Optimal. Leaf size=32 \[ \frac{b \sqrt{c x^2} \log (x)}{x}-\frac{a \sqrt{c x^2}}{x^2} \]
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Rubi [A] time = 0.0067455, antiderivative size = 32, 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} \[ \frac{b \sqrt{c x^2} \log (x)}{x}-\frac{a \sqrt{c x^2}}{x^2} \]
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^3} \, dx &=\frac{\sqrt{c x^2} \int \frac{a+b x}{x^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{a}{x^2}+\frac{b}{x}\right ) \, dx}{x}\\ &=-\frac{a \sqrt{c x^2}}{x^2}+\frac{b \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0050649, size = 20, normalized size = 0.62 \[ \frac{c (b x \log (x)-a)}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 0.7 \begin{align*}{\frac{b\ln \left ( x \right ) x-a}{{x}^{2}}\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 = 1.52064, size = 46, normalized size = 1.44 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{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{\sqrt{c x^{2}} \left (a + b x\right )}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05994, size = 27, normalized size = 0.84 \begin{align*}{\left (b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) - \frac{a \mathrm{sgn}\left (x\right )}{x}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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