Optimal. Leaf size=32 \[ 2 a^2 \sqrt{x}-\frac{4 a b}{\sqrt{x}}-\frac{2 b^2}{3 x^{3/2}} \]
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Rubi [A] time = 0.0091209, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ 2 a^2 \sqrt{x}-\frac{4 a b}{\sqrt{x}}-\frac{2 b^2}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^2}{\sqrt{x}} \, dx &=\int \frac{(b+a x)^2}{x^{5/2}} \, dx\\ &=\int \left (\frac{b^2}{x^{5/2}}+\frac{2 a b}{x^{3/2}}+\frac{a^2}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 b^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 a^2 \sqrt{x}\\ \end{align*}
Mathematica [A] time = 0.0077322, size = 26, normalized size = 0.81 \[ -\frac{2 \left (-3 a^2 x^2+6 a b x+b^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.8 \begin{align*}{\frac{6\,{a}^{2}{x}^{2}-12\,xab-2\,{b}^{2}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982192, size = 32, normalized size = 1. \begin{align*} 2 \, a^{2} \sqrt{x} - \frac{4 \, a b}{\sqrt{x}} - \frac{2 \, b^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97556, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (3 \, a^{2} x^{2} - 6 \, a b x - b^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.487387, size = 31, normalized size = 0.97 \begin{align*} 2 a^{2} \sqrt{x} - \frac{4 a b}{\sqrt{x}} - \frac{2 b^{2}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10572, size = 31, normalized size = 0.97 \begin{align*} 2 \, a^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + b^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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