Integrand size = 25, antiderivative size = 236 \[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-3 n/4}}{3 b n}+\frac {\sqrt {2} c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac {\sqrt {2} c^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac {c^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1598, 369, 352, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\frac {\sqrt {2} c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac {\sqrt {2} c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}+1\right )}{b^{7/4} n}+\frac {c^{3/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {b}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {b}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {4 x^{-3 n/4}}{3 b n} \]
[In]
[Out]
Rule 210
Rule 217
Rule 352
Rule 369
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{-1-\frac {3 n}{4}}}{b+c x^n} \, dx \\ & = -\frac {4 x^{-3 n/4}}{3 b n}-\frac {c \int \frac {x^{\frac {1}{4} (-4+n)}}{b+c x^n} \, dx}{b} \\ & = -\frac {4 x^{-3 n/4}}{3 b n}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b n} \\ & = -\frac {4 x^{-3 n/4}}{3 b n}-\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}-\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n} \\ & = -\frac {4 x^{-3 n/4}}{3 b n}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} b^{7/4} n}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} b^{7/4} n} \\ & = -\frac {4 x^{-3 n/4}}{3 b n}+\frac {c^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {\left (\sqrt {2} c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac {\left (\sqrt {2} c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{b}}\right )}{b^{7/4} n} \\ & = -\frac {4 x^{-3 n/4}}{3 b n}+\frac {\sqrt {2} c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac {\sqrt {2} c^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac {c^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.14 \[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-3 n/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {c x^n}{b}\right )}{3 b n} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.69 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.23
method | result | size |
risch | \(-\frac {4 x^{-\frac {3 n}{4}}}{3 b n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{7} n^{4} \textit {\_Z}^{4}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}-\frac {b^{2} n \textit {\_R}}{c}\right )\right )\) | \(54\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.14 \[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {3 \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} n \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} + c x x^{\frac {1}{4} \, n - 1}}{x}\right ) - 3 \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} n \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} - c x x^{\frac {1}{4} \, n - 1}}{x}\right ) + 3 i \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} n \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} + c x x^{\frac {1}{4} \, n - 1}}{x}\right ) - 3 i \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} n \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} + c x x^{\frac {1}{4} \, n - 1}}{x}\right ) + 4}{3 \, b n x^{3} x^{\frac {3}{4} \, n - 3}} \]
[In]
[Out]
\[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{- n} x^{\frac {n}{4} - 1}}{b + c x^{n}}\, dx \]
[In]
[Out]
\[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.86 \[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {\frac {6 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, {\left (x^{n}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {6 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, {\left (x^{n}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (x^{\frac {1}{2} \, n} + \sqrt {2} {\left (x^{n}\right )}^{\frac {1}{4}} \left (\frac {b}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{c}}\right )}{b^{2}} - \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (x^{\frac {1}{2} \, n} - \sqrt {2} {\left (x^{n}\right )}^{\frac {1}{4}} \left (\frac {b}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{c}}\right )}{b^{2}} + \frac {8}{b x^{\frac {3}{4} \, n}}}{6 \, n} \]
[In]
[Out]
Timed out. \[ \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{4}-1}}{b\,x^n+c\,x^{2\,n}} \,d x \]
[In]
[Out]