Integrand size = 33, antiderivative size = 65 \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (4,1+m,2+m,-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {133} \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\frac {(a+b x)^{m+1} (c+d x)^{-m-1} \operatorname {Hypergeometric2F1}\left (4,m+1,m+2,-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]
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Rule 133
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (4,1+m;2+m;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (1+m)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (4,1+m,2+m,-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (1+m)} \]
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\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{2-m}}{\left (2 b d x +a d +b c \right )^{4}}d x\]
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\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{4}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{4}} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx=\int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m}}{{\left (a\,d+b\,c+2\,b\,d\,x\right )}^4} \,d x \]
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