Integrand size = 23, antiderivative size = 271 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {2 a b d^2 n x}{e^3}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {2 b d^3 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^3 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2395, 2333, 2332, 2342, 2341, 2354, 2421, 6724} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {2 b d^3 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}-\frac {2 a b d^2 n x}{e^3}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {2 b^2 d^3 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e} \]
[In]
[Out]
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2354
Rule 2395
Rule 2421
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x \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 {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {d^2 \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^3}-\frac {d^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^3}-\frac {d \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^2}+\frac {\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e} \\ & = \frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (2 b d^3 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}-\frac {\left (2 b d^2 n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}+\frac {(b d n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {(2 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e} \\ & = -\frac {2 a b d^2 n x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {2 b d^3 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {\left (2 b^2 d^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e^3}+\frac {\left (2 b^2 d^3 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^4} \\ & = -\frac {2 a b d^2 n x}{e^3}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {2 b d^3 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^3 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {-108 d^2 e x \left (a+b \log \left (c x^n\right )\right )^2+54 d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-36 e^3 x^3 \left (a+b \log \left (c x^n\right )\right )^2+216 b d^2 e n x \left (a-b n+b \log \left (c x^n\right )\right )-8 b e^3 n x^3 \left (b n-3 \left (a+b \log \left (c x^n\right )\right )\right )+27 b d e^2 n x^2 \left (b n-2 \left (a+b \log \left (c x^n\right )\right )\right )+108 d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+216 b d^3 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{108 e^4} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.57 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.70
method | result | size |
risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} x^{3}}{3 e}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d \,x^{2}}{2 e^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} x \,d^{2}}{e^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x^{3}}{9 e}+\frac {b^{2} n \ln \left (x^{n}\right ) d \,x^{2}}{2 e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x \,d^{2}}{e^{3}}+\frac {2 b^{2} n^{2} x^{3}}{27 e}-\frac {b^{2} d \,n^{2} x^{2}}{4 e^{2}}+\frac {2 b^{2} d^{2} n^{2} x}{e^{3}}-\frac {2 b^{2} d^{3} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{e^{4}}-\frac {2 b^{2} d^{3} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{e^{4}}+\frac {2 b^{2} n \,d^{3} \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}+\frac {2 b^{2} n \,d^{3} \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{4}}+\frac {b^{2} d^{3} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{4}}-\frac {b^{2} d^{3} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{4}}-\frac {2 b^{2} d^{3} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e^{4}}+\frac {2 b^{2} d^{3} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e^{4}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (x^{n}\right ) x^{3}}{3 e}-\frac {\ln \left (x^{n}\right ) d \,x^{2}}{2 e^{2}}+\frac {\ln \left (x^{n}\right ) x \,d^{2}}{e^{3}}-\frac {\ln \left (x^{n}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-n \left (\frac {\frac {2 \left (e x +d \right )^{3}}{3}-\frac {7 d \left (e x +d \right )^{2}}{2}+11 d^{2} \left (e x +d \right )}{6 e^{4}}-\frac {d^{3} \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e^{4}}\right )\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (\frac {\frac {1}{3} e^{2} x^{3}-\frac {1}{2} d e \,x^{2}+d^{2} x}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right )}{e^{4}}\right )}{4}\) | \(732\) |
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\int \frac {x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \]
[In]
[Out]