Integrand size = 20, antiderivative size = 67 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\frac {16 b^2 n^2 \sqrt {d x}}{d}-\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2342, 2341} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=-\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {16 b^2 n^2 \sqrt {d x}}{d} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}-(4 b n) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx \\ & = \frac {16 b^2 n^2 \sqrt {d x}}{d}-\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\frac {2 x \left (a^2-4 a b n+8 b^2 n^2+2 b (a-2 b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right )}{\sqrt {d x}} \]
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Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d x}\, a^{2}+2 b^{2} \sqrt {d x}\, \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+16 b^{2} n^{2} \sqrt {d x}-8 b^{2} n \sqrt {d x}\, \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+4 \sqrt {d x}\, a b \ln \left (c \,x^{n}\right )-8 a b n \sqrt {d x}}{d}\) | \(92\) |
default | \(\frac {2 \sqrt {d x}\, a^{2}+2 b^{2} \sqrt {d x}\, \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+16 b^{2} n^{2} \sqrt {d x}-8 b^{2} n \sqrt {d x}\, \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+4 \sqrt {d x}\, a b \ln \left (c \,x^{n}\right )-8 a b n \sqrt {d x}}{d}\) | \(92\) |
risch | \(\frac {2 b^{2} x \ln \left (x^{n}\right )^{2}}{\sqrt {d x}}+\frac {2 b \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 \ln \left (x^{n}\right )}{\sqrt {d x}}+\frac {\left (4 a^{2}+8 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+32 b^{2} n^{2}+8 \ln \left (c \right ) a b +4 \ln \left (c \right )^{2} b^{2}-16 b^{2} \ln \left (c \right ) n -16 a b n -\pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-8 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-8 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}\right ) x}{2 \sqrt {d x}}\) | \(701\) |
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Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\frac {2 \, {\left (b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} - 4 \, a b n + a^{2} - 2 \, {\left (2 \, b^{2} n - a b\right )} \log \left (c\right ) - 2 \, {\left (2 \, b^{2} n^{2} - b^{2} n \log \left (c\right ) - a b n\right )} \log \left (x\right )\right )} \sqrt {d x}}{d} \]
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Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\frac {2 a^{2} x}{\sqrt {d x}} - \frac {8 a b n x}{\sqrt {d x}} + \frac {4 a b x \log {\left (c x^{n} \right )}}{\sqrt {d x}} + \frac {16 b^{2} n^{2} x}{\sqrt {d x}} - \frac {8 b^{2} n x \log {\left (c x^{n} \right )}}{\sqrt {d x}} + \frac {2 b^{2} x \log {\left (c x^{n} \right )}^{2}}{\sqrt {d x}} \]
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Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\frac {2 \, \sqrt {d x} b^{2} \log \left (c x^{n}\right )^{2}}{d} + 8 \, {\left (\frac {2 \, \sqrt {d x} n^{2}}{d} - \frac {\sqrt {d x} n \log \left (c x^{n}\right )}{d}\right )} b^{2} - \frac {8 \, \sqrt {d x} a b n}{d} + \frac {4 \, \sqrt {d x} a b \log \left (c x^{n}\right )}{d} + \frac {2 \, \sqrt {d x} a^{2}}{d} \]
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Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\frac {2 \, {\left ({\left (\sqrt {d x} \log \left (x\right )^{2} - 4 \, \sqrt {d x} \log \left (x\right ) + 8 \, \sqrt {d x}\right )} b^{2} n^{2} + 2 \, {\left (\sqrt {d x} \log \left (x\right ) - 2 \, \sqrt {d x}\right )} b^{2} n \log \left (c\right ) + \sqrt {d x} b^{2} \log \left (c\right )^{2} + 2 \, {\left (\sqrt {d x} \log \left (x\right ) - 2 \, \sqrt {d x}\right )} a b n + 2 \, \sqrt {d x} a b \log \left (c\right ) + \sqrt {d x} a^{2}\right )}}{d} \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {d x}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{\sqrt {d\,x}} \,d x \]
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