Optimal. Leaf size=25 \[ 2 \sqrt{x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]
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Rubi [A] time = 0.0112607, antiderivative size = 25, 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} \[ 2 \sqrt{x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]
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{x}} \, dx &=\int \frac{1}{\left (1+\sqrt [4]{x}\right ) \sqrt [4]{x}} \, dx\\ &=4 \operatorname{Subst}\left (\int \frac{x^2}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \operatorname{Subst}\left (\int \left (-1+x+\frac{1}{1+x}\right ) \, dx,x,\sqrt [4]{x}\right )\\ &=-4 \sqrt [4]{x}+2 \sqrt{x}+4 \log \left (1+\sqrt [4]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0089011, size = 25, normalized size = 1. \[ 2 \sqrt{x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.8 \begin{align*} -4\,\sqrt [4]{x}+4\,\ln \left ( 1+\sqrt [4]{x} \right ) +2\,\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78604, size = 26, normalized size = 1.04 \begin{align*} 2 \, \sqrt{x} - 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16614, size = 59, normalized size = 2.36 \begin{align*} 2 \, \sqrt{x} - 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.210391, size = 22, normalized size = 0.88 \begin{align*} - 4 \sqrt [4]{x} + 2 \sqrt{x} + 4 \log{\left (\sqrt [4]{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09892, size = 26, normalized size = 1.04 \begin{align*} 2 \, \sqrt{x} - 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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