Optimal. Leaf size=33 \[ \frac{b x \log (x)}{c \sqrt{c x^2}}-\frac{a}{c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0067799, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 43} \[ \frac{b x \log (x)}{c \sqrt{c x^2}}-\frac{a}{c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{a+b x}{x^2} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^2}+\frac{b}{x}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a}{c \sqrt{c x^2}}+\frac{b x \log (x)}{c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0029206, size = 22, normalized size = 0.67 \[ \frac{x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*}{{x}^{2} \left ( b\ln \left ( x \right ) x-a \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06939, size = 28, normalized size = 0.85 \begin{align*} \frac{b \log \left (x\right )}{c^{\frac{3}{2}}} - \frac{a}{\sqrt{c x^{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61938, size = 54, normalized size = 1.64 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{c^{2} 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{x \left (a + b x\right )}{\left (c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0683, size = 63, normalized size = 1.91 \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}}}}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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