Integrand size = 21, antiderivative size = 171 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=-\frac {b n}{d^3 x}+\frac {b e n}{2 d^3 (d+e x)}+\frac {b e n \log (x)}{2 d^4}-\frac {a+b \log \left (c x^n\right )}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {5 b e n \log (d+e x)}{2 d^4}-\frac {3 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4} \]
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Time = 0.17 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {46, 2393, 2341, 2356, 2351, 31, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac {3 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {a+b \log \left (c x^n\right )}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 (d+e x)^2}-\frac {3 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4}+\frac {b e n \log (x)}{2 d^4}-\frac {5 b e n \log (d+e x)}{2 d^4}+\frac {b e n}{2 d^3 (d+e x)}-\frac {b n}{d^3 x} \]
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Rule 31
Rule 46
Rule 2341
Rule 2351
Rule 2356
Rule 2379
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log \left (c x^n\right )}{d^3 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)^3}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^3 x (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^3}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^2} \\ & = -\frac {b n}{d^3 x}-\frac {a+b \log \left (c x^n\right )}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {(3 b e n) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^4}+\frac {(b e n) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^2}-\frac {\left (2 b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{d^4} \\ & = -\frac {b n}{d^3 x}-\frac {a+b \log \left (c x^n\right )}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {2 b e n \log (d+e x)}{d^4}-\frac {3 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}+\frac {(b e n) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 d^2} \\ & = -\frac {b n}{d^3 x}+\frac {b e n}{2 d^3 (d+e x)}+\frac {b e n \log (x)}{2 d^4}-\frac {a+b \log \left (c x^n\right )}{d^3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 (d+e x)^2}+\frac {2 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac {3 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {5 b e n \log (d+e x)}{2 d^4}-\frac {3 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\frac {-\frac {2 b d n}{x}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {4 d e \left (a+b \log \left (c x^n\right )\right )}{d+e x}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+4 b e n (\log (x)-\log (d+e x))+b e n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+6 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+6 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.89
method | result | size |
risch | \(-\frac {b \ln \left (x^{n}\right ) e}{2 d^{2} \left (e x +d \right )^{2}}+\frac {3 b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{4}}-\frac {2 b \ln \left (x^{n}\right ) e}{d^{3} \left (e x +d \right )}-\frac {b \ln \left (x^{n}\right )}{d^{3} x}-\frac {3 b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{4}}+\frac {3 b n e \ln \left (x \right )^{2}}{2 d^{4}}-\frac {3 b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{4}}-\frac {3 b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{4}}+\frac {b e n}{2 d^{3} \left (e x +d \right )}-\frac {5 b e n \ln \left (e x +d \right )}{2 d^{4}}-\frac {b n}{d^{3} x}+\frac {5 b e n \ln \left (x \right )}{2 d^{4}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e}{2 d^{2} \left (e x +d \right )^{2}}+\frac {3 e \ln \left (e x +d \right )}{d^{4}}-\frac {2 e}{d^{3} \left (e x +d \right )}-\frac {1}{d^{3} x}-\frac {3 e \ln \left (x \right )}{d^{4}}\right )\) | \(324\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \]
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Time = 38.43 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.60 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\frac {a e^{2} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {2 a e^{2} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {a}{d^{3} x} + \frac {3 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^{4}} - \frac {3 a e \log {\left (x \right )}}{d^{4}} - \frac {b e^{2} n \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 d^{2} e + 2 d e^{2} x} - \frac {\log {\left (x \right )}}{2 d^{2} e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} - \frac {2 b e^{2} n \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {2 b e^{2} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} - \frac {b n}{d^{3} x} - \frac {b \log {\left (c x^{n} \right )}}{d^{3} x} - \frac {3 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^{4}} + \frac {3 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^{4}} + \frac {3 b e n \log {\left (x \right )}^{2}}{2 d^{4}} - \frac {3 b e \log {\left (x \right )} \log {\left (c x^{n} \right )}}{d^{4}} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (d+e\,x\right )}^3} \,d x \]
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