Optimal. Leaf size=158 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} n}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} n}-\frac{3 x^{-n/3}}{a n} \]
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Rubi [A] time = 0.108181, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} n}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} n}-\frac{3 x^{-n/3}}{a n} \]
Antiderivative was successfully verified.
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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}}}{a+b x^n} \, dx &=-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^3}} \, dx,x,x^{-n/3}\right )}{n}\\ &=-\frac{3 \operatorname{Subst}\left (\int \frac{x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{n}\\ &=-\frac{3 x^{-n/3}}{a n}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a n}\\ &=-\frac{3 x^{-n/3}}{a n}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a n}+\frac{\sqrt [3]{b} \operatorname{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 n}\\ &=-\frac{3 x^{-n/3}}{a n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} n}-\frac{\sqrt [3]{b} \operatorname{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^{4/3} n}+\frac{\left (3 b^{2/3}\right ) \operatorname{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 n}\\ &=-\frac{3 x^{-n/3}}{a n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} n}-\frac{\sqrt [3]{b} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{4/3} n}+\frac{\left (3 \sqrt [3]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}\right )}{a^{4/3} n}\\ &=-\frac{3 x^{-n/3}}{a n}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} n}-\frac{\sqrt [3]{b} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{4/3} n}\\ \end{align*}
Mathematica [C] time = 0.0065309, size = 32, normalized size = 0.2 \[ -\frac{3 x^{-n/3} \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};-\frac{b x^n}{a}\right )}{a n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.047, size = 57, normalized size = 0.4 \begin{align*} -3\,{\frac{1}{an{x}^{n/3}}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{4}{n}^{3}{{\it \_Z}}^{3}-b \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+{\frac{{a}^{3}{n}^{2}{{\it \_R}}^{2}}{b}} \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*} -b \int \frac{x^{\frac{2}{3} \, n}}{a b x x^{n} + a^{2} x}\,{d x} - \frac{3}{a n x^{\frac{1}{3} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06053, size = 366, normalized size = 2.32 \begin{align*} -\frac{6 \, x x^{-\frac{1}{3} \, n - 1} - 2 \, \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x x^{-\frac{1}{3} \, n - 1} \left (\frac{b}{a}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - 2 \, \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (\frac{x x^{-\frac{1}{3} \, n - 1} + \left (\frac{b}{a}\right )^{\frac{1}{3}}}{x}\right ) + \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} x^{-\frac{2}{3} \, n - 2} - x x^{-\frac{1}{3} \, n - 1} \left (\frac{b}{a}\right )^{\frac{1}{3}} + \left (\frac{b}{a}\right )^{\frac{2}{3}}}{x^{2}}\right )}{2 \, a 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}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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