Integrand size = 19, antiderivative size = 36 \[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d) (1+n)} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\frac {(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]
[In]
[Out]
Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d) (1+n)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d) (1+n)} \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (d x +c \right )^{-1-n}}{a d n -b c n +a d -b c}\) | \(42\) |
parallelrisch | \(-\frac {x^{2} \left (b x +a \right )^{n} \left (d x +c \right )^{-2-n} b^{2} d^{2}+x \left (b x +a \right )^{n} \left (d x +c \right )^{-2-n} a b \,d^{2}+x \left (b x +a \right )^{n} \left (d x +c \right )^{-2-n} b^{2} c d +\left (b x +a \right )^{n} \left (d x +c \right )^{-2-n} a b c d}{\left (a d n -b c n +a d -b c \right ) b d}\) | \(130\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\frac {{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}}{b c - a d + {\left (b c - a d\right )} n} \]
[In]
[Out]
Exception generated. \[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
[In]
[Out]
\[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2} \,d x } \]
[In]
[Out]
\[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2} \,d x } \]
[In]
[Out]
Time = 0.56 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.72 \[ \int (a+b x)^n (c+d x)^{-2-n} \, dx=-\frac {\frac {a\,c\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {x\,\left (a\,d+b\,c\right )\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}}{{\left (c+d\,x\right )}^{n+2}} \]
[In]
[Out]