Integrand size = 19, antiderivative size = 93 \[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\frac {c (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {c (d+e x)}{c d-b e}\right )}{b (c d-b e) (1+m)}-\frac {(d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )}{b d (1+m)} \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {725, 67, 70} \[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\frac {c (d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {c (d+e x)}{c d-b e}\right )}{b (m+1) (c d-b e)}-\frac {(d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {e x}{d}+1\right )}{b d (m+1)} \]
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Rule 67
Rule 70
Rule 725
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(d+e x)^m}{b x}-\frac {c (d+e x)^m}{b (b+c x)}\right ) \, dx \\ & = \frac {\int \frac {(d+e x)^m}{x} \, dx}{b}-\frac {c \int \frac {(d+e x)^m}{b+c x} \, dx}{b} \\ & = \frac {c (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {c (d+e x)}{c d-b e}\right )}{b (c d-b e) (1+m)}-\frac {(d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac {e x}{d}\right )}{b d (1+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=-\frac {(d+e x)^{1+m} \left (c d \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {c (d+e x)}{c d-b e}\right )+(-c d+b e) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )\right )}{b d (-c d+b e) (1+m)} \]
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\[\int \frac {\left (e x +d \right )^{m}}{c \,x^{2}+b x}d x\]
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\[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{c x^{2} + b x} \,d x } \]
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\[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\int \frac {\left (d + e x\right )^{m}}{x \left (b + c x\right )}\, dx \]
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\[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{c x^{2} + b x} \,d x } \]
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\[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{c x^{2} + b x} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m}{b x+c x^2} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{c\,x^2+b\,x} \,d x \]
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