Integrand size = 18, antiderivative size = 63 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=-\frac {b p}{a \sqrt {x}}+\frac {b^2 p \log \left (a+b \sqrt {x}\right )}{a^2}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x}-\frac {b^2 p \log (x)}{2 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2504, 2442, 46} \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=\frac {b^2 p \log \left (a+b \sqrt {x}\right )}{a^2}-\frac {b^2 p \log (x)}{2 a^2}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x}-\frac {b p}{a \sqrt {x}} \]
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Rule 46
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^3} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x}+(b p) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x}+(b p) \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {b p}{a \sqrt {x}}+\frac {b^2 p \log \left (a+b \sqrt {x}\right )}{a^2}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x}-\frac {b^2 p \log (x)}{2 a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x}-\frac {b p \left (\frac {2 a}{\sqrt {x}}-2 b \log \left (a+b \sqrt {x}\right )+b \log (x)\right )}{2 a^2} \]
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Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{x}+\frac {p b \left (-\frac {2}{a \sqrt {x}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {2 b \ln \left (a +b \sqrt {x}\right )}{a^{2}}\right )}{2}\) | \(54\) |
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none
Time = 0.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=-\frac {b^{2} p x \log \left (\sqrt {x}\right ) + a b p \sqrt {x} + a^{2} \log \left (c\right ) - {\left (b^{2} p x - a^{2} p\right )} \log \left (b \sqrt {x} + a\right )}{a^{2} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (56) = 112\).
Time = 11.00 (sec) , antiderivative size = 352, normalized size of antiderivative = 5.59 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=\begin {cases} - \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {p}{2 x} - \frac {\log {\left (c \left (b \sqrt {x}\right )^{p} \right )}}{x} & \text {for}\: a = 0 \\- \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = - b \sqrt {x} \\- \frac {2 a^{3} \sqrt {x} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} - \frac {2 a^{2} b p x}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} - \frac {2 a^{2} b x \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} - \frac {a b^{2} p x^{\frac {3}{2}} \log {\left (x \right )}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} - \frac {2 a b^{2} p x^{\frac {3}{2}}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} + \frac {2 a b^{2} x^{\frac {3}{2}} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} - \frac {b^{3} p x^{2} \log {\left (x \right )}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} + \frac {2 b^{3} x^{2} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{2 a^{3} x^{\frac {3}{2}} + 2 a^{2} b x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=\frac {1}{2} \, b p {\left (\frac {2 \, b \log \left (b \sqrt {x} + a\right )}{a^{2}} - \frac {b \log \left (x\right )}{a^{2}} - \frac {2}{a \sqrt {x}}\right )} - \frac {\log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right )}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (55) = 110\).
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.10 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=-\frac {\frac {b^{3} p \log \left (b \sqrt {x} + a\right )}{{\left (b \sqrt {x} + a\right )}^{2} - 2 \, {\left (b \sqrt {x} + a\right )} a + a^{2}} - \frac {b^{3} p \log \left (b \sqrt {x} + a\right )}{a^{2}} + \frac {b^{3} p \log \left (b \sqrt {x}\right )}{a^{2}} + \frac {{\left (b \sqrt {x} + a\right )} b^{3} p - a b^{3} p + a b^{3} \log \left (c\right )}{{\left (b \sqrt {x} + a\right )}^{2} a - 2 \, {\left (b \sqrt {x} + a\right )} a^{2} + a^{3}}}{b} \]
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Time = 1.75 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^2} \, dx=\frac {2\,b^2\,p\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^2}-\frac {\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{x}-\frac {b\,p}{a\,\sqrt {x}} \]
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