Optimal. Leaf size=29 \[ \frac{a x \log (x)}{\sqrt{c x^2}}+\frac{b x^2}{\sqrt{c x^2}} \]
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Rubi [A] time = 0.0051277, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac{a x \log (x)}{\sqrt{c x^2}}+\frac{b x^2}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{\sqrt{c x^2}} \, dx &=\frac{x \int \frac{a+b x}{x} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (b+\frac{a}{x}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{b x^2}{\sqrt{c x^2}}+\frac{a x \log (x)}{\sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0016435, size = 19, normalized size = 0.66 \[ \frac{x (a \log (x)+b x)}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 0.6 \begin{align*}{x \left ( bx+a\ln \left ( x \right ) \right ){\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 = 1.04629, size = 27, normalized size = 0.93 \begin{align*} \frac{a \log \left (x\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79276, size = 49, normalized size = 1.69 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c x} \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}{\sqrt{c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07634, size = 47, normalized size = 1.62 \begin{align*} -\frac{a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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