Integrand size = 22, antiderivative size = 66 \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\frac {2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac {(1+2 m) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )}{a c^2 e (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {79, 66} \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b x}{a}\right )}{a c^2 e (m+1)} \]
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Rule 66
Rule 79
Rubi steps \begin{align*} \text {integral}& = \frac {2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac {(1+2 m) \int \frac {(e x)^m}{a c-b c x} \, dx}{c} \\ & = \frac {2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac {(1+2 m) (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{a}\right )}{a c^2 e (1+m)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=-\frac {x (e x)^m \left (2 a (1+m)-(1+2 m) (a-b x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{a c^2 (1+m) (-a+b x)} \]
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\[\int \frac {\left (e x \right )^{m} \left (b x +a \right )}{\left (-b c x +a c \right )^{2}}d x\]
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\[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int { \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 2.80 (sec) , antiderivative size = 790, normalized size of antiderivative = 11.97 \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=a \left (\frac {a e^{m} m^{2} x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} + \frac {a e^{m} m x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} m x^{m + 1} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} x^{m + 1} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m^{2} x x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m x x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )}\right ) + b \left (\frac {a e^{m} m^{2} x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {3 a e^{m} m x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {a e^{m} m x^{m + 2} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {2 a e^{m} x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 a e^{m} x^{m + 2} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {b e^{m} m^{2} x x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {3 b e^{m} m x x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 b e^{m} x x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )}\right ) \]
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\[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int { \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}} \,d x } \]
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\[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int { \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (a+b\,x\right )}{{\left (a\,c-b\,c\,x\right )}^2} \,d x \]
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