Optimal. Leaf size=246 \[ -\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} c n}+\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} c n}+\frac{\sqrt{3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{7/3} c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{3 (c x)^{-4 n/3}}{4 a c n} \]
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Rubi [A] time = 0.15086, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {363, 362, 345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} c n}+\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} c n}+\frac{\sqrt{3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{7/3} c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{3 (c x)^{-4 n/3}}{4 a c n} \]
Antiderivative was successfully verified.
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Rule 363
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{(c x)^{-1-\frac{4 n}{3}}}{a+b x^n} \, dx &=\frac{\left (x^{4 n/3} (c x)^{-4 n/3}\right ) \int \frac{x^{-1-\frac{4 n}{3}}}{a+b x^n} \, dx}{c}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}-\frac{\left (b x^{4 n/3} (c x)^{-4 n/3}\right ) \int \frac{x^{-1-\frac{n}{3}}}{a+b x^n} \, dx}{a c}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{\left (3 b x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^3}} \, dx,x,x^{-n/3}\right )}{a c n}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{\left (3 b x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname{Subst}\left (\int \frac{x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a c n}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{\left (3 b^2 x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a^2 c n}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{\left (b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a^2 c n}-\frac{\left (b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\right ) \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^2 c n}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} c n}+\frac{\left (b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\right ) \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^{7/3} c n}-\frac{\left (3 b^{5/3} x^{4 n/3} (c x)^{-4 n/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^2 c n}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} c n}+\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/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} c n}-\frac{\left (3 b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\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^{7/3} c n}\\ &=-\frac{3 (c x)^{-4 n/3}}{4 a c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}+\frac{\sqrt{3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{a^{7/3} c n}-\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} c n}+\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/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} c n}\\ \end{align*}
Mathematica [C] time = 0.0102427, size = 39, normalized size = 0.16 \[ -\frac{3 x (c x)^{-\frac{4 n}{3}-1} \, _2F_1\left (-\frac{4}{3},1;-\frac{1}{3};-\frac{b x^n}{a}\right )}{4 a n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{4\,n}{3}}}}\, dx \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^{2} \int \frac{x^{\frac{2}{3} \, n}}{a^{2} b c^{\frac{4}{3} \, n + 1} x x^{n} + a^{3} c^{\frac{4}{3} \, n + 1} x}\,{d x} + \frac{3 \,{\left (4 \, b x^{n} - a\right )} c^{-\frac{4}{3} \, n - 1}}{4 \, a^{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 = 4.45635, size = 980, normalized size = 3.98 \begin{align*} \frac{4 \, \sqrt{3} b c^{-n - \frac{3}{4}} \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{2}{3}} x^{\frac{1}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} - \sqrt{3} b c^{-n - \frac{3}{4}}}{3 \, b c^{-n - \frac{3}{4}}}\right ) - 2 \, b c^{-n - \frac{3}{4}} \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} \log \left (\frac{\left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} x^{\frac{3}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} + x e^{\left (-\frac{1}{6} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} + \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{2}{3}} \sqrt{x}}{x}\right ) + 4 \, b c^{-n - \frac{3}{4}} \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} \log \left (\frac{x e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} - \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} x^{\frac{3}{4}}}{x}\right ) + 12 \, b c^{-n - \frac{3}{4}} x^{\frac{1}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} - 3 \, a x e^{\left (-\frac{1}{3} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )}}{4 \, a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.20668, size = 272, normalized size = 1.11 \begin{align*} \frac{c^{- \frac{4 n}{3}} x^{- \frac{4 n}{3}} \Gamma \left (- \frac{4}{3}\right )}{a c n \Gamma \left (- \frac{1}{3}\right )} - \frac{4 b c^{- \frac{4 n}{3}} x^{- \frac{n}{3}} \Gamma \left (- \frac{4}{3}\right )}{a^{2} c n \Gamma \left (- \frac{1}{3}\right )} + \frac{4 b^{\frac{4}{3}} c^{- \frac{4 n}{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}} c n \Gamma \left (- \frac{1}{3}\right )} + \frac{4 b^{\frac{4}{3}} c^{- \frac{4 n}{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}} c n \Gamma \left (- \frac{1}{3}\right )} + \frac{4 b^{\frac{4}{3}} c^{- \frac{4 n}{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}} c n \Gamma \left (- \frac{1}{3}\right )} \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{\left (c x\right )^{-\frac{4}{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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