Optimal. Leaf size=20 \[ \frac {x \log (a+b x)}{b \sqrt {c x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 31} \begin {gather*} \frac {x \log (a+b x)}{b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 31
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {c x^2} (a+b x)} \, dx &=\frac {x \int \frac {1}{a+b x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \log (a+b x)}{b \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \log (a+b x)}{b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 25, normalized size = 1.25 \begin {gather*} \frac {\sqrt {c x^2} \log (a+b x)}{b c x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 23, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c x^{2}} \log \left (b x + a\right )}{b c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 36, normalized size = 1.80 \begin {gather*} -\frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} \frac {x \ln \left (b x +a \right )}{\sqrt {c \,x^{2}}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 46, normalized size = 2.30 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b \sqrt {c}} + \frac {\log \left (b x\right )}{b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \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 {x}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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