Integrand size = 21, antiderivative size = 74 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=-\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}+\frac {e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2} \]
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Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2380, 2341, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\frac {e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {a+b \log \left (c x^n\right )}{d x}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2}-\frac {b n}{d x} \]
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Rule 2341
Rule 2379
Rule 2380
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d} \\ & = -\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}+\frac {e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {(b e n) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^2} \\ & = -\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}+\frac {e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=-\frac {\frac {2 b d n}{x}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.76 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.99
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {b \ln \left (x^{n}\right )}{d x}-\frac {b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{2}}-\frac {b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b n}{d x}+\frac {b n e \ln \left (x \right )^{2}}{2 d^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\) | \(221\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Time = 32.45 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.92 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=- \frac {a}{d x} + \frac {a e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e \log {\left (x \right )}}{d^{2}} - \frac {b n}{d x} - \frac {b \log {\left (c x^{n} \right )}}{d x} - \frac {b e^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b e n \log {\left (x \right )}^{2}}{2 d^{2}} - \frac {b e \log {\left (x \right )} \log {\left (c x^{n} \right )}}{d^{2}} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
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