Optimal. Leaf size=30 \[ \frac{2 a}{b^2 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^2} \]
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Rubi [A] time = 0.0076305, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 a}{b^2 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^{3/2}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}\right ) \, dx\\ &=\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0379939, size = 21, normalized size = 0.7 \[ \frac{2 (2 a+b x)}{b^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 20, normalized size = 0.7 \begin{align*} 2\,{\frac{bx+2\,a}{{b}^{2}\sqrt{bx+a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02808, size = 35, normalized size = 1.17 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b^{2}} + \frac{2 \, a}{\sqrt{b x + a} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46682, size = 61, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (b x + 2 \, a\right )} \sqrt{b x + a}}{b^{3} x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.737788, size = 37, normalized size = 1.23 \begin{align*} \begin{cases} \frac{4 a}{b^{2} \sqrt{a + b x}} + \frac{2 x}{b \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1926, size = 39, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{b x + a}}{b} + \frac{a}{\sqrt{b x + a} b}\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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