Integrand size = 18, antiderivative size = 82 \[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\frac {b x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a (b c-a d) (1+m)}-\frac {d x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )}{c (b c-a d) (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {88, 66} \[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\frac {b x^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a (m+1) (b c-a d)}-\frac {d x^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d x}{c}\right )}{c (m+1) (b c-a d)} \]
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Rule 66
Rule 88
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {x^m}{a+b x} \, dx}{b c-a d}-\frac {d \int \frac {x^m}{c+d x} \, dx}{b c-a d} \\ & = \frac {b x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a (b c-a d) (1+m)}-\frac {d x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d x}{c}\right )}{c (b c-a d) (1+m)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\frac {x^{1+m} \left (-b c \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+a d \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d x}{c}\right )\right )}{a c (-b c+a d) (1+m)} \]
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\[\int \frac {x^{m}}{\left (b x +a \right ) \left (d x +c \right )}d x\]
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\[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=- \frac {b^{m} m x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m\right ) \Gamma \left (- m\right )}{a b^{m} d \Gamma \left (1 - m\right ) - b b^{m} c \Gamma \left (1 - m\right )} + \frac {b^{m} m x^{m} \Phi \left (\frac {c e^{i \pi }}{d x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{a b^{m} d \Gamma \left (1 - m\right ) - b b^{m} c \Gamma \left (1 - m\right )} \]
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\[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{(a+b x) (c+d x)} \, dx=\int \frac {x^m}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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