Integrand size = 15, antiderivative size = 67 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=\frac {2 b^2}{a^3 \left (a+b \sqrt {x}\right )}-\frac {1}{a^2 x}+\frac {4 b}{a^3 \sqrt {x}}-\frac {6 b^2 \log \left (a+b \sqrt {x}\right )}{a^4}+\frac {3 b^2 \log (x)}{a^4} \]
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Time = 0.03 (sec) , antiderivative size = 67, 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} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=-\frac {6 b^2 \log \left (a+b \sqrt {x}\right )}{a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {2 b^2}{a^3 \left (a+b \sqrt {x}\right )}+\frac {4 b}{a^3 \sqrt {x}}-\frac {1}{a^2 x} \]
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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)^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 b^2}{a^3 \left (a+b \sqrt {x}\right )}-\frac {1}{a^2 x}+\frac {4 b}{a^3 \sqrt {x}}-\frac {6 b^2 \log \left (a+b \sqrt {x}\right )}{a^4}+\frac {3 b^2 \log (x)}{a^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=\frac {\frac {a \left (-a^2+3 a b \sqrt {x}+6 b^2 x\right )}{\left (a+b \sqrt {x}\right ) x}-6 b^2 \log \left (a+b \sqrt {x}\right )+3 b^2 \log (x)}{a^4} \]
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Time = 3.56 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {1}{a^{2} x}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {6 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{4}}+\frac {4 b}{a^{3} \sqrt {x}}+\frac {2 b^{2}}{a^{3} \left (a +b \sqrt {x}\right )}\) | \(62\) |
default | \(-\frac {1}{a^{2} x}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {6 b^{2} \ln \left (a +b \sqrt {x}\right )}{a^{4}}+\frac {4 b}{a^{3} \sqrt {x}}+\frac {2 b^{2}}{a^{3} \left (a +b \sqrt {x}\right )}\) | \(62\) |
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=-\frac {3 \, a^{2} b^{2} x - a^{4} + 6 \, {\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 6 \, {\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (3 \, a b^{3} x - 2 \, a^{3} b\right )} \sqrt {x}}{a^{4} b^{2} x^{2} - a^{6} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (65) = 130\).
Time = 0.42 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.55 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{2}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{2} x} & \text {for}\: b = 0 \\- \frac {1}{2 b^{2} x^{2}} & \text {for}\: a = 0 \\- \frac {a^{3} \sqrt {x}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {3 a^{2} b x}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {3 a b^{2} x^{\frac {3}{2}} \log {\left (x \right )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} - \frac {6 a b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {6 a b^{2} x^{\frac {3}{2}}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {3 b^{3} x^{2} \log {\left (x \right )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} - \frac {6 b^{3} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=\frac {6 \, b^{2} x + 3 \, a b \sqrt {x} - a^{2}}{a^{3} b x^{\frac {3}{2}} + a^{4} x} - \frac {6 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{4}} + \frac {3 \, b^{2} \log \left (x\right )}{a^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=-\frac {6 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{4}} + \frac {3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {6 \, a b^{2} x + 3 \, a^{2} b \sqrt {x} - a^{3}}{{\left (b \sqrt {x} + a\right )} a^{4} x} \]
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Time = 5.69 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^2 x^2} \, dx=\frac {\frac {3\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {6\,b^2\,x}{a^3}}{a\,x+b\,x^{3/2}}-\frac {12\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^4} \]
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