Optimal. Leaf size=16 \[ \frac {x}{a \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197}
\begin {gather*} \frac {x}{a \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rubi steps
\begin {align*} \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx &=\frac {x}{a \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{a \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 15, normalized size = 0.94
method | result | size |
gosper | \(\frac {x}{a \sqrt {c \,x^{2}+a}}\) | \(15\) |
default | \(\frac {x}{a \sqrt {c \,x^{2}+a}}\) | \(15\) |
trager | \(\frac {x}{a \sqrt {c \,x^{2}+a}}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {c x^{2} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.10, size = 23, normalized size = 1.44 \begin {gather*} \frac {\sqrt {c x^{2} + a} x}{a c x^{2} + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 17, normalized size = 1.06 \begin {gather*} \frac {x}{a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {c x^{2} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{a\,\sqrt {c\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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