Integrand size = 18, antiderivative size = 37 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=-\frac {4 b n \sqrt {d x}}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2341} \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {4 b n \sqrt {d x}}{d} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {4 b n \sqrt {d x}}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=\frac {2 x \left (a-2 b n+b \log \left (c x^n\right )\right )}{\sqrt {d x}} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d x}\, a +2 \sqrt {d x}\, b \ln \left (c \,x^{n}\right )-4 b n \sqrt {d x}}{d}\) | \(36\) |
default | \(\frac {2 \sqrt {d x}\, a +2 \sqrt {d x}\, b \ln \left (c \,x^{n}\right )-4 b n \sqrt {d x}}{d}\) | \(36\) |
parts | \(\frac {2 a \sqrt {d x}}{d}+\frac {2 b \sqrt {d x}\, \ln \left (c \,x^{n}\right )}{d}-\frac {4 b n \sqrt {d x}}{d}\) | \(42\) |
risch | \(\frac {2 b x \ln \left (x^{n}\right )}{\sqrt {d x}}+\frac {\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )-4 b n +2 a \right ) x}{\sqrt {d x}}\) | \(117\) |
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=\frac {2 \, {\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {d x}}{d} \]
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Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=\frac {2 a x}{\sqrt {d x}} - \frac {4 b n x}{\sqrt {d x}} + \frac {2 b x \log {\left (c x^{n} \right )}}{\sqrt {d x}} \]
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Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=-\frac {4 \, \sqrt {d x} b n}{d} + \frac {2 \, \sqrt {d x} b \log \left (c x^{n}\right )}{d} + \frac {2 \, \sqrt {d x} a}{d} \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=\frac {2 \, {\left ({\left (\sqrt {d x} \log \left (x\right ) - 2 \, \sqrt {d x}\right )} b n + \sqrt {d x} b \log \left (c\right ) + \sqrt {d x} a\right )}}{d} \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{\sqrt {d\,x}} \,d x \]
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