Optimal. Leaf size=164 \[ \frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 \sqrt {d} e^{3/2}} \]
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Rubi [A]
time = 0.20, 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 = {294, 211,
2393, 2360, 2361, 12, 4940, 2438} \begin {gather*} -\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 {\text {ArcTan}\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 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 294
Rule 2360
Rule 2361
Rule 2393
Rule 2438
Rule 4940
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 = 0.32, size = 258, normalized size = 1.57 \begin {gather*} \frac {\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 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}}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{\sqrt {-d}}+\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}}}{4 e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.14, size = 752, normalized size = 4.59
method | result | size |
risch | \(-\frac {b x \ln \left (x^{n}\right )}{2 e \left (e \,x^{2}+d \right )}-\frac {b \arctan \left (\frac {x e}{\sqrt {e d}}\right ) n \ln \left (x \right )}{2 e \sqrt {e d}}+\frac {b \arctan \left (\frac {x e}{\sqrt {e d}}\right ) \ln \left (x^{n}\right )}{2 e \sqrt {e d}}+\frac {b n \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e \sqrt {e d}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right ) x^{2}}{4 \left (e \,x^{2}+d \right ) \sqrt {-e d}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right ) x^{2}}{4 \left (e \,x^{2}+d \right ) \sqrt {-e d}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 e \left (e \,x^{2}+d \right ) \sqrt {-e d}}+\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 e \left (e \,x^{2}+d \right ) \sqrt {-e d}}+\frac {b n \dilog \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 e \sqrt {-e d}}-\frac {b n \dilog \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{4 e \sqrt {-e d}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 e \sqrt {-e d}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 e \sqrt {-e d}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{4 e \left (e \,x^{2}+d \right )}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{4 e \left (e \,x^{2}+d \right )}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 e \sqrt {e d}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{4 e \left (e \,x^{2}+d \right )}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 e \sqrt {e d}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 e \sqrt {e d}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x}{4 e \left (e \,x^{2}+d \right )}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{4 e \sqrt {e d}}-\frac {b \ln \left (c \right ) x}{2 e \left (e \,x^{2}+d \right )}+\frac {b \ln \left (c \right ) \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e \sqrt {e d}}-\frac {a x}{2 e \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e \sqrt {e d}}\) | \(752\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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