Integrand size = 19, antiderivative size = 176 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}+\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{7/3} n} \]
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Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {369, 352, 199, 327, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} n}-\frac {b^{4/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {3 x^{-4 n/3}}{4 a n} \]
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Rule 31
Rule 199
Rule 206
Rule 210
Rule 327
Rule 352
Rule 369
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {3 x^{-4 n/3}}{4 a n}-\frac {b \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n} \, dx}{a} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{a+\frac {b}{x^3}} \, dx,x,x^{-n/3}\right )}{a n} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {(3 b) \text {Subst}\left (\int \frac {x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a n} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a^2 n} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {b^{4/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a^2 n}-\frac {b^{4/3} \text {Subst}\left (\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{a^2 n} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 a^{7/3} n}-\frac {\left (3 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 a^2 n} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{7/3} n}-\frac {\left (3 b^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}\right )}{a^{7/3} n} \\ & = -\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}+\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{7/3} n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{7/3} n} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.19 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=-\frac {3 x^{-4 n/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},1,-\frac {1}{3},-\frac {b x^n}{a}\right )}{4 a n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.70 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\frac {3 b \,x^{-\frac {n}{3}}}{a^{2} n}-\frac {3 x^{-\frac {4 n}{3}}}{4 a n}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} n^{3} \textit {\_Z}^{3}+b^{4}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\frac {a^{5} n^{2} \textit {\_R}^{2}}{b^{3}}\right )\right )\) | \(73\) |
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none
Time = 0.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=-\frac {3 \, a x x^{-\frac {4}{3} \, n - 1} - 4 \, \sqrt {3} b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}} \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x^{\frac {3}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + x x^{-\frac {2}{3} \, n - \frac {1}{2}} + \sqrt {x} \left (-\frac {b}{a}\right )^{\frac {2}{3}}}{x}\right ) - 4 \, b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x x^{-\frac {1}{3} \, n - \frac {1}{4}} - x^{\frac {3}{4}} \left (-\frac {b}{a}\right )^{\frac {1}{3}}}{x}\right ) - 12 \, b x^{\frac {1}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}}}{4 \, a^{2} n} \]
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Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.31 \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\frac {x^{- \frac {4 n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a n \Gamma \left (- \frac {1}{3}\right )} - \frac {4 b x^{- \frac {n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a^{2} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} \]
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\[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {4}{3} \, n - 1}}{b x^{n} + a} \,d x } \]
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\[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {4}{3} \, n - 1}}{b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {4\,n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \]
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