Integrand size = 23, antiderivative size = 219 \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {b p x}{2 a e}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {b d p \log (b+a x)}{a e^2}-\frac {b^2 p \log (b+a x)}{2 a^2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^3}+\frac {d^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^3} \]
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Time = 0.17 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2516, 2498, 269, 31, 2505, 199, 45, 2512, 266, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=-\frac {b^2 p \log (a x+b)}{2 a^2 e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e^3}-\frac {b d p \log (a x+b)}{a e^2}+\frac {b p x}{2 a e}+\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^3}+\frac {d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3} \]
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Rule 31
Rule 45
Rule 199
Rule 266
Rule 269
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2498
Rule 2505
Rule 2512
Rule 2516
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {d \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx}{e^2}+\frac {d^2 \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{e^2}+\frac {\int x \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx}{e} \\ & = -\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {\left (b d^2 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{e^3}-\frac {(b d p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x} \, dx}{e^2}+\frac {(b p) \int \frac {1}{a+\frac {b}{x}} \, dx}{2 e} \\ & = -\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {\left (b d^2 p\right ) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{e^3}-\frac {(b d p) \int \frac {1}{b+a x} \, dx}{e^2}+\frac {(b p) \int \frac {x}{b+a x} \, dx}{2 e} \\ & = -\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {b d p \log (b+a x)}{a e^2}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {\left (d^2 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{e^3}-\frac {\left (a d^2 p\right ) \int \frac {\log (d+e x)}{b+a x} \, dx}{e^3}+\frac {(b p) \int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx}{2 e} \\ & = \frac {b p x}{2 a e}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {b d p \log (b+a x)}{a e^2}-\frac {b^2 p \log (b+a x)}{2 a^2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}-\frac {\left (d^2 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e^2}+\frac {\left (d^2 p\right ) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{e^2} \\ & = \frac {b p x}{2 a e}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {b d p \log (b+a x)}{a e^2}-\frac {b^2 p \log (b+a x)}{2 a^2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}+\frac {d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (d^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{e^3} \\ & = \frac {b p x}{2 a e}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {b d p \log (b+a x)}{a e^2}-\frac {b^2 p \log (b+a x)}{2 a^2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{e^3}+\frac {d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=-\frac {b d p \log \left (a+\frac {b}{x}\right )}{a e^2}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e}-\frac {b d p \log (x)}{a e^2}+\frac {b p \left (\frac {x}{a}-\frac {b \log (b+a x)}{a^2}\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac {d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}+\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^3} \]
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Time = 1.71 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.10
method | result | size |
parts | \(\frac {x^{2} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{2 e}-\frac {d x \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{e^{2}}+\frac {d^{2} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{3}}+p b e \left (-\frac {d^{2} \operatorname {dilog}\left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{e^{4} b}-\frac {d^{2} \ln \left (e x +d \right ) \ln \left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{e^{4} b}+\frac {d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4} b}+\frac {d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4} b}+\frac {\frac {e x +d}{a}+\frac {\left (-2 a d -b e \right ) \ln \left (a d -a \left (e x +d \right )-b e \right )}{a^{2}}}{2 e^{3}}\right )\) | \(240\) |
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\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d} \,d x } \]
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\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\int \frac {x^{2} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{d + e x}\, dx \]
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\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d} \,d x } \]
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\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\int \frac {x^2\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{d+e\,x} \,d x \]
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