Integrand size = 27, antiderivative size = 140 \[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=-\frac {(d e-c f) (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f) (f g-e h) (1+m)}+\frac {(d g-c h) (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {h (a+b x)}{b g-a h}\right )}{(b g-a h) (f g-e h) (1+m)} \]
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Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {162, 70} \[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {(a+b x)^{m+1} (d g-c h) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (f g-e h)}-\frac {(a+b x)^{m+1} (d e-c f) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (f g-e h)} \]
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Rule 70
Rule 162
Rubi steps \begin{align*} \text {integral}& = -\frac {(d e-c f) \int \frac {(a+b x)^m}{e+f x} \, dx}{f g-e h}+\frac {(d g-c h) \int \frac {(a+b x)^m}{g+h x} \, dx}{f g-e h} \\ & = -\frac {(d e-c f) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f) (f g-e h) (1+m)}+\frac {(d g-c h) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {h (a+b x)}{b g-a h}\right )}{(b g-a h) (f g-e h) (1+m)} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {(a+b x)^{1+m} \left (-\frac {(d e-c f) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {f (a+b x)}{-b e+a f}\right )}{b e-a f}+\frac {(d g-c h) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {h (a+b x)}{-b g+a h}\right )}{b g-a h}\right )}{(f g-e h) (1+m)} \]
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\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )}{\left (f x +e \right ) \left (h x +g \right )}d x\]
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\[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\int { \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (h x + g\right )}} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\int \frac {\left (a + b x\right )^{m} \left (c + d x\right )}{\left (e + f x\right ) \left (g + h x\right )}\, dx \]
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\[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\int { \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (h x + g\right )}} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\int { \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (h x + g\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx=\int \frac {{\left (a+b\,x\right )}^m\,\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (g+h\,x\right )} \,d x \]
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