Optimal. Leaf size=26 \[ -\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0036094, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 37} \[ -\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^3 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^4} \, dx}{\sqrt{c x^2}}\\ &=-\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0096495, size = 33, normalized size = 1.27 \[ \frac{c \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 1.2 \begin{align*} -{\frac{3\,{b}^{2}{x}^{2}+3\,abx+{a}^{2}}{3\,{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 = 1.07142, size = 45, normalized size = 1.73 \begin{align*} -\frac{b^{2}}{\sqrt{c} x} - \frac{a b}{\sqrt{c} x^{2}} - \frac{a^{2}}{3 \, \sqrt{c} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49316, size = 73, normalized size = 2.81 \begin{align*} -\frac{{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt{c x^{2}}}{3 \, c x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.608357, size = 53, normalized size = 2.04 \begin{align*} - \frac{a^{2}}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} - \frac{a b}{\sqrt{c} x \sqrt{x^{2}}} - \frac{b^{2}}{\sqrt{c} \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{{\left (b x + a\right )}^{2}}{\sqrt{c x^{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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