Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=\frac {1}{a \left (a+b \sqrt {x}\right )^2}+\frac {2}{a^2 \left (a+b \sqrt {x}\right )}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^3}+\frac {\log (x)}{a^3} \]
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Time = 0.02 (sec) , antiderivative size = 53, 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} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^3}+\frac {\log (x)}{a^3}+\frac {2}{a^2 \left (a+b \sqrt {x}\right )}+\frac {1}{a \left (a+b \sqrt {x}\right )^2} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{a \left (a+b \sqrt {x}\right )^2}+\frac {2}{a^2 \left (a+b \sqrt {x}\right )}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^3}+\frac {\log (x)}{a^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=\frac {\frac {a \left (3 a+2 b \sqrt {x}\right )}{\left (a+b \sqrt {x}\right )^2}-2 \log \left (a+b \sqrt {x}\right )+\log (x)}{a^3} \]
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Time = 3.60 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {2 \ln \left (a +b \sqrt {x}\right )}{a^{3}}+\frac {1}{a \left (a +b \sqrt {x}\right )^{2}}+\frac {2}{a^{2} \left (a +b \sqrt {x}\right )}\) | \(48\) |
default | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {2 \ln \left (a +b \sqrt {x}\right )}{a^{3}}+\frac {1}{a \left (a +b \sqrt {x}\right )^{2}}+\frac {2}{a^{2} \left (a +b \sqrt {x}\right )}\) | \(48\) |
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (47) = 94\).
Time = 0.25 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.17 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=-\frac {a^{2} b^{2} x - 3 \, a^{4} + 2 \, {\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (a b^{3} x - 2 \, a^{3} b\right )} \sqrt {x}}{a^{3} b^{4} x^{2} - 2 \, a^{5} b^{2} x + a^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (48) = 96\).
Time = 0.52 (sec) , antiderivative size = 364, normalized size of antiderivative = 6.87 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (x \right )}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{3} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {a^{2} \sqrt {x} \log {\left (x \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} - \frac {2 a^{2} \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {3 a^{2} \sqrt {x}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {2 a b x \log {\left (x \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} - \frac {4 a b x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {2 a b x}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {b^{2} x^{\frac {3}{2}} \log {\left (x \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} - \frac {2 b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=\frac {2 \, b \sqrt {x} + 3 \, a}{a^{2} b^{2} x + 2 \, a^{3} b \sqrt {x} + a^{4}} - \frac {2 \, \log \left (b \sqrt {x} + a\right )}{a^{3}} + \frac {\log \left (x\right )}{a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=-\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b \sqrt {x} + 3 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{2} a^{3}} \]
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Time = 5.82 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^3 x} \, dx=\frac {\frac {3}{a}+\frac {2\,b\,\sqrt {x}}{a^2}}{b^2\,x+a^2+2\,a\,b\,\sqrt {x}}-\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^3} \]
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