Optimal. Leaf size=21 \[ \frac {2 \sqrt {c+d (a+b x)}}{b d} \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32}
\begin {gather*} \frac {2 \sqrt {d (a+b x)+c}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 33
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c+d (a+b x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {c+d x}} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 \sqrt {c+d (a+b x)}}{b d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c+a d+b d x}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 20, normalized size = 0.95
method | result | size |
gosper | \(\frac {2 \sqrt {b d x +a d +c}}{b d}\) | \(20\) |
derivativedivides | \(\frac {2 \sqrt {b d x +a d +c}}{b d}\) | \(20\) |
default | \(\frac {2 \sqrt {b d x +a d +c}}{b d}\) | \(20\) |
trager | \(\frac {2 \sqrt {b d x +a d +c}}{b d}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {{\left (b x + a\right )} d + c}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b d x + a d + c}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.61, size = 31, normalized size = 1.48 \begin {gather*} \begin {cases} \frac {x}{\sqrt {a d + c}} & \text {for}\: b = 0 \\\frac {x}{\sqrt {c}} & \text {for}\: d = 0 \\\frac {2 \sqrt {c + d \left (a + b x\right )}}{b d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.21, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {b d x + a d + c}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 19, normalized size = 0.90 \begin {gather*} \frac {2\,\sqrt {c+d\,\left (a+b\,x\right )}}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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