Integrand size = 16, antiderivative size = 56 \[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\frac {d x^{1+m}}{b (1+m)}+\frac {(b c-a d) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a b (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 56, 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 = {81, 66} \[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\frac {x^{m+1} (b c-a d) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a b (m+1)}+\frac {d x^{m+1}}{b (m+1)} \]
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Rule 66
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {d x^{1+m}}{b (1+m)}+\frac {(b c (1+m)-a d (1+m)) \int \frac {x^m}{a+b x} \, dx}{b (1+m)} \\ & = \frac {d x^{1+m}}{b (1+m)}+\frac {(b c-a d) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b (1+m)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\frac {x^{1+m} \left (a d+(b c-a d) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )\right )}{a b (1+m)} \]
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\[\int \frac {x^{m} \left (d x +c \right )}{b x +a}d x\]
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\[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\int { \frac {{\left (d x + c\right )} x^{m}}{b x + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.38 \[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\frac {c m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {c x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {d m x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {2 d x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} \]
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\[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\int { \frac {{\left (d x + c\right )} x^{m}}{b x + a} \,d x } \]
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\[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\int { \frac {{\left (d x + c\right )} x^{m}}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {x^m (c+d x)}{a+b x} \, dx=\int \frac {x^m\,\left (c+d\,x\right )}{a+b\,x} \,d x \]
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