Optimal. Leaf size=20 \[ \frac {d x}{b c \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {21, 197}
\begin {gather*} \frac {d x}{b c \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 197
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx &=\frac {d \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{b}\\ &=\frac {d x}{b c \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} \frac {d x}{b c \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 19, normalized size = 0.95
method | result | size |
gosper | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
default | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
trager | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 27, normalized size = 1.35 \begin {gather*} \frac {\sqrt {d x^{2} + c} d x}{b c d x^{2} + b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {d \int \frac {1}{c \sqrt {c + d x^{2}} + d x^{2} \sqrt {c + d x^{2}}}\, dx}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 18, normalized size = 0.90 \begin {gather*} \frac {d x}{\sqrt {d x^{2} + c} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.77, size = 18, normalized size = 0.90 \begin {gather*} \frac {d\,x}{b\,c\,\sqrt {d\,x^2+c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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