Optimal. Leaf size=73 \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]
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Rubi [A] time = 0.0282093, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1593, 266, 43} \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx &=\int \frac{1}{\left (1+\sqrt [12]{x}\right ) \sqrt [4]{x}} \, dx\\ &=12 \operatorname{Subst}\left (\int \frac{x^8}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname{Subst}\left (\int \left (-1+x-x^2+x^3-x^4+x^5-x^6+x^7+\frac{1}{1+x}\right ) \, dx,x,\sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}-\frac{12 x^{7/12}}{7}+\frac{3 x^{2/3}}{2}+12 \log \left (1+\sqrt [12]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0240054, size = 73, normalized size = 1. \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.087, size = 173, normalized size = 2.4 \begin{align*} -\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) +2\,\ln \left ( \sqrt [3]{x}-1 \right ) +\ln \left ( -1+\sqrt{x} \right ) -\ln \left ( 1+\sqrt{x} \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) +\ln \left ( 1-\sqrt [6]{x}+\sqrt [3]{x} \right ) +2\,\ln \left ( \sqrt [6]{x}-1 \right ) -\ln \left ( \sqrt [3]{x}+\sqrt [6]{x}+1 \right ) -2\,\ln \left ( \sqrt [4]{x}-1 \right ) -{\frac{12}{7}{x}^{{\frac{7}{12}}}}-{\frac{12}{5}{x}^{{\frac{5}{12}}}}-12\,{x}^{1/12}+2\,\ln \left ( 1+\sqrt [4]{x} \right ) +\ln \left ( x-1 \right ) +{\frac{3}{2}{x}^{{\frac{2}{3}}}}+6\,\sqrt [6]{x}-4\,\sqrt [4]{x}+3\,\sqrt [3]{x}+2\,\sqrt{x}+4\,\ln \left ( 1+{x}^{1/12} \right ) -2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) -4\,\ln \left ({x}^{1/12}-1 \right ) +2\,\ln \left ( \sqrt [6]{x}+{x}^{1/12}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.29995, size = 66, normalized size = 0.9 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18504, size = 176, normalized size = 2.41 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\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}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1225, size = 66, normalized size = 0.9 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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