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 = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1979, 620, 206} \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 206
Rule 620
Rule 1979
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c x (a+b x)}} \, dx &=\int \frac {1}{\sqrt {a c x+b c x^2}} \, dx\\ &=2 \operatorname {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 = 0.00, size = 58, normalized size = 1.45 \begin {gather*} \frac {2 \sqrt {a} \sqrt {x} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {c x (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.24, size = 133, normalized size = 3.32 \begin {gather*} -\frac {\sqrt {b c} \log \left (a^2 c+8 b x \sqrt {b c} \sqrt {a c x+b c x^2}-4 a b c x-8 b^2 c x^2\right )}{2 b c}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {b} x \sqrt {b c}}{a \sqrt {c}}-\frac {2 \sqrt {b} \sqrt {a c x+b c x^2}}{a \sqrt {c}}\right )}{\sqrt {b} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, 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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giac [A] time = 0.49, size = 50, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {b c} \log \left ({\left | -2 \, {\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c x}\right )} b - \sqrt {b c} a \right |}\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.92 \begin {gather*} \frac {\ln \left (\frac {b c x +\frac {1}{2} a c}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c x}\right )}{\sqrt {b c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, 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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mupad [B] time = 3.47, 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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c x \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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