Integrand size = 25, antiderivative size = 181 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=-\frac {a f x}{g^2}+\frac {b f n x}{g^2}+\frac {b d n x}{2 e g}-\frac {b n x^2}{4 g}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]
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Time = 0.14 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac {a f x}{g^2}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {b d n x}{2 e g}+\frac {b f n x}{g^2}-\frac {b n x^2}{4 g} \]
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Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}\right ) \, dx \\ & = -\frac {f \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}+\frac {f^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g^2}+\frac {\int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g} \\ & = -\frac {a f x}{g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}-\frac {(b f) \int \log \left (c (d+e x)^n\right ) \, dx}{g^2}-\frac {\left (b e f^2 n\right ) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^3}-\frac {(b e n) \int \frac {x^2}{d+e x} \, dx}{2 g} \\ & = -\frac {a f x}{g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}-\frac {(b f) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}-\frac {\left (b f^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^3}-\frac {(b e n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{2 g} \\ & = -\frac {a f x}{g^2}+\frac {b f n x}{g^2}+\frac {b d n x}{2 e g}-\frac {b n x^2}{4 g}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b f^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=-\frac {a f x}{g^2}+\frac {b f n x}{g^2}+\frac {b n \left (\frac {2 d x}{e}-x^2-\frac {2 d^2 \log (d+e x)}{e^2}\right )}{4 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.80 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.14
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{2}}{2 g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f x}{g^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f^{2} \ln \left (g x +f \right )}{g^{3}}-\frac {b n \,f^{2} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{3}}-\frac {b n \,f^{2} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{3}}-\frac {b n \,x^{2}}{4 g}+\frac {b f n x}{g^{2}}+\frac {5 b \,f^{2} n}{4 g^{3}}+\frac {b d n x}{2 e g}+\frac {b d f n}{2 e \,g^{2}}-\frac {b n \,d^{2} \ln \left (\left (g x +f \right ) e +d g -e f \right )}{2 e^{2} g}-\frac {b n d \ln \left (\left (g x +f \right ) e +d g -e f \right ) f}{e \,g^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} g \,x^{2}-f x}{g^{2}}+\frac {f^{2} \ln \left (g x +f \right )}{g^{3}}\right )\) | \(388\) |
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\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{g x + f} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{f + g x}\, dx \]
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\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{g x + f} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \]
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