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 = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 43
Rule 190
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.67, size = 50, normalized size = 1.52 \[ \frac {2 \, {\left (a b \sqrt {x} - a^{2} + {\left (b^{2} x - a^{2}\right )} \log \left (b \sqrt {x} + a\right )\right )}}{b^{4} x - a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 30, normalized size = 0.91 \[ \frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{2}} + \frac {2 \, a}{{\left (b \sqrt {x} + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 96, normalized size = 2.91 \[ -\frac {2 a^{2}}{\left (b^{2} x -a^{2}\right ) b^{2}}+\frac {a}{\left (b \sqrt {x}+a \right ) b^{2}}+\frac {a}{\left (b \sqrt {x}-a \right ) b^{2}}+\frac {\ln \left (b \sqrt {x}+a \right )}{b^{2}}-\frac {\ln \left (b \sqrt {x}-a \right )}{b^{2}}+\frac {\ln \left (b^{2} x -a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 29, normalized size = 0.88 \[ \frac {2 \, \log \left (b \sqrt {x} + a\right )}{b^{2}} + \frac {2 \, a}{{\left (b \sqrt {x} + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 29, normalized size = 0.88 \[ \frac {2\,\ln \left (a+b\,\sqrt {x}\right )}{b^2}+\frac {2\,a}{b^2\,\left (a+b\,\sqrt {x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 80, normalized size = 2.42 \[ \begin {cases} \frac {2 a \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a b^{2} + b^{3} \sqrt {x}} + \frac {2 a}{a b^{2} + b^{3} \sqrt {x}} + \frac {2 b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a b^{2} + b^{3} \sqrt {x}} & \text {for}\: b \neq 0 \\\frac {x}{a^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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