Optimal. Leaf size=21 \[ a^2 \log (x)+4 a b \sqrt{x}+b^2 x \]
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Rubi [A] time = 0.0119502, antiderivative size = 21, 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} \[ a^2 \log (x)+4 a b \sqrt{x}+b^2 x \]
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} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,\sqrt{x}\right )\\ &=4 a b \sqrt{x}+b^2 x+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0079906, size = 21, normalized size = 1. \[ a^2 \log (x)+4 a b \sqrt{x}+b^2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 1. \begin{align*}{b}^{2}x+{a}^{2}\ln \left ( x \right ) +4\,ab\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950582, size = 26, normalized size = 1.24 \begin{align*} b^{2} x + a^{2} \log \left (x\right ) + 4 \, a b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51504, size = 59, normalized size = 2.81 \begin{align*} b^{2} x + 2 \, a^{2} \log \left (\sqrt{x}\right ) + 4 \, a b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.17154, size = 20, normalized size = 0.95 \begin{align*} a^{2} \log{\left (x \right )} + 4 a b \sqrt{x} + b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10201, size = 27, normalized size = 1.29 \begin{align*} b^{2} x + a^{2} \log \left ({\left | x \right |}\right ) + 4 \, a b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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