Optimal. Leaf size=41 \[ -\frac{a}{4 c x^3 \sqrt{c x^2}}-\frac{b}{3 c x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0074609, antiderivative size = 41, 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} \[ -\frac{a}{4 c x^3 \sqrt{c x^2}}-\frac{b}{3 c x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{x^2 \left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{a+b x}{x^5} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^5}+\frac{b}{x^4}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a}{4 c x^3 \sqrt{c x^2}}-\frac{b}{3 c x^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0082576, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c x^2} (3 a+4 b x)}{12 c^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.5 \begin{align*} -{\frac{4\,bx+3\,a}{12\,x} \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 = 1.04431, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{3 \, c^{\frac{3}{2}} x^{3}} - \frac{a}{4 \, c^{\frac{3}{2}} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58013, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (4 \, b x + 3 \, a\right )}}{12 \, c^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.712332, size = 32, normalized size = 0.78 \begin{align*} - \frac{a}{4 c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b}{3 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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