Optimal. Leaf size=29 \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0043576, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 37} \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin{align*} \int \frac{a+b x}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{a+b x}{x^3} \, dx}{c \sqrt{c x^2}}\\ &=-\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0027527, size = 22, normalized size = 0.76 \[ \frac{x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 0.6 \begin{align*} -{\frac{x \left ( 2\,bx+a \right ) }{2} \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 = 1.06174, size = 31, normalized size = 1.07 \begin{align*} -\frac{b}{\sqrt{c x^{2}} c} - \frac{a}{2 \, c^{\frac{3}{2}} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62874, size = 54, normalized size = 1.86 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, c^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.505787, size = 34, normalized size = 1.17 \begin{align*} - \frac{a x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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