Integrand size = 19, antiderivative size = 15 \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=\frac {2 \text {arcsinh}\left (\sqrt {1+b x}\right )}{b} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {65, 221} \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=\frac {2 \text {arcsinh}\left (\sqrt {b x+1}\right )}{b} \]
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Rule 65
Rule 221
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {1+b x}\right )}{b} \\ & = \frac {2 \sinh ^{-1}\left (\sqrt {1+b x}\right )}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {2+b x}}{\sqrt {1+b x}}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(13)=26\).
Time = 0.52 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.40
method | result | size |
default | \(\frac {\sqrt {\left (b x +1\right ) \left (b x +2\right )}\, \ln \left (\frac {\frac {3}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+3 b x +2}\right )}{\sqrt {b x +1}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=-\frac {\log \left (-2 \, b x + 2 \, \sqrt {b x + 2} \sqrt {b x + 1} - 3\right )}{b} \]
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\[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=\int \frac {1}{\sqrt {b x + 1} \sqrt {b x + 2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20 \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 3 \, b x + 2} b + 3 \, b\right )}{b} \]
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none
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=-\frac {2 \, \log \left (\sqrt {b x + 2} - \sqrt {b x + 1}\right )}{b} \]
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.87 \[ \int \frac {1}{\sqrt {1+b x} \sqrt {2+b x}} \, dx=\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\right )}{\left (\sqrt {b\,x+1}-1\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
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