Optimal. Leaf size=8 \[ \tan ^{-1}\left (2 \sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1593, 63, 203} \begin {gather*} \tan ^{-1}\left (2 \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}+4 x^{3/2}} \, dx &=\int \frac {1}{\sqrt {x} (1+4 x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\sqrt {x}\right )\\ &=\tan ^{-1}\left (2 \sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (2 \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*} \tan ^{-1}\left (2 \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 6, normalized size = 0.75 \begin {gather*} \arctan \left (2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 6, normalized size = 0.75 \begin {gather*} \arctan \left (2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 7, normalized size = 0.88 \begin {gather*} \arctan \left (2 \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 6, normalized size = 0.75 \begin {gather*} \arctan \left (2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 6, normalized size = 0.75 \begin {gather*} \mathrm {atan}\left (2\,\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 7, normalized size = 0.88 \begin {gather*} \operatorname {atan}{\left (2 \sqrt {x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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