Optimal. Leaf size=40 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, 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 {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a c x+b c x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{1-b c x^2} \, dx,x,\frac {x}{\sqrt {a c x+b c x^2}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 56, normalized size = 1.40 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b} \sqrt {c x (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 37, normalized size = 0.92
method | result | size |
default | \(\frac {\ln \left (\frac {\frac {1}{2} a c +b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c x}\right )}{\sqrt {b c}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 36, normalized size = 0.90 \begin {gather*} \frac {\log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{\sqrt {b c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 87, normalized size = 2.18 \begin {gather*} \left [\frac {\sqrt {b c} \log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{b c}, -\frac {2 \, \sqrt {-b c} \arctan \left (\frac {\sqrt {b c x^{2} + a c x} \sqrt {-b c}}{b c x}\right )}{b c}\right ] \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 {a c x + b c x^{2}}}\, 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. 76 vs.
\(2 (30) = 60\).
time = 3.15, size = 76, normalized size = 1.90 \begin {gather*} \frac {a^{2} c \log \left ({\left | -a c - 2 \, \sqrt {b c} {\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c x}\right )} \right |}\right )}{8 \, \sqrt {b c} b} + \frac {1}{4} \, \sqrt {b c x^{2} + a c x} {\left (2 \, x + \frac {a}{b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.90, size = 33, normalized size = 0.82 \begin {gather*} \frac {\ln \left (a\,c+2\,\sqrt {b\,c}\,\sqrt {c\,x\,\left (a+b\,x\right )}+2\,b\,c\,x\right )}{\sqrt {b\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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