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 = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 43
Rule 266
Rule 1593
Rule 2012
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*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ 2 \sqrt {x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 25, normalized size = 0.81 \[ 2 \, \sqrt {x + 1} - 4 \, {\left (x + 1\right )}^{\frac {1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 25, normalized size = 0.81 \[ 2 \, \sqrt {x + 1} - 4 \, {\left (x + 1\right )}^{\frac {1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 0.84 \[ 4 \ln \left (1+\left (x +1\right )^{\frac {1}{4}}\right )-4 \left (x +1\right )^{\frac {1}{4}}+2 \sqrt {x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 25, normalized size = 0.81 \[ 2 \, \sqrt {x + 1} - 4 \, {\left (x + 1\right )}^{\frac {1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac {1}{4}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 25, normalized size = 0.81 \[ 4\,\ln \left ({\left (x+1\right )}^{1/4}+1\right )+2\,\sqrt {x+1}-4\,{\left (x+1\right )}^{1/4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 27, normalized size = 0.87 \[ - 4 \sqrt [4]{x + 1} + 2 \sqrt {x + 1} + 4 \log {\left (\sqrt [4]{x + 1} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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