Integrand size = 19, antiderivative size = 84 \[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (-2+m,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{b^3 (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {72, 71} \[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m-2,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{b^3 (m+1)} \]
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Rule 71
Rule 72
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((b c-a d)^2 (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{2-m} \, dx}{b^2} \\ & = \frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-2+m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{b^3 (1+m)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (-2+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{b^3 (1+m)} \]
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\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{2-m}d x\]
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\[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2} \,d x } \]
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Exception generated. \[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2} \,d x } \]
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\[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^{2-m} \, dx=\int {\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m} \,d x \]
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