Integrand size = 13, antiderivative size = 45 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\frac {\sqrt {x}}{a (a+b x)}+\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {44, 65, 211} \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)} \]
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Rule 44
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x}}{a (a+b x)}+\frac {\int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a} \\ & = \frac {\sqrt {x}}{a (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\frac {\sqrt {x}}{a (a+b x)}+\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{a \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(36\) |
default | \(\frac {\sqrt {x}}{a \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(36\) |
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Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\left [\frac {2 \, a b \sqrt {x} - \sqrt {-a b} {\left (b x + a\right )} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac {a b \sqrt {x} - \sqrt {a b} {\left (b x + a\right )} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right )}{a^{2} b^{2} x + a^{3} b}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (37) = 74\).
Time = 3.19 (sec) , antiderivative size = 277, normalized size of antiderivative = 6.16 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} + \frac {2 b \sqrt {x} \sqrt {- \frac {a}{b}}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} + \frac {b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} - \frac {b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\frac {\sqrt {x}}{a b x + a^{2}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {\sqrt {x}}{{\left (b x + a\right )} a} \]
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Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx=\frac {\sqrt {x}}{a\,\left (a+b\,x\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {b}} \]
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