Integrand size = 25, antiderivative size = 252 \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {\sqrt {2} c^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n} \]
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Time = 0.15 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1598, 369, 352, 199, 327, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\frac {\sqrt {2} c^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}+1\right )}{b^{9/4} n}+\frac {c^{5/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}+\sqrt {b} x^{-n/2}+\sqrt {c}\right )}{\sqrt {2} b^{9/4} n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {4 x^{-5 n/4}}{5 b n} \]
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Rule 199
Rule 210
Rule 217
Rule 327
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 {5 n}{4}}}{b+c x^n} \, dx \\ & = -\frac {4 x^{-5 n/4}}{5 b n}-\frac {c \int \frac {x^{-1-\frac {n}{4}}}{b+c x^n} \, dx}{b} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {(4 c) \text {Subst}\left (\int \frac {1}{b+\frac {c}{x^4}} \, dx,x,x^{-n/4}\right )}{b n} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {(4 c) \text {Subst}\left (\int \frac {x^4}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b n} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {\left (4 c^2\right ) \text {Subst}\left (\int \frac {1}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b^2 n} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}-\frac {\left (2 c^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {b} x^2}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b^2 n}-\frac {\left (2 c^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {b} x^2}{c+b x^4} \, dx,x,x^{-n/4}\right )}{b^2 n} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {c^{5/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {c}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}+\frac {c^{5/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {c}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{-n/4}\right )}{b^{5/2} n}-\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^{-n/4}\right )}{b^{5/2} n} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {\left (\sqrt {2} c^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {\left (\sqrt {2} c^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n} \\ & = -\frac {4 x^{-5 n/4}}{5 b n}+\frac {4 c x^{-n/4}}{b^2 n}+\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}-\frac {\sqrt {2} c^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^{-n/4}}{\sqrt [4]{c}}\right )}{b^{9/4} n}+\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} n}-\frac {c^{5/4} \log \left (\sqrt {c}+\sqrt {b} x^{-n/2}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{-n/4}\right )}{\sqrt {2} b^{9/4} 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 = 0.13 \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 x^{-5 n/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\frac {c x^n}{b}\right )}{5 b n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.69 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.29
method | result | size |
risch | \(\frac {4 c \,x^{-\frac {n}{4}}}{b^{2} n}-\frac {4 x^{-\frac {5 n}{4}}}{5 b n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{9} n^{4} \textit {\_Z}^{4}+c^{5}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}+\frac {b^{7} n^{3} \textit {\_R}^{3}}{c^{4}}\right )\right )\) | \(73\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.99 \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=-\frac {4 \, b x^{5} x^{-\frac {5}{4} \, n - 5} + 5 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 5 \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} - c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) + 5 i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 5 i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} n \left (-\frac {c^{5}}{b^{9} n^{4}}\right )^{\frac {1}{4}} + c x x^{-\frac {1}{4} \, n - 1}}{x}\right ) - 20 \, c x x^{-\frac {1}{4} \, n - 1}}{5 \, b^{2} n} \]
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\[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{- n} x^{- \frac {n}{4} - 1}}{b + c x^{n}}\, dx \]
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\[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
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\[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
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Timed out. \[ \int \frac {x^{-1-\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{\frac {n}{4}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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