Integrand size = 25, antiderivative size = 50 \[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=-\frac {2 x^{-n/2}}{b n}+\frac {2 \sqrt {c} \arctan \left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{3/2} n} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 352, 199, 327, 211} \[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=\frac {2 \sqrt {c} \arctan \left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{3/2} n}-\frac {2 x^{-n/2}}{b n} \]
[In]
[Out]
Rule 199
Rule 211
Rule 327
Rule 352
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{-1-\frac {n}{2}}}{b+c x^n} \, dx \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{b+\frac {c}{x^2}} \, dx,x,x^{-n/2}\right )}{n} \\ & = -\frac {2 \text {Subst}\left (\int \frac {x^2}{c+b x^2} \, dx,x,x^{-n/2}\right )}{n} \\ & = -\frac {2 x^{-n/2}}{b n}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{c+b x^2} \, dx,x,x^{-n/2}\right )}{b n} \\ & = -\frac {2 x^{-n/2}}{b n}+\frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {b} x^{-n/2}}{\sqrt {c}}\right )}{b^{3/2} n} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.64 \[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=-\frac {2 x^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {c x^n}{b}\right )}{b n} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.58
method | result | size |
risch | \(-\frac {2 x^{-\frac {n}{2}}}{b n}+\frac {\sqrt {-b c}\, \ln \left (x^{\frac {n}{2}}-\frac {\sqrt {-b c}}{c}\right )}{b^{2} n}-\frac {\sqrt {-b c}\, \ln \left (x^{\frac {n}{2}}+\frac {\sqrt {-b c}}{c}\right )}{b^{2} n}\) | \(79\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.02 \[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=\left [\frac {x x^{\frac {1}{2} \, n - 1} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} x^{n - 2} - 2 \, b x x^{\frac {1}{2} \, n - 1} \sqrt {-\frac {c}{b}} - b}{c x^{2} x^{n - 2} + b}\right ) - 2}{b n x x^{\frac {1}{2} \, n - 1}}, \frac {2 \, {\left (x x^{\frac {1}{2} \, n - 1} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c x x^{\frac {1}{2} \, n - 1}}\right ) - 1\right )}}{b n x x^{\frac {1}{2} \, n - 1}}\right ] \]
[In]
[Out]
\[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{- n} x^{\frac {n}{2} - 1}}{b + c x^{n}}\, dx \]
[In]
[Out]
\[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{2} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=-\frac {2 \, {\left (\frac {c \arctan \left (\frac {c \sqrt {x^{n}}}{\sqrt {b c}}\right )}{\sqrt {b c} b} + \frac {1}{b \sqrt {x^{n}}}\right )}}{n} \]
[In]
[Out]
Timed out. \[ \int \frac {x^{-1+\frac {n}{2}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{2}-1}}{b\,x^n+c\,x^{2\,n}} \,d x \]
[In]
[Out]