Integrand size = 18, antiderivative size = 125 \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=-\frac {d x^{1+m}}{c (b c-a d) (c+d x)}+\frac {b^2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a (b c-a d)^2 (1+m)}-\frac {d (b c (1-m)+a d m) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{c^2 (b c-a d)^2 (1+m)} \]
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Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {105, 162, 66} \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\frac {b^2 x^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a (m+1) (b c-a d)^2}-\frac {d x^{m+1} (a d m+b c (1-m)) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d x}{c}\right )}{c^2 (m+1) (b c-a d)^2}-\frac {d x^{m+1}}{c (c+d x) (b c-a d)} \]
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Rule 66
Rule 105
Rule 162
Rubi steps \begin{align*} \text {integral}& = -\frac {d x^{1+m}}{c (b c-a d) (c+d x)}-\frac {\int \frac {x^m (-b c-a d m-b d m x)}{(a+b x) (c+d x)} \, dx}{c (b c-a d)} \\ & = -\frac {d x^{1+m}}{c (b c-a d) (c+d x)}+\frac {b^2 \int \frac {x^m}{a+b x} \, dx}{(b c-a d)^2}-\frac {(d (a d m+b (c-c m))) \int \frac {x^m}{c+d x} \, dx}{c (b c-a d)^2} \\ & = -\frac {d x^{1+m}}{c (b c-a d) (c+d x)}+\frac {b^2 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a (b c-a d)^2 (1+m)}-\frac {d (a d m+b (c-c m)) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d x}{c}\right )}{c^2 (b c-a d)^2 (1+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\frac {x^{1+m} \left (b^2 c^2 (c+d x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+a d \left (-c (b c-a d) (1+m)+(b c (-1+m)-a d m) (c+d x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )\right )\right )}{a c^2 (b c-a d)^2 (1+m) (c+d x)} \]
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\[\int \frac {x^{m}}{\left (b x +a \right ) \left (d x +c \right )^{2}}d x\]
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\[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}^{2}} \,d x } \]
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\[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx=\int \frac {x^m}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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