Optimal. Leaf size=8 \[ \tan ^{-1}\left (2 \sqrt {x}\right ) \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 65, 209}
\begin {gather*} \text {ArcTan}\left (2 \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x}+4 x^{3/2}} \, dx &=\int \frac {1}{\sqrt {x} (1+4 x)} \, dx\\ &=2 \text {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 = 0.01, 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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Maple [A]
time = 0.20, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(\arctan \left (2 \sqrt {x}\right )\) | \(7\) |
default | \(\arctan \left (2 \sqrt {x}\right )\) | \(7\) |
meijerg | \(\arctan \left (2 \sqrt {x}\right )\) | \(7\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {4 x \RootOf \left (\textit {\_Z}^{2}+1\right )-\RootOf \left (\textit {\_Z}^{2}+1\right )-4 \sqrt {x}}{1+4 x}\right )}{2}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, 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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Fricas [A]
time = 0.35, 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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Sympy [A]
time = 0.07, 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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Giac [A]
time = 4.21, 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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