Integrand size = 19, antiderivative size = 75 \[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,-\frac {d (a+b x)}{b c-a d}\right )}{b n} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {72, 71} \[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,-\frac {d (a+b x)}{b c-a d}\right )}{b n} \]
[In]
[Out]
Rule 71
Rule 72
Rubi steps \begin{align*} \text {integral}& = \left ((c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{-1-n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \, dx \\ & = -\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac {d (a+b x)}{b c-a d}\right )}{b n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=-\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {d (a+b x)}{-b c+a d}\right )}{b n} \]
[In]
[Out]
\[\int \left (b x +a \right )^{-1-n} \left (d x +c \right )^{n}d x\]
[In]
[Out]
\[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 1} {\left (d x + c\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=\int \left (a + b x\right )^{- n - 1} \left (c + d x\right )^{n}\, dx \]
[In]
[Out]
\[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 1} {\left (d x + c\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{-n - 1} {\left (d x + c\right )}^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+b x)^{-1-n} (c+d x)^n \, dx=\int \frac {{\left (c+d\,x\right )}^n}{{\left (a+b\,x\right )}^{n+1}} \,d x \]
[In]
[Out]