Optimal. Leaf size=8 \[ 2 \tan ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 63, 203} \begin {gather*} 2 \tan ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x}+x^{3/2}} \, dx &=\int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 7, normalized size = 0.88 \begin {gather*} 2 \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.14, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 6, normalized size = 0.75 \begin {gather*} 2\,\mathrm {atan}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 7, normalized size = 0.88 \begin {gather*} 2 \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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