Integrand size = 23, antiderivative size = 152 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b d n}{8 e^3 \left (d+e x^2\right )}+\frac {b n \log (x)}{4 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {3 b n \log \left (d+e x^2\right )}{8 e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^3}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^3} \]
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Time = 0.19 (sec) , antiderivative size = 152, 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 = {272, 45, 2393, 2376, 46, 2373, 266, 2375, 2438} \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^3}+\frac {b d n}{8 e^3 \left (d+e x^2\right )}+\frac {3 b n \log \left (d+e x^2\right )}{8 e^3}+\frac {b n \log (x)}{4 e^3} \]
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Rule 45
Rule 46
Rule 266
Rule 272
Rule 2373
Rule 2375
Rule 2376
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^3}-\frac {2 d x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{e^2}-\frac {(2 d) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac {d^2 \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx}{e^2} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^3}-\frac {(b n) \int \frac {\log \left (1+\frac {e x^2}{d}\right )}{x} \, dx}{2 e^3}+\frac {\left (b d^2 n\right ) \int \frac {1}{x \left (d+e x^2\right )^2} \, dx}{4 e^3}+\frac {(b n) \int \frac {x}{d+e x^2} \, dx}{e^2} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{2 e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^3}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^3}+\frac {\left (b d^2 n\right ) \text {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^2\right )}{8 e^3} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{2 e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^3}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^3}+\frac {\left (b d^2 n\right ) \text {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^2\right )}{8 e^3} \\ & = \frac {b d n}{8 e^3 \left (d+e x^2\right )}+\frac {b n \log (x)}{4 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {3 b n \log \left (d+e x^2\right )}{8 e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^3}+\frac {b n \text {Li}_2\left (-\frac {e x^2}{d}\right )}{4 e^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.28 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {-2 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+8 d \left (d+e x^2\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+4 \left (d+e x^2\right )^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+b n \left (d^2+d e x^2-4 d e x^2 \log (x)-6 e^2 x^4 \log (x)+3 d^2 \log \left (i \sqrt {d}-\sqrt {e} x\right )+6 d e x^2 \log \left (i \sqrt {d}-\sqrt {e} x\right )+3 e^2 x^4 \log \left (i \sqrt {d}-\sqrt {e} x\right )+3 d^2 \log \left (i \sqrt {d}+\sqrt {e} x\right )+6 d e x^2 \log \left (i \sqrt {d}+\sqrt {e} x\right )+3 e^2 x^4 \log \left (i \sqrt {d}+\sqrt {e} x\right )+4 d^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 d e x^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 e^2 x^4 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 d^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 d e x^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 e^2 x^4 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 \left (d+e x^2\right )^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 \left (d+e x^2\right )^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{8 e^3 \left (d+e x^2\right )^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.70 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.36
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}+\frac {b \ln \left (x^{n}\right ) d}{e^{3} \left (e \,x^{2}+d \right )}-\frac {b \ln \left (x^{n}\right ) d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )^{2}}+\frac {3 b n \ln \left (e \,x^{2}+d \right )}{8 e^{3}}+\frac {b d n}{8 e^{3} \left (e \,x^{2}+d \right )}-\frac {3 b n \ln \left (x \right )}{4 e^{3}}-\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\ln \left (e \,x^{2}+d \right )}{2 e^{3}}+\frac {d}{e^{3} \left (e \,x^{2}+d \right )}-\frac {d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )^{2}}\right )\) | \(359\) |
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Time = 63.33 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.65 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {a d^{2} \left (\begin {cases} \frac {x^{2}}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x^{2}\right )^{2}} & \text {otherwise} \end {cases}\right )}{2 e^{2}} - \frac {a d \left (\begin {cases} \frac {x^{2}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{2}} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {a \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right )}{2 e^{2}} - \frac {b d^{2} n \left (\begin {cases} \frac {x^{2}}{2 d^{3}} & \text {for}\: e = 0 \\- \frac {1}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac {\log {\left (x \right )}}{2 d^{2} e} + \frac {\log {\left (\frac {d}{e} + x^{2} \right )}}{4 d^{2} e} & \text {otherwise} \end {cases}\right )}{2 e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x^{2}}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x^{2}\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 e^{2}} + \frac {b d n \left (\begin {cases} \frac {x^{2}}{2 d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x^{2} \right )}}{2 d e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {b d \left (\begin {cases} \frac {x^{2}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n \left (\begin {cases} \frac {x^{2}}{2 d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{2 e^{2}} + \frac {b \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 e^{2}} \]
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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