Optimal. Leaf size=24 \[ -\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x) \]
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Rubi [A] time = 0.0132717, antiderivative size = 24, 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 = {266, 43} \[ -\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^2}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{x^3}+\frac{2 a b}{x^2}+\frac{b^2}{x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0136935, size = 24, normalized size = 1. \[ -\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{x}}+{b}^{2}\ln \left ( x \right ) -4\,{\frac{ab}{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954224, size = 31, normalized size = 1.29 \begin{align*} b^{2} \log \left (x\right ) - \frac{4 \, a b \sqrt{x} + a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43834, size = 65, normalized size = 2.71 \begin{align*} \frac{2 \, b^{2} x \log \left (\sqrt{x}\right ) - 4 \, a b \sqrt{x} - a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.413131, size = 20, normalized size = 0.83 \begin{align*} - \frac{a^{2}}{x} - \frac{4 a b}{\sqrt{x}} + b^{2} \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11112, size = 32, normalized size = 1.33 \begin{align*} b^{2} \log \left ({\left | x \right |}\right ) - \frac{4 \, a b \sqrt{x} + a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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