \\- m - \frac {1}{2}, - m, - \frac {m}{2}, \frac {1}{2} - m, \frac {1}{2} - \frac {m}{2} & 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )} e^{- i \pi m}}{4 \pi b \Gamma \left (- m\right )} - \frac {a^{2 m + 1} c^{m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, 1 & \\- \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} & - \frac {1}{2}, 0, - m - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi b \Gamma \left (- m\right )} \]
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\[ \int (a+b x)^m (a c-b c x)^m \, dx=\int { {\left (-b c x + a c\right )}^{m} {\left (b x + a\right )}^{m} \,d x } \]
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\[ \int (a+b x)^m (a c-b c x)^m \, dx=\int { {\left (-b c x + a c\right )}^{m} {\left (b x + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+b x)^m (a c-b c x)^m \, dx=\int {\left (a\,c-b\,c\,x\right )}^m\,{\left (a+b\,x\right )}^m \,d x \]
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