Optimal. Leaf size=51 \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]
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Rubi [A] time = 0.0279938, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0222261, size = 51, normalized size = 1. \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 44, normalized size = 0.9 \begin{align*} -{\frac{ax}{{b}^{2}}}+{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}+2\,{\frac{{a}^{2}\sqrt{x}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981197, size = 82, normalized size = 1.61 \begin{align*} -\frac{2 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{4}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33225, size = 107, normalized size = 2.1 \begin{align*} -\frac{3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt{x}}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.224569, size = 54, normalized size = 1.06 \begin{align*} \begin{cases} - \frac{2 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{4}} + \frac{2 a^{2} \sqrt{x}}{b^{3}} - \frac{a x}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11229, size = 61, normalized size = 1.2 \begin{align*} -\frac{2 \, a^{3} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, b^{2} x^{\frac{3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt{x}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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