Integrand size = 23, antiderivative size = 191 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {3 i b \sqrt {d} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}} \]
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Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {294, 327, 211, 2393, 2332, 2360, 2361, 12, 4940, 2438} \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {3 \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}+\frac {a x}{e^2}-\frac {b \sqrt {d} n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {3 i b \sqrt {d} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {b n x}{e^2} \]
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Rule 12
Rule 211
Rule 294
Rule 327
Rule 2332
Rule 2360
Rule 2361
Rule 2393
Rule 2438
Rule 4940
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log \left (c x^n\right )}{e^2}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^2}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {(2 d) \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e^2} \\ & = \frac {a x}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}+\frac {b \int \log \left (c x^n\right ) \, dx}{e^2}+\frac {d \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 e^2}-\frac {(b d n) \int \frac {1}{d+e x^2} \, dx}{2 e^2}+\frac {(2 b d n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{e^2} \\ & = \frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {\left (2 b \sqrt {d} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}-\frac {(b d n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 e^2} \\ & = \frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}-\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}-\frac {\left (b \sqrt {d} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{5/2}} \\ & = \frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {i b \sqrt {d} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}-\frac {i b \sqrt {d} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}-\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 e^{5/2}}+\frac {\left (i b \sqrt {d} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 e^{5/2}} \\ & = \frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {3 i b \sqrt {d} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.55 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {4 a \sqrt {e} x-4 b \sqrt {e} n x+4 b \sqrt {e} x \log \left (c x^n\right )-\frac {d \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}-\sqrt {e} x}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}+\sqrt {e} x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{\sqrt {-d}}+b \sqrt {-d} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )-3 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+3 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+3 b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )-3 b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{4 e^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.52 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.93
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) d x}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {3 b d \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{2 e^{2} \sqrt {d e}}-\frac {3 b d \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{2 e^{2} \sqrt {d e}}+\frac {b \ln \left (x^{n}\right ) x}{e^{2}}-\frac {b n x}{e^{2}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e^{2} \sqrt {-d e}}+\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e^{2} \sqrt {-d e}}-\frac {3 b n d \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \sqrt {-d e}}+\frac {3 b n d \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \sqrt {-d e}}-\frac {b n d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 e^{2} \sqrt {d e}}+\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\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 {d \left (-\frac {x}{2 \left (e \,x^{2}+d \right )}+\frac {3 \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{e^{2}}+\frac {x}{e^{2}}\right )\) | \(559\) |
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\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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