Integrand size = 22, antiderivative size = 124 \[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=-\frac {b (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f) (1+n)}+\frac {d (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d) (d e-c f) (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {88, 70} \[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {d (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)}-\frac {b (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)} \]
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Rule 70
Rule 88
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {(e+f x)^n}{a+b x} \, dx}{b c-a d}-\frac {d \int \frac {(e+f x)^n}{c+d x} \, dx}{b c-a d} \\ & = -\frac {b (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f) (1+n)}+\frac {d (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d) (d e-c f) (1+n)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {(e+f x)^{1+n} \left (b (d e-c f) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )+d (-b e+a f) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )\right )}{(b c-a d) (b e-a f) (-d e+c f) (1+n)} \]
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\[\int \frac {\left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}d x\]
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\[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
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\[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^n}{(a+b x) (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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