Optimal. Leaf size=164 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}+\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {288, 205, 2351, 2323, 2324, 12, 4848, 2391} \[ -\frac {i b n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}+\frac {i b n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 288
Rule 2323
Rule 2324
Rule 2351
Rule 2391
Rule 4848
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^2}+\frac {a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} e^{3/2}}-\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 e}+\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{2 e}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{e}\\ &=\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{\sqrt {d} e^{3/2}}+\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 e}\\ &=\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {(i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 \sqrt {d} e^{3/2}}+\frac {(i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 \sqrt {d} e^{3/2}}+\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 \sqrt {d} e^{3/2}}\\ &=\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {(i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 \sqrt {d} e^{3/2}}-\frac {(i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 \sqrt {d} e^{3/2}}\\ &=\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 258, normalized size = 1.57 \[ \frac {\frac {d \log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac {\log \left (\frac {d \sqrt {e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d}-\sqrt {e} x}-\frac {a+b \log \left (c x^n\right )}{\sqrt {-d}+\sqrt {e} x}+\frac {b n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d}}+\frac {b d n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{3/2}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{\sqrt {-d}}}{4 e^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left (c x^{n}\right ) + a x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 752, normalized size = 4.59 \[ -\frac {b n \,x^{2} \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \,x^{2} \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b d n \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}\, e}+\frac {b d n \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}\, e}+\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 \left (e \,x^{2}+d \right ) e}-\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (e \,x^{2}+d \right ) e}-\frac {i \pi b x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (e \,x^{2}+d \right ) e}+\frac {i \pi b x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 \left (e \,x^{2}+d \right ) e}-\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 \sqrt {d e}\, e}+\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \sqrt {d e}\, e}+\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \sqrt {d e}\, e}-\frac {i \pi b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 \sqrt {d e}\, e}-\frac {b n \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \ln \relax (x )}{2 \sqrt {d e}\, e}+\frac {b n \ln \relax (x ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}\, e}-\frac {b n \ln \relax (x ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}\, e}+\frac {b n \dilog \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \sqrt {-d e}\, e}-\frac {b n \dilog \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \sqrt {-d e}\, e}+\frac {b n \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, e}-\frac {b x \ln \relax (c )}{2 \left (e \,x^{2}+d \right ) e}-\frac {b x \ln \left (x^{n}\right )}{2 \left (e \,x^{2}+d \right ) e}+\frac {b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \ln \relax (c )}{2 \sqrt {d e}\, e}+\frac {b \arctan \left (\frac {e x}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{2 \sqrt {d e}\, e}-\frac {a x}{2 \left (e \,x^{2}+d \right ) e}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a {\left (\frac {x}{e^{2} x^{2} + d e} - \frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e}\right )} + b \int \frac {x^{2} \log \relax (c) + x^{2} \log \left (x^{n}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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