Optimal. Leaf size=27 \[ \frac {b x \log (x)}{\sqrt {c x^2}}-\frac {a}{\sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \begin {gather*} \frac {b x \log (x)}{\sqrt {c x^2}}-\frac {a}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{x \sqrt {c x^2}} \, dx &=\frac {x \int \frac {a+b x}{x^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {a}{\sqrt {c x^2}}+\frac {b x \log (x)}{\sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.85 \begin {gather*} \frac {c x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 30, normalized size = 1.11 \begin {gather*} \sqrt {c x^2} \left (\frac {b \log (x)}{c x}-\frac {a}{c x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 23, normalized size = 0.85 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \relax (x) - a\right )}}{c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 47, normalized size = 1.74 \begin {gather*} -\frac {b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {2 \, a \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.67 \begin {gather*} \frac {b x \ln \relax (x )-a}{\sqrt {c \,x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 17, normalized size = 0.63 \begin {gather*} \frac {b \log \relax (x)}{\sqrt {c}} - \frac {a}{\sqrt {c} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 22, normalized size = 0.81 \begin {gather*} -\frac {\frac {a}{\sqrt {x^2}}-b\,\ln \left (c\,x\right )\,\mathrm {sign}\relax (x)}{\sqrt {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 {a + b x}{x \sqrt {c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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