Integrand size = 28, antiderivative size = 125 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=-\frac {12 a b n^3 x^m}{m^4}-\frac {a^2 n x^{2 m}}{4 m^2}+\frac {12 a b n^2 x^m \log \left (c x^n\right )}{m^3}+\frac {a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac {6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac {2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac {b^2 \log ^6\left (c x^n\right )}{6 n} \]
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Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2619, 2341, 2342, 2339, 30} \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=\frac {a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac {a^2 n x^{2 m}}{4 m^2}+\frac {12 a b n^2 x^m \log \left (c x^n\right )}{m^3}-\frac {6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac {2 a b x^m \log ^3\left (c x^n\right )}{m}-\frac {12 a b n^3 x^m}{m^4}+\frac {b^2 \log ^6\left (c x^n\right )}{6 n} \]
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Rule 30
Rule 2339
Rule 2341
Rule 2342
Rule 2619
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x^{-1+2 m} \log \left (c x^n\right )+2 a b x^{-1+m} \log ^3\left (c x^n\right )+\frac {b^2 \log ^5\left (c x^n\right )}{x}\right ) \, dx \\ & = a^2 \int x^{-1+2 m} \log \left (c x^n\right ) \, dx+(2 a b) \int x^{-1+m} \log ^3\left (c x^n\right ) \, dx+b^2 \int \frac {\log ^5\left (c x^n\right )}{x} \, dx \\ & = -\frac {a^2 n x^{2 m}}{4 m^2}+\frac {a^2 x^{2 m} \log \left (c x^n\right )}{2 m}+\frac {2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac {b^2 \text {Subst}\left (\int x^5 \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac {(6 a b n) \int x^{-1+m} \log ^2\left (c x^n\right ) \, dx}{m} \\ & = -\frac {a^2 n x^{2 m}}{4 m^2}+\frac {a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac {6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac {2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac {b^2 \log ^6\left (c x^n\right )}{6 n}+\frac {\left (12 a b n^2\right ) \int x^{-1+m} \log \left (c x^n\right ) \, dx}{m^2} \\ & = -\frac {12 a b n^3 x^m}{m^4}-\frac {a^2 n x^{2 m}}{4 m^2}+\frac {12 a b n^2 x^m \log \left (c x^n\right )}{m^3}+\frac {a^2 x^{2 m} \log \left (c x^n\right )}{2 m}-\frac {6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac {2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac {b^2 \log ^6\left (c x^n\right )}{6 n} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=-\frac {a n x^m \left (48 b n^2+a m^2 x^m\right )}{4 m^4}+\frac {a x^m \left (24 b n^2+a m^2 x^m\right ) \log \left (c x^n\right )}{2 m^3}-\frac {6 a b n x^m \log ^2\left (c x^n\right )}{m^2}+\frac {2 a b x^m \log ^3\left (c x^n\right )}{m}+\frac {b^2 \log ^6\left (c x^n\right )}{6 n} \]
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Time = 2.97 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(-\frac {-2 b^{2} \ln \left (c \,x^{n}\right )^{6} m^{4}-24 x^{m} \ln \left (c \,x^{n}\right )^{3} a b \,m^{3} n -6 x^{2 m} \ln \left (c \,x^{n}\right ) a^{2} m^{3} n +72 a b \,n^{2} \ln \left (c \,x^{n}\right )^{2} x^{m} m^{2}+3 a^{2} n^{2} x^{2 m} m^{2}-144 a b \,n^{3} \ln \left (c \,x^{n}\right ) x^{m} m +144 a b \,n^{4} x^{m}}{12 m^{4} n}\) | \(129\) |
risch | \(\text {Expression too large to display}\) | \(14983\) |
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Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (119) = 238\).
