Optimal. Leaf size=34 \[ \sqrt {x} \sqrt {1+a x}+\frac {\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, 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 = {52, 56, 221}
\begin {gather*} \sqrt {x} \sqrt {a x+1}+\frac {\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {1+a x}+\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx\\ &=\sqrt {x} \sqrt {1+a x}+\text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {1+a x}+\frac {\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 46, normalized size = 1.35 \begin {gather*} \sqrt {x} \sqrt {1+a x}-\frac {\log \left (-\sqrt {a} \sqrt {x}+\sqrt {1+a x}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(24)=48\).
time = 0.08, size = 57, normalized size = 1.68
method | result | size |
meijerg | \(-\frac {-2 \sqrt {\pi }\, \sqrt {a}\, \sqrt {x}\, \sqrt {a x +1}-2 \sqrt {\pi }\, \arcsinh \left (\sqrt {a}\, \sqrt {x}\right )}{2 \sqrt {a}\, \sqrt {\pi }}\) | \(41\) |
default | \(\sqrt {x}\, \sqrt {a x +1}+\frac {\sqrt {\left (a x +1\right ) x}\, \ln \left (\frac {\frac {1}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+x}\right )}{2 \sqrt {a x +1}\, \sqrt {x}\, \sqrt {a}}\) | \(57\) |
risch | \(\sqrt {x}\, \sqrt {a x +1}+\frac {\sqrt {\left (a x +1\right ) x}\, \ln \left (\frac {\frac {1}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+x}\right )}{2 \sqrt {a x +1}\, \sqrt {x}\, \sqrt {a}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (24) = 48\).
time = 0.49, size = 68, normalized size = 2.00 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {a} - \frac {\sqrt {a x + 1}}{\sqrt {x}}}{\sqrt {a} + \frac {\sqrt {a x + 1}}{\sqrt {x}}}\right )}{2 \, \sqrt {a}} - \frac {\sqrt {a x + 1}}{{\left (a - \frac {a x + 1}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.08, size = 90, normalized size = 2.65 \begin {gather*} \left [\frac {2 \, \sqrt {a x + 1} a \sqrt {x} + \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a x + 1} \sqrt {a} \sqrt {x} + 1\right )}{2 \, a}, \frac {\sqrt {a x + 1} a \sqrt {x} - \sqrt {-a} \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a}}{a \sqrt {x}}\right )}{a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.80, size = 29, normalized size = 0.85 \begin {gather*} \sqrt {x} \sqrt {a x + 1} + \frac {\operatorname {asinh}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.00, size = 36, normalized size = 1.06 \begin {gather*} \sqrt {x}\,\sqrt {a\,x+1}+\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {a\,x+1}-1}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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