Integrand size = 15, antiderivative size = 47 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=-\frac {1}{a x}+\frac {2 b}{a^2 \sqrt {x}}-\frac {2 b^2 \log \left (a+b \sqrt {x}\right )}{a^3}+\frac {b^2 \log (x)}{a^3} \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46} \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=-\frac {2 b^2 \log \left (a+b \sqrt {x}\right )}{a^3}+\frac {b^2 \log (x)}{a^3}+\frac {2 b}{a^2 \sqrt {x}}-\frac {1}{a x} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{a x}+\frac {2 b}{a^2 \sqrt {x}}-\frac {2 b^2 \log \left (a+b \sqrt {x}\right )}{a^3}+\frac {b^2 \log (x)}{a^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=\frac {-a \left (a-2 b \sqrt {x}\right )-2 b^2 x \log \left (a+b \sqrt {x}\right )+b^2 x \log (x)}{a^3 x} \]
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Time = 5.85 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {1}{a x}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {2 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{3}}+\frac {2 b}{a^{2} \sqrt {x}}\) | \(44\) |
default | \(-\frac {1}{a x}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {2 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{3}}+\frac {2 b}{a^{2} \sqrt {x}}\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=-\frac {2 \, b^{2} x \log \left (b \sqrt {x} + a\right ) - 2 \, b^{2} x \log \left (\sqrt {x}\right ) - 2 \, a b \sqrt {x} + a^{2}}{a^{3} x} \]
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Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {1}{a x} & \text {for}\: b = 0 \\- \frac {1}{a x} + \frac {2 b}{a^{2} \sqrt {x}} + \frac {b^{2} \log {\left (x \right )}}{a^{3}} - \frac {2 b^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=-\frac {2 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{3}} + \frac {b^{2} \log \left (x\right )}{a^{3}} + \frac {2 \, b \sqrt {x} - a}{a^{2} x} \]
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Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=-\frac {2 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b \sqrt {x} - a^{2}}{a^{3} x} \]
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Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2} \, dx=-\frac {\frac {1}{a}-\frac {2\,b\,\sqrt {x}}{a^2}}{x}-\frac {4\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^3} \]
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