Integrand size = 21, antiderivative size = 110 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=-\frac {b n}{4 d x^2}+\frac {b e n}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^3} \]
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Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2380, 2341, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=-\frac {e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^3}+\frac {b e n}{d^2 x}-\frac {b n}{4 d x^2} \]
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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^3} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx}{d} \\ & = -\frac {b n}{4 d x^2}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^2} \\ & = -\frac {b n}{4 d x^2}+\frac {b e n}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {\left (b e^2 n\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^3} \\ & = -\frac {b n}{4 d x^2}+\frac {b e n}{d^2 x}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=-\frac {\frac {b d^2 n}{x^2}-\frac {4 b d e n}{x}+\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {4 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+4 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+4 b e^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 d^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.45 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.40
method | result | size |
risch | \(-\frac {b \ln \left (x^{n}\right ) e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {b \ln \left (x^{n}\right )}{2 d \,x^{2}}+\frac {b \ln \left (x^{n}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {b \ln \left (x^{n}\right ) e}{d^{2} x}+\frac {b e n}{d^{2} x}-\frac {b n}{4 d \,x^{2}}-\frac {b n \,e^{2} \ln \left (x \right )^{2}}{2 d^{3}}+\frac {b n \,e^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{3}}+\frac {b n \,e^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )\) | \(264\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Time = 35.16 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.41 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=- \frac {a}{2 d x^{2}} + \frac {a e}{d^{2} x} - \frac {a e^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {a e^{2} \log {\left (x \right )}}{d^{3}} - \frac {b n}{4 d x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 d x^{2}} + \frac {b e n}{d^{2} x} + \frac {b e \log {\left (c x^{n} \right )}}{d^{2} x} + \frac {b e^{3} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {b e^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} - \frac {b e^{2} n \log {\left (x \right )}^{2}}{2 d^{3}} + \frac {b e^{2} \log {\left (x \right )} \log {\left (c x^{n} \right )}}{d^{3}} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\left (d+e\,x\right )} \,d x \]
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