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 = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 43
Rule 266
Rule 1593
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ 2 \sqrt {x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 19, normalized size = 0.76 \[ 2 \, \sqrt {x} - 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 19, normalized size = 0.76 \[ 2 \, \sqrt {x} - 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.80 \[ 4 \ln \left (x^{\frac {1}{4}}+1\right )+2 \sqrt {x}-4 x^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 19, normalized size = 0.76 \[ 2 \, \sqrt {x} - 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 19, normalized size = 0.76 \[ 4\,\ln \left (x^{1/4}+1\right )+2\,\sqrt {x}-4\,x^{1/4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 22, normalized size = 0.88 \[ - 4 \sqrt [4]{x} + 2 \sqrt {x} + 4 \log {\left (\sqrt [4]{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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