Time = 0.32 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.14 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=\frac {2 \, b^{2} m^{4} n^{5} \log \left (x\right )^{6} + 12 \, b^{2} m^{4} n^{4} \log \left (c\right ) \log \left (x\right )^{5} + 30 \, b^{2} m^{4} n^{3} \log \left (c\right )^{2} \log \left (x\right )^{4} + 40 \, b^{2} m^{4} n^{2} \log \left (c\right )^{3} \log \left (x\right )^{3} + 30 \, b^{2} m^{4} n \log \left (c\right )^{4} \log \left (x\right )^{2} + 12 \, b^{2} m^{4} \log \left (c\right )^{5} \log \left (x\right ) + 3 \, {\left (2 \, a^{2} m^{3} n \log \left (x\right ) + 2 \, a^{2} m^{3} \log \left (c\right ) - a^{2} m^{2} n\right )} x^{2 \, m} + 24 \, {\left (a b m^{3} n^{3} \log \left (x\right )^{3} + a b m^{3} \log \left (c\right )^{3} - 3 \, a b m^{2} n \log \left (c\right )^{2} + 6 \, a b m n^{2} \log \left (c\right ) - 6 \, a b n^{3} + 3 \, {\left (a b m^{3} n^{2} \log \left (c\right ) - a b m^{2} n^{3}\right )} \log \left (x\right )^{2} + 3 \, {\left (a b m^{3} n \log \left (c\right )^{2} - 2 \, a b m^{2} n^{2} \log \left (c\right ) + 2 \, a b m n^{3}\right )} \log \left (x\right )\right )} x^{m}}{12 \, m^{4}} \]
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Time = 13.32 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.73 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=- a^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 m}}{2 m} & \text {for}\: m \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 m} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + a^{2} \left (\begin {cases} \frac {x^{2 m}}{2 m} & \text {for}\: m \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} + 2 a b \left (\begin {cases} \frac {x^{m} \log {\left (c x^{n} \right )}^{3}}{m} - \frac {3 n x^{m} \log {\left (c x^{n} \right )}^{2}}{m^{2}} + \frac {6 n^{2} x^{m} \log {\left (c x^{n} \right )}}{m^{3}} - \frac {6 n^{3} x^{m}}{m^{4}} & \text {for}\: m \neq 0 \\\begin {cases} 0 & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \wedge \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{4}}{4 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {6 {G_{5, 5}^{5, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {6 {G_{5, 5}^{0, 5}\left (\begin {matrix} 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases}\right ) - b^{2} \left (\begin {cases} - \log {\left (c \right )}^{5} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{6}}{6 n} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (119) = 238\).
Time = 0.23 (sec) , antiderivative size = 530, normalized size of antiderivative = 4.24 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=\frac {1}{10} \, {\left (\frac {2 \, b^{2} \log \left (c x^{n}\right )^{5}}{n} + \frac {20 \, a b x^{m} \log \left (c x^{n}\right )^{2}}{m} - 40 \, a b {\left (\frac {n x^{m} \log \left (c x^{n}\right )}{m^{2}} - \frac {n^{2} x^{m}}{m^{3}}\right )} + \frac {5 \, a^{2} x^{2 \, m}}{m}\right )} \log \left (c x^{n}\right ) + \frac {2 \, b^{2} m^{4} n^{5} \log \left (x\right )^{6} - 12 \, b^{2} m^{4} n^{4} \log \left (c\right ) \log \left (x\right )^{5} + 30 \, b^{2} m^{4} n^{3} \log \left (c\right )^{2} \log \left (x\right )^{4} - 40 \, b^{2} m^{4} n^{2} \log \left (c\right )^{3} \log \left (x\right )^{3} + 30 \, b^{2} m^{4} n \log \left (c\right )^{4} \log \left (x\right )^{2} - 12 \, b^{2} m^{4} \log \left (c\right )^{5} \log \left (x\right ) - 12 \, b^{2} m^{4} \log \left (x\right ) \log \left (x^{n}\right )^{5} - 15 \, a^{2} m^{2} n x^{2 \, m} + 30 \, {\left (b^{2} m^{4} n \log \left (x\right )^{2} - 2 \, b^{2} m^{4} \log \left (c\right ) \log \left (x\right )\right )} \log \left (x^{n}\right )^{4} - 120 \, {\left (m^{2} n \log \left (c\right )^{2} - 4 \, m n^{2} \log \left (c\right ) + 6 \, n^{3}\right )} a b x^{m} - 40 \, {\left (b^{2} m^{4} n^{2} \log \left (x\right )^{3} - 3 \, b^{2} m^{4} n \log \left (c\right ) \log \left (x\right )^{2} + 3 \, b^{2} m^{4} \log \left (c\right )^{2} \log \left (x\right )\right )} \log \left (x^{n}\right )^{3} + 30 \, {\left (b^{2} m^{4} n^{3} \log \left (x\right )^{4} - 4 \, b^{2} m^{4} n^{2} \log \left (c\right ) \log \left (x\right )^{3} + 6 \, b^{2} m^{4} n \log \left (c\right )^{2} \log \left (x\right )^{2} - 4 \, b^{2} m^{4} \log \left (c\right )^{3} \log \left (x\right ) - 4 \, a b m^{2} n x^{m}\right )} \log \left (x^{n}\right )^{2} - 12 \, {\left (b^{2} m^{4} n^{4} \log \left (x\right )^{5} - 5 \, b^{2} m^{4} n^{3} \log \left (c\right ) \log \left (x\right )^{4} + 10 \, b^{2} m^{4} n^{2} \log \left (c\right )^{2} \log \left (x\right )^{3} - 10 \, b^{2} m^{4} n \log \left (c\right )^{3} \log \left (x\right )^{2} + 5 \, b^{2} m^{4} \log \left (c\right )^{4} \log \left (x\right ) + 20 \, {\left (m^{2} n \log \left (c\right ) - 2 \, m n^{2}\right )} a b x^{m}\right )} \log \left (x^{n}\right )}{60 \, m^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (119) = 238\).
Time = 0.32 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.29 \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=\frac {1}{6} \, b^{2} n^{5} \log \left (x\right )^{6} + b^{2} n^{4} \log \left (c\right ) \log \left (x\right )^{5} + \frac {5}{2} \, b^{2} n^{3} \log \left (c\right )^{2} \log \left (x\right )^{4} + \frac {10}{3} \, b^{2} n^{2} \log \left (c\right )^{3} \log \left (x\right )^{3} + \frac {5}{2} \, b^{2} n \log \left (c\right )^{4} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{5} \log \left (x\right ) + \frac {2 \, a b n^{3} x^{m} \log \left (x\right )^{3}}{m} + \frac {6 \, a b n^{2} x^{m} \log \left (c\right ) \log \left (x\right )^{2}}{m} + \frac {6 \, a b n x^{m} \log \left (c\right )^{2} \log \left (x\right )}{m} - \frac {6 \, a b n^{3} x^{m} \log \left (x\right )^{2}}{m^{2}} + \frac {2 \, a b x^{m} \log \left (c\right )^{3}}{m} - \frac {12 \, a b n^{2} x^{m} \log \left (c\right ) \log \left (x\right )}{m^{2}} - \frac {6 \, a b n x^{m} \log \left (c\right )^{2}}{m^{2}} + \frac {a^{2} n x^{2 \, m} \log \left (x\right )}{2 \, m} + \frac {12 \, a b n^{3} x^{m} \log \left (x\right )}{m^{3}} + \frac {a^{2} x^{2 \, m} \log \left (c\right )}{2 \, m} + \frac {12 \, a b n^{2} x^{m} \log \left (c\right )}{m^{3}} - \frac {a^{2} n x^{2 \, m}}{4 \, m^{2}} - \frac {12 \, a b n^{3} x^{m}}{m^{4}} \]
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Timed out. \[ \int \frac {\log \left (c x^n\right ) \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {\ln \left (c\,x^n\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^2\right )}^2}{x} \,d x \]
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