Integrand size = 16, antiderivative size = 73 \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\frac {(A b-a B) x^{1+m}}{a b (a+b x)}-\frac {(A b m-a B (1+m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a^2 b (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 66} \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\frac {x^{m+1} (A b-a B)}{a b (a+b x)}-\frac {x^{m+1} (A b m-a B (m+1)) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a^2 b (m+1)} \]
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Rule 66
Rule 79
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{1+m}}{a b (a+b x)}-\frac {(A b m-a B (1+m)) \int \frac {x^m}{a+b x} \, dx}{a b} \\ & = \frac {(A b-a B) x^{1+m}}{a b (a+b x)}-\frac {(A b m-a B (1+m)) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a^2 b (1+m)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\frac {x^{1+m} \left (\frac {a (A b-a B)}{a+b x}+\frac {(-A b m+a B (1+m)) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{1+m}\right )}{a^2 b} \]
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\[\int \frac {x^{m} \left (B x +A \right )}{\left (b x +a \right )^{2}}d x\]
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\[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 631, normalized size of antiderivative = 8.64 \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=A \left (- \frac {a m^{2} x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {a m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a m x^{m + 1} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a x^{m + 1} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m^{2} x x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m x x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )}\right ) + B \left (- \frac {a m^{2} x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {3 a m x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} + \frac {a m x^{m + 2} \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {2 a x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} + \frac {2 a x^{m + 2} \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {b m^{2} x x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {3 b m x x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {2 b x x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )}\right ) \]
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\[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int \frac {x^m\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
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