Optimal. Leaf size=23 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {7, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 7
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {x^{\frac {1-n}{2}+\frac {1}{2} (-3+n)}}{\sqrt {a+b x}} \, dx &=\int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 56, normalized size = 2.43 \[ \left [\frac {\log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right )}{\sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 21, normalized size = 0.91 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.78 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 32, normalized size = 1.39 \[ \frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 17, normalized size = 0.74 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.09, size = 24, normalized size = 1.04 \[ - \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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