Optimal. Leaf size=27 \[ \frac {2 \sqrt {x}}{b}-\frac {2 a \log \left (a+b \sqrt {x}\right )}{b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {2 \sqrt {x}}{b}-\frac {2 a \log \left (a+b \sqrt {x}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{a+b \sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x}}{b}-\frac {2 a \log \left (a+b \sqrt {x}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 1.00 \[ \frac {2 \sqrt {x}}{b}-\frac {2 a \log \left (a+b \sqrt {x}\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 22, normalized size = 0.81 \[ -\frac {2 \, {\left (a \log \left (b \sqrt {x} + a\right ) - b \sqrt {x}\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 24, normalized size = 0.89 \[ -\frac {2 \, a \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{2}} + \frac {2 \, \sqrt {x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 57, normalized size = 2.11 \[ -\frac {a \ln \left (b \sqrt {x}+a \right )}{b^{2}}+\frac {a \ln \left (b \sqrt {x}-a \right )}{b^{2}}-\frac {a \ln \left (b^{2} x -a^{2}\right )}{b^{2}}+\frac {2 \sqrt {x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 27, normalized size = 1.00 \[ -\frac {2 \, a \log \left (b \sqrt {x} + a\right )}{b^{2}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 23, normalized size = 0.85 \[ \frac {2\,\sqrt {x}}{b}-\frac {2\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 27, normalized size = 1.00 \[ \begin {cases} - \frac {2 a \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{2}} + \frac {2 \sqrt {x}}{b} & \text {for}\: b \neq 0 \\\frac {x}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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