Optimal. Leaf size=33 \[ \frac{2 a}{b^2 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^2} \]
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Rubi [A] time = 0.0178957, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{2 a}{b^2 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a}{b^2 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0178077, size = 29, normalized size = 0.88 \[ \frac{2 \left (\frac{a}{a+b \sqrt{x}}+\log \left (a+b \sqrt{x}\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 96, normalized size = 2.9 \begin{align*} -2\,{\frac{{a}^{2}}{ \left ({b}^{2}x-{a}^{2} \right ){b}^{2}}}+{\frac{\ln \left ({b}^{2}x-{a}^{2} \right ) }{{b}^{2}}}+{\frac{a}{{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-1}}+{\frac{1}{{b}^{2}}\ln \left ( a+b\sqrt{x} \right ) }+{\frac{a}{{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-1}}-{\frac{1}{{b}^{2}}\ln \left ( b\sqrt{x}-a \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948855, size = 39, normalized size = 1.18 \begin{align*} \frac{2 \, \log \left (b \sqrt{x} + a\right )}{b^{2}} + \frac{2 \, a}{{\left (b \sqrt{x} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30208, size = 103, normalized size = 3.12 \begin{align*} \frac{2 \,{\left (a b \sqrt{x} - a^{2} +{\left (b^{2} x - a^{2}\right )} \log \left (b \sqrt{x} + a\right )\right )}}{b^{4} x - a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.54428, size = 80, normalized size = 2.42 \begin{align*} \begin{cases} \frac{2 a \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{2} + b^{3} \sqrt{x}} + \frac{2 a}{a b^{2} + b^{3} \sqrt{x}} + \frac{2 b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{2} + b^{3} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x}{a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09635, size = 41, normalized size = 1.24 \begin{align*} \frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{2}} + \frac{2 \, a}{{\left (b \sqrt{x} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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