Optimal. Leaf size=31 \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]
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Rubi [A] time = 0.0152038, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2012, 1593, 266, 43} \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2012
Rule 1593
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{1+x}+\sqrt{1+x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{x}+\sqrt{x}} \, dx,x,1+x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (1+\sqrt [4]{x}\right ) \sqrt [4]{x}} \, dx,x,1+x\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^2}{1+x} \, dx,x,\sqrt [4]{1+x}\right )\\ &=4 \operatorname{Subst}\left (\int \left (-1+x+\frac{1}{1+x}\right ) \, dx,x,\sqrt [4]{1+x}\right )\\ &=-4 \sqrt [4]{1+x}+2 \sqrt{1+x}+4 \log \left (1+\sqrt [4]{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.011783, size = 31, normalized size = 1. \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 26, normalized size = 0.8 \begin{align*} -4\,\sqrt [4]{1+x}+4\,\ln \left ( 1+\sqrt [4]{1+x} \right ) +2\,\sqrt{1+x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1067, size = 34, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47412, size = 81, normalized size = 2.61 \begin{align*} 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.220459, size = 27, normalized size = 0.87 \begin{align*} - 4 \sqrt [4]{x + 1} + 2 \sqrt{x + 1} + 4 \log{\left (\sqrt [4]{x + 1} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11456, size = 34, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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