Optimal. Leaf size=32 \[ -\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2} \]
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Rubi [A] time = 0.0067661, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ -\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^{3/2}} \, dx &=\int \left (\frac{a^2}{x^{3/2}}+\frac{2 a b}{\sqrt{x}}+b^2 \sqrt{x}\right ) \, dx\\ &=-\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0094845, size = 27, normalized size = 0.84 \[ \frac{2 \left (-3 a^2+6 a b x+b^2 x^2\right )}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{2}-12\,abx+6\,{a}^{2}}{3}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02959, size = 32, normalized size = 1. \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46386, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.730087, size = 31, normalized size = 0.97 \begin{align*} - \frac{2 a^{2}}{\sqrt{x}} + 4 a b \sqrt{x} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20122, size = 32, normalized size = 1. \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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