Integrand size = 23, antiderivative size = 83 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=-\frac {b n x^2}{4 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}-\frac {b d n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2} \]
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Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {272, 45, 2393, 2341, 2375, 2438} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=-\frac {d \log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {b d n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2}-\frac {b n x^2}{4 e} \]
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Rule 45
Rule 272
Rule 2341
Rule 2375
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {d x \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}-\frac {d \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e} \\ & = -\frac {b n x^2}{4 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}+\frac {(b d n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 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 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}-\frac {b d n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.63 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=-\frac {b e n x^2-2 e x^2 \left (a+b \log \left (c x^n\right )\right )+2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+2 b d n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 b d n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{4 e^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.44 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.42
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e}-\frac {b \ln \left (x^{n}\right ) d \ln \left (e \,x^{2}+d \right )}{2 e^{2}}-\frac {b n \,x^{2}}{4 e}+\frac {b n d \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n d \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n d \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x^{2}}{2 e}-\frac {d \ln \left (e \,x^{2}+d \right )}{2 e^{2}}\right )\) | \(284\) |
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x^{2} + d} \,d x } \]
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Time = 17.42 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.43 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=- \frac {a d \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right )}{2 e} + \frac {a x^{2}}{2 e} + \frac {b d n \left (\begin {cases} \frac {x^{2}}{2 d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{2 e} - \frac {b d \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 e} - \frac {b n x^{2}}{4 e} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e} \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x^{2} + d} \,d x } \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x^{2} + d} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{e\,x^2+d} \,d x \]
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