Integrand size = 18, antiderivative size = 124 \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}-\frac {(a d (1-n)+b c (1+n)) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (1+n)} \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 72, 71} \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (\frac {a (1-n)}{n+1}+\frac {b c}{d}\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2} \]
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Rule 71
Rule 72
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}+\frac {1}{2} \left (-\frac {a (1-n)}{b}-\frac {c (1+n)}{d}\right ) \int (a+b x)^n (c+d x)^{-n} \, dx \\ & = \frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}+\frac {1}{2} \left (\left (-\frac {a (1-n)}{b}-\frac {c (1+n)}{d}\right ) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx \\ & = \frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}-\frac {\left (\frac {b c}{d}+\frac {a (1-n)}{1+n}\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.81 \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-n} \left (b (c+d x)-\frac {(-a d (-1+n)+b c (1+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{2 b^2 d} \]
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\[\int x \left (b x +a \right )^{n} \left (d x +c \right )^{-n}d x\]
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\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{n}} \,d x } \]
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Exception generated. \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{n}} \,d x } \]
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\[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{{\left (d x + c\right )}^{n}} \,d x } \]
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Timed out. \[ \int x (a+b x)^n (c+d x)^{-n} \, dx=\int \frac {x\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \]
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