Integrand size = 25, antiderivative size = 176 \[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}+\frac {\sqrt {3} c^{4/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt {3} \sqrt [3]{c}}\right )}{b^{7/3} n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (c^{2/3}+b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}\right )}{2 b^{7/3} n} \]
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Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1598, 369, 352, 199, 327, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=\frac {\sqrt {3} c^{4/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt {3} \sqrt [3]{c}}\right )}{b^{7/3} n}-\frac {c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {3 x^{-4 n/3}}{4 b n} \]
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Rule 31
Rule 199
Rule 206
Rule 210
Rule 327
Rule 352
Rule 369
Rule 631
Rule 642
Rule 648
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{-1-\frac {4 n}{3}}}{b+c x^n} \, dx \\ & = -\frac {3 x^{-4 n/3}}{4 b n}-\frac {c \int \frac {x^{-1-\frac {n}{3}}}{b+c x^n} \, dx}{b} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {(3 c) \text {Subst}\left (\int \frac {1}{b+\frac {c}{x^3}} \, dx,x,x^{-n/3}\right )}{b n} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {(3 c) \text {Subst}\left (\int \frac {x^3}{c+b x^3} \, dx,x,x^{-n/3}\right )}{b n} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{c+b x^3} \, dx,x,x^{-n/3}\right )}{b^2 n} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {c^{4/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{c}+\sqrt [3]{b} x} \, dx,x,x^{-n/3}\right )}{b^2 n}-\frac {c^{4/3} \text {Subst}\left (\int \frac {2 \sqrt [3]{c}-\sqrt [3]{b} x}{c^{2/3}-\sqrt [3]{b} \sqrt [3]{c} x+b^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{b^2 n} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \text {Subst}\left (\int \frac {-\sqrt [3]{b} \sqrt [3]{c}+2 b^{2/3} x}{c^{2/3}-\sqrt [3]{b} \sqrt [3]{c} x+b^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 b^{7/3} n}-\frac {\left (3 c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{c^{2/3}-\sqrt [3]{b} \sqrt [3]{c} x+b^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 b^2 n} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (c^{2/3}+b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}\right )}{2 b^{7/3} n}-\frac {\left (3 c^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^{-n/3}}{\sqrt [3]{c}}\right )}{b^{7/3} n} \\ & = -\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}+\frac {\sqrt {3} c^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x^{-n/3}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{b^{7/3} n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (c^{2/3}+b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}\right )}{2 b^{7/3} 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 = 0.19 \[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=-\frac {3 x^{-4 n/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},1,-\frac {1}{3},-\frac {c x^n}{b}\right )}{4 b n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.67 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\frac {3 c \,x^{-\frac {n}{3}}}{b^{2} n}-\frac {3 x^{-\frac {4 n}{3}}}{4 b n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{7} n^{3} \textit {\_Z}^{3}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\frac {b^{5} n^{2} \textit {\_R}^{2}}{c^{3}}\right )\right )\) | \(73\) |
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Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.97 \[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=-\frac {3 \, b x^{4} x^{-\frac {4}{3} \, n - 4} - 12 \, c x x^{-\frac {1}{3} \, n - 1} - 4 \, \sqrt {3} c \left (-\frac {c}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x x^{-\frac {1}{3} \, n - 1} \left (-\frac {c}{b}\right )^{\frac {2}{3}} - \sqrt {3} c}{3 \, c}\right ) - 4 \, c \left (-\frac {c}{b}\right )^{\frac {1}{3}} \log \left (\frac {x x^{-\frac {1}{3} \, n - 1} - \left (-\frac {c}{b}\right )^{\frac {1}{3}}}{x}\right ) + 2 \, c \left (-\frac {c}{b}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} x^{-\frac {2}{3} \, n - 2} + x x^{-\frac {1}{3} \, n - 1} \left (-\frac {c}{b}\right )^{\frac {1}{3}} + \left (-\frac {c}{b}\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 \, b^{2} n} \]
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\[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=\int \frac {x^{- n} x^{- \frac {n}{3} - 1}}{b + c x^{n}}\, dx \]
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\[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
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\[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n}} \,d x } \]
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Timed out. \[ \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{\frac {n}{3}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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