Optimal. Leaf size=29 \[ -\frac{(a+b x)^3}{3 a c x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0043833, antiderivative size = 29, 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} \[ -\frac{(a+b x)^3}{3 a c x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^4} \, dx}{c \sqrt{c x^2}}\\ &=-\frac{(a+b x)^3}{3 a c x^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0117217, size = 36, normalized size = 1.24 \[ \frac{c x^2 \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 0.9 \begin{align*} -{\frac{3\,{b}^{2}{x}^{2}+3\,abx+{a}^{2}}{3} \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 = 1.05268, size = 50, normalized size = 1.72 \begin{align*} -\frac{b^{2}}{\sqrt{c x^{2}} c} - \frac{a b}{c^{\frac{3}{2}} x^{2}} - \frac{a^{2}}{3 \, c^{\frac{3}{2}} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48277, size = 76, normalized size = 2.62 \begin{align*} -\frac{{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt{c x^{2}}}{3 \, c^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.610677, size = 53, normalized size = 1.83 \begin{align*} - \frac{a^{2}}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{a b x}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b^{2} x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{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}}{\left (c x^{2}\right )^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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