Optimal. Leaf size=183 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.0769335, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {254, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 254
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{a+b \left (c x^n\right )^{3/n}} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3}}+\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3}}\\ &=\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{2 \sqrt [3]{a}}-\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{6 a^{2/3} \sqrt [3]{b}}\\ &=\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}\\ &=-\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}}+\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0618857, size = 133, normalized size = 0.73 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.681, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-1}\, 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*} \int \frac{1}{\left (c x^{n}\right )^{\frac{3}{n}} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3986, size = 1127, normalized size = 6.16 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b c^{\frac{3}{n}} \sqrt{-\frac{\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}}{b c^{\frac{3}{n}}}} \log \left (\frac{2 \, a b c^{\frac{3}{n}} x^{3} - 3 \, \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b c^{\frac{3}{n}} x^{2} + \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}}{b c^{\frac{3}{n}}}}}{b c^{\frac{3}{n}} x^{3} + a}\right ) - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} \log \left (a b c^{\frac{3}{n}} x^{2} - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x + \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} \log \left (a b c^{\frac{3}{n}} x + \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b c^{\frac{3}{n}}}, \frac{6 \, \sqrt{\frac{1}{3}} a b c^{\frac{3}{n}} \sqrt{\frac{\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}}{b c^{\frac{3}{n}}}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}}{b c^{\frac{3}{n}}}}}{a^{2}}\right ) - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} \log \left (a b c^{\frac{3}{n}} x^{2} - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x + \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} \log \left (a b c^{\frac{3}{n}} x + \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b c^{\frac{3}{n}}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \left (c x^{n}\right )^{\frac{3}{n}}}\, dx \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{1}{\left (c x^{n}\right )^{\frac{3}{n}} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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