Optimal. Leaf size=331 \[ -\frac {b e n}{2 d f x}-\frac {b e^2 n \log (x)}{2 d^2 f}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}-\frac {b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2} \]
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Rubi [A]
time = 0.28, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 46, 2463,
2442, 2441, 2352, 266, 2440, 2438} \begin {gather*} \frac {b g n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 f^2}-\frac {b g n \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {b e^2 n \log (x)}{2 d^2 f}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}-\frac {b e n}{2 d f x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 266
Rule 272
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 \left (f+g x^2\right )} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f x^3}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac {g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3} \, dx}{f}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^2}+\frac {g^2 \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{f^2}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g^2 \int \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f^2}+\frac {(b e n) \int \frac {1}{x^2 (d+e x)} \, dx}{2 f}+\frac {(b e g n) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{f^2}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}-\frac {g^{3/2} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f^2}+\frac {g^{3/2} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f^2}+\frac {(b e n) \int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx}{2 f}\\ &=-\frac {b e n}{2 d f x}-\frac {b e^2 n \log (x)}{2 d^2 f}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}-\frac {b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}-\frac {(b e g n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 f^2}-\frac {(b e g n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 f^2}\\ &=-\frac {b e n}{2 d f x}-\frac {b e^2 n \log (x)}{2 d^2 f}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}-\frac {b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}-\frac {(b g n) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 f^2}-\frac {(b g n) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 f^2}\\ &=-\frac {b e n}{2 d f x}-\frac {b e^2 n \log (x)}{2 d^2 f}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}-\frac {b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 279, normalized size = 0.84 \begin {gather*} \frac {-\frac {b e f n (d+e x \log (x)-e x \log (d+e x))}{d^2 x}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2}-2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )+b g n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+b g n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )-2 b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{2 f^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 = 0.32, size = 841, normalized size = 2.54
method | result | size |
risch | \(-\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} g \ln \left (x \right )}{2 f^{2}}-\frac {a}{2 f \,x^{2}}-\frac {b \ln \left (c \right ) g \ln \left (x \right )}{f^{2}}+\frac {b n g \dilog \left (\frac {e x +d}{d}\right )}{f^{2}}-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 f \,x^{2}}-\frac {b \ln \left (c \right )}{2 f \,x^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{4 f \,x^{2}}+\frac {b n g \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{f^{2}}-\frac {a g \ln \left (x \right )}{f^{2}}+\frac {a g \ln \left (g \,x^{2}+f \right )}{2 f^{2}}-\frac {b n g \ln \left (e x +d \right ) \ln \left (g \,x^{2}+f \right )}{2 f^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g \ln \left (g \,x^{2}+f \right )}{2 f^{2}}+\frac {b n g \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 f^{2}}+\frac {b n g \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 f^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} g \ln \left (x \right )}{2 f^{2}}+\frac {b n g \dilog \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 f^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} g \ln \left (g \,x^{2}+f \right )}{4 f^{2}}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g \ln \left (x \right )}{f^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} g \ln \left (g \,x^{2}+f \right )}{4 f^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{4 f \,x^{2}}+\frac {b \ln \left (c \right ) g \ln \left (g \,x^{2}+f \right )}{2 f^{2}}+\frac {b n g \dilog \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 f^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4 f \,x^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} g \ln \left (g \,x^{2}+f \right )}{4 f^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} g \ln \left (x \right )}{2 f^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4 f \,x^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) g \ln \left (x \right )}{2 f^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) g \ln \left (g \,x^{2}+f \right )}{4 f^{2}}-\frac {b \,e^{2} n \ln \left (x \right )}{2 d^{2} f}+\frac {b \,e^{2} n \ln \left (e x +d \right )}{2 d^{2} f}-\frac {b e n}{2 d f x}\) | \(841\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x^3\,\left (g\,x^2+f\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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