Optimal. Leaf size=35 \[ -\frac{a}{3 x^2 \sqrt{c x^2}}-\frac{b}{2 x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0067594, antiderivative size = 35, 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} \[ -\frac{a}{3 x^2 \sqrt{c x^2}}-\frac{b}{2 x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{x^3 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{a+b x}{x^4} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^4}+\frac{b}{x^3}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a}{3 x^2 \sqrt{c x^2}}-\frac{b}{2 x \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0061041, size = 22, normalized size = 0.63 \[ \frac{c (-2 a-3 b x)}{6 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{3\,bx+2\,a}{6\,{x}^{2}}{\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 = 1.02184, size = 26, normalized size = 0.74 \begin{align*} -\frac{b}{2 \, \sqrt{c} x^{2}} - \frac{a}{3 \, \sqrt{c} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54425, size = 54, normalized size = 1.54 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592965, size = 36, normalized size = 1.03 \begin{align*} - \frac{a}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} - \frac{b}{2 \sqrt{c} x \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{\sqrt{c x^{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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