Integrand size = 21, antiderivative size = 152 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {3 b d n x}{e^3}-\frac {d (3 a+b n) x}{e^3}-\frac {3 b n x^2}{4 e^2}-\frac {3 b d x \log \left (c x^n\right )}{e^3}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {3 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4} \]
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Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2384, 45, 2393, 2332, 2341, 2354, 2438} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e^4}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e^2}-\frac {d x (3 a+b n)}{e^3}-\frac {3 b d x \log \left (c x^n\right )}{e^3}+\frac {3 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}+\frac {3 b d n x}{e^3}-\frac {3 b n x^2}{4 e^2} \]
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Rule 45
Rule 2332
Rule 2341
Rule 2354
Rule 2384
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\int \frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{d+e x} \, dx}{e} \\ & = -\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {\int \left (-\frac {d \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{e^2}+\frac {x \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{e}+\frac {d^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{e^2 (d+e x)}\right ) \, dx}{e} \\ & = -\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}-\frac {d \int \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \, dx}{e^3}+\frac {d^2 \int \frac {3 a+b n+3 b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}+\frac {\int x \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \, dx}{e^2} \\ & = -\frac {d (3 a+b n) x}{e^3}-\frac {3 b n x^2}{4 e^2}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {(3 b d) \int \log \left (c x^n\right ) \, dx}{e^3}-\frac {\left (3 b d^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4} \\ & = \frac {3 b d n x}{e^3}-\frac {d (3 a+b n) x}{e^3}-\frac {3 b n x^2}{4 e^2}-\frac {3 b d x \log \left (c x^n\right )}{e^3}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{2 e^2}+\frac {d^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {3 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {-8 a d e x+8 b d e n x-b e^2 n x^2-8 b d e x \log \left (c x^n\right )+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-4 b d^2 n (\log (x)-\log (d+e x))+12 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+12 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.41 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.96
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e^{2}}-\frac {2 b \ln \left (x^{n}\right ) d x}{e^{3}}+\frac {3 b \ln \left (x^{n}\right ) d^{2} \ln \left (e x +d \right )}{e^{4}}+\frac {b \ln \left (x^{n}\right ) d^{3}}{e^{4} \left (e x +d \right )}-\frac {3 b n \,d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {3 b n \,d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4}}-\frac {b n \,x^{2}}{4 e^{2}}+\frac {2 b d n x}{e^{3}}+\frac {9 b n \,d^{2}}{4 e^{4}}+\frac {b n \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {b n \,d^{2} \ln \left (e x \right )}{e^{4}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} e \,x^{2}-2 d x}{e^{3}}+\frac {3 d^{2} \ln \left (e x +d \right )}{e^{4}}+\frac {d^{3}}{e^{4} \left (e x +d \right )}\right )\) | \(298\) |
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Time = 24.56 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.12 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=- \frac {a d^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 a d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {2 a d x}{e^{3}} + \frac {a x^{2}}{2 e^{2}} + \frac {b d^{3} 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 )}{e^{3}} - \frac {b d^{3} \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 )}}{e^{3}} - \frac {3 b d^{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 )}{e^{3}} + \frac {3 b d^{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 )}}{e^{3}} + \frac {2 b d n x}{e^{3}} - \frac {2 b d x \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b n x^{2}}{4 e^{2}} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e^{2}} \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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