Optimal. Leaf size=176 \[ -\frac{c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac{c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac{\sqrt{3} c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt{3} \sqrt [3]{c}}\right )}{b^{7/3} n}+\frac{3 c x^{-n/3}}{b^2 n}-\frac{3 x^{-4 n/3}}{4 b n} \]
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Rubi [A] time = 0.142492, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {1584, 362, 345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac{c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac{c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac{\sqrt{3} c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt{3} \sqrt [3]{c}}\right )}{b^{7/3} n}+\frac{3 c x^{-n/3}}{b^2 n}-\frac{3 x^{-4 n/3}}{4 b n} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 362
Rule 345
Rule 193
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{-1-\frac{n}{3}}}{b x^n+c x^{2 n}} \, dx &=\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) \operatorname{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) \operatorname{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 ) \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.0078412, size = 34, normalized size = 0.19 \[ -\frac{3 x^{-4 n/3} \, _2F_1\left (-\frac{4}{3},1;-\frac{1}{3};-\frac{c x^n}{b}\right )}{4 b n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.065, size = 73, normalized size = 0.4 \begin{align*} 3\,{\frac{c}{{b}^{2}n{x}^{n/3}}}-{\frac{3}{4\,bn} \left ({x}^{{\frac{n}{3}}} \right ) ^{-4}}+\sum _{{\it \_R}={\it RootOf} \left ({b}^{7}{n}^{3}{{\it \_Z}}^{3}+{c}^{4} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+{\frac{{b}^{5}{n}^{2}{{\it \_R}}^{2}}{{c}^{3}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \int \frac{x^{\frac{2}{3} \, n}}{b^{2} c x x^{n} + b^{3} x}\,{d x} + \frac{3 \,{\left (4 \, c x^{n} - b\right )}}{4 \, b^{2} n x^{\frac{4}{3} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67486, size = 427, normalized size = 2.43 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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