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 = 0.0060676, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ \frac{b x \log (x)}{\sqrt{c x^2}}-\frac{a}{\sqrt{c x^2}} \]
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*}
Mathematica [A] time = 0.0060521, size = 23, normalized size = 0.85 \[ \frac{c x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 0.7 \begin{align*}{(b\ln \left ( x \right ) x-a){\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07141, size = 23, normalized size = 0.85 \begin{align*} \frac{b \log \left (x\right )}{\sqrt{c}} - \frac{a}{\sqrt{c} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82753, size = 51, normalized size = 1.89 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x}{x \sqrt{c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07331, size = 63, normalized size = 2.33 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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