Optimal. Leaf size=24 \[ \frac {2 \tanh ^{-1}\left (\frac {b x}{\sqrt {b x+b^2 x^2}}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {634, 212}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {b x}{\sqrt {b^2 x^2+b x}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b x+b^2 x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{1-b^2 x^2} \, dx,x,\frac {x}{\sqrt {b x+b^2 x^2}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {b x}{\sqrt {b x+b^2 x^2}}\right )}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(24)=48\).
time = 0.04, size = 56, normalized size = 2.33 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {1+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {1+b x}\right )}{\sqrt {b} \sqrt {b x (1+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 37, normalized size = 1.54
method | result | size |
default | \(\frac {\ln \left (\frac {\frac {1}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+b x}\right )}{\sqrt {b^{2}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 29, normalized size = 1.21 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + b x} b + b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.56, size = 27, normalized size = 1.12 \begin {gather*} -\frac {\log \left (-2 \, b x + 2 \, \sqrt {b^{2} x^{2} + b x} - 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b^{2} x^{2} + b x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (22) = 44\).
time = 2.22, size = 59, normalized size = 2.46 \begin {gather*} \frac {1}{4} \, \sqrt {b^{2} x^{2} + b x} {\left (2 \, x + \frac {1}{b}\right )} + \frac {\log \left ({\left | -2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + b x}\right )} {\left | b \right |} - b \right |}\right )}{8 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 36, normalized size = 1.50 \begin {gather*} \frac {\ln \left (\frac {x\,b^2+\frac {b}{2}}{\sqrt {b^2}}+\sqrt {b^2\,x^2+b\,x}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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