Integrand size = 21, antiderivative size = 107 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=-\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^2}{4 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3} \]
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Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {45, 2393, 2332, 2341, 2354, 2438} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {a d x}{e^2}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {b d n x}{e^2}-\frac {b n x^2}{4 e} \]
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Rule 45
Rule 2332
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {d \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e} \\ & = -\frac {a d x}{e^2}-\frac {b n x^2}{4 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {(b d) \int \log \left (c x^n\right ) \, dx}{e^2}-\frac {\left (b d^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3} \\ & = -\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^2}{4 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\frac {-4 a d e x+4 b d e n x+2 a e^2 x^2-b e^2 n x^2+4 a d^2 \log \left (1+\frac {e x}{d}\right )+2 b \log \left (c x^n\right ) \left (e x (-2 d+e x)+2 d^2 \log \left (1+\frac {e x}{d}\right )\right )+4 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.40 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.18
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e}-\frac {b \ln \left (x^{n}\right ) d x}{e^{2}}+\frac {b \ln \left (x^{n}\right ) d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {b n \,d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}-\frac {b n \,d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}-\frac {b n \,x^{2}}{4 e}+\frac {b d n x}{e^{2}}+\frac {5 b n \,d^{2}}{4 e^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} e \,x^{2}-d x}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right )}{e^{3}}\right )\) | \(233\) |
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
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Time = 12.90 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.04 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\frac {a d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {a d x}{e^{2}} + \frac {a x^{2}}{2 e} - \frac {b d^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} + \frac {b d n x}{e^{2}} - \frac {b d x \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n x^{2}}{4 e} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e} \]
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x} \,d x \]
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