Optimal. Leaf size=28 \[ \frac{a \sqrt{c x^2} \log (x)}{x}+b \sqrt{c x^2} \]
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Rubi [A] time = 0.0049333, antiderivative size = 28, 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} \[ \frac{a \sqrt{c x^2} \log (x)}{x}+b \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)}{x^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{a+b x}{x} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (b+\frac{a}{x}\right ) \, dx}{x}\\ &=b \sqrt{c x^2}+\frac{a \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0042371, size = 20, normalized size = 0.71 \[ \frac{c x (a \log (x)+b x)}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.7 \begin{align*}{\frac{bx+a\ln \left ( x \right ) }{x}\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52884, size = 43, normalized size = 1.54 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{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{\sqrt{c x^{2}} \left (a + b x\right )}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06821, size = 23, normalized size = 0.82 \begin{align*}{\left (b x \mathrm{sgn}\left (x\right ) + a \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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