Integrand size = 23, antiderivative size = 167 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {i b d^{3/2} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {i b d^{3/2} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}} \]
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Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {308, 211, 2393, 2332, 2341, 2361, 12, 4940, 2438} \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {a d x}{e^2}-\frac {b d x \log \left (c x^n\right )}{e^2}-\frac {i b d^{3/2} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {i b d^{3/2} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e} \]
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Rule 12
Rule 211
Rule 308
Rule 2332
Rule 2341
Rule 2361
Rule 2393
Rule 2438
Rule 4940
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = -\frac {d \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e^2}+\frac {\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{e} \\ & = -\frac {a d x}{e^2}-\frac {b n x^3}{9 e}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \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 d) \int \log \left (c x^n\right ) \, dx}{e^2}-\frac {\left (b d^2 n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{e^2} \\ & = -\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {\left (b d^{3/2} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}} \\ & = -\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {\left (i b d^{3/2} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{5/2}}+\frac {\left (i b d^{3/2} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{5/2}} \\ & = -\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {i b d^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {i b d^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.25 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {-18 a d \sqrt {e} x+18 b d \sqrt {e} n x-2 b e^{3/2} n x^3-18 b d \sqrt {e} x \log \left (c x^n\right )+6 e^{3/2} x^3 \left (a+b \log \left (c x^n\right )\right )+9 \sqrt {-d} d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+9 (-d)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+9 b (-d)^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )-9 b (-d)^{3/2} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{18 e^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.49 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.19
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x^{3}}{3 e}-\frac {b \ln \left (x^{n}\right ) d x}{e^{2}}-\frac {b \,d^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{e^{2} \sqrt {d e}}+\frac {b \,d^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{e^{2} \sqrt {d e}}-\frac {b n \,x^{3}}{9 e}+\frac {b d n x}{e^{2}}+\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2} \sqrt {-d e}}-\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2} \sqrt {-d e}}+\frac {b n \,d^{2} \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2} \sqrt {-d e}}-\frac {b n \,d^{2} \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2} \sqrt {-d e}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{3} e \,x^{3}-d x}{e^{2}}+\frac {d^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}\right )\) | \(365\) |
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\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{e x^{2} + d} \,d x } \]
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\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{e x^{2} + d} \,d x } \]
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Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {x^4\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{e\,x^2+d} \,d x \]
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