Optimal. Leaf size=397 \[ -\frac {b d f n x}{2 e g^2}+\frac {b d^3 n x}{4 e^3 g}+\frac {b f n x^2}{4 g^2}-\frac {b d^2 n x^2}{8 e^2 g}+\frac {b d n x^3}{12 e g}-\frac {b n x^4}{16 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3} \]
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Rubi [A]
time = 0.36, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {272, 45, 2463,
2442, 266, 2441, 2440, 2438} \begin {gather*} \frac {b f^2 n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 g^3}+\frac {f^2 \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 g^3}+\frac {f^2 \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 g^3}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}+\frac {b d^3 n x}{4 e^3 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^2 n x^2}{8 e^2 g}-\frac {b d f n x}{2 e g^2}+\frac {b d n x^3}{12 e g}+\frac {b f n x^2}{4 g^2}-\frac {b n x^4}{16 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 266
Rule 272
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx &=\int \left (-\frac {f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac {f \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}+\frac {f^2 \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{g^2}+\frac {\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}\\ &=-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac {f^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}{g^2}+\frac {(b e f n) \int \frac {x^2}{d+e x} \, dx}{2 g^2}-\frac {(b e n) \int \frac {x^4}{d+e x} \, dx}{4 g}\\ &=-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}-\frac {f^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 g^{5/2}}+\frac {f^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 g^{5/2}}+\frac {(b e f n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{2 g^2}-\frac {(b e n) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx}{4 g}\\ &=-\frac {b d f n x}{2 e g^2}+\frac {b d^3 n x}{4 e^3 g}+\frac {b f n x^2}{4 g^2}-\frac {b d^2 n x^2}{8 e^2 g}+\frac {b d n x^3}{12 e g}-\frac {b n x^4}{16 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac {f^2 \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 g^3}+\frac {f^2 \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 g^3}-\frac {\left (b e f^2 n\right ) \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 g^3}-\frac {\left (b e f^2 n\right ) \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 g^3}\\ &=-\frac {b d f n x}{2 e g^2}+\frac {b d^3 n x}{4 e^3 g}+\frac {b f n x^2}{4 g^2}-\frac {b d^2 n x^2}{8 e^2 g}+\frac {b d n x^3}{12 e g}-\frac {b n x^4}{16 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac {f^2 \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 g^3}+\frac {f^2 \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 g^3}-\frac {\left (b f^2 n\right ) \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 g^3}-\frac {\left (b f^2 n\right ) \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 g^3}\\ &=-\frac {b d f n x}{2 e g^2}+\frac {b d^3 n x}{4 e^3 g}+\frac {b f n x^2}{4 g^2}-\frac {b d^2 n x^2}{8 e^2 g}+\frac {b d n x^3}{12 e g}-\frac {b n x^4}{16 g}+\frac {b d^2 f n \log (d+e x)}{2 e^2 g^2}-\frac {b d^4 n \log (d+e x)}{4 e^4 g}-\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g}+\frac {f^2 \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 g^3}+\frac {f^2 \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 g^3}+\frac {b f^2 n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 331, normalized size = 0.83 \begin {gather*} \frac {\frac {12 b f g n \left (e x (-2 d+e x)+2 d^2 \log (d+e x)\right )}{e^2}-\frac {b g^2 n \left (e x \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+12 d^4 \log (d+e x)\right )}{e^4}-24 f g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+12 g^2 x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+24 f^2 \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 )+24 f^2 \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 )+24 b f^2 n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+24 b f^2 n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{48 g^3} \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 = 0.70, size = 905, normalized size = 2.28
method | result | size |
risch | \(-\frac {b \ln \left (c \right ) f \,x^{2}}{2 g^{2}}+\frac {b \ln \left (c \right ) x^{4}}{4 g}+\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 ) f \,x^{2}}{4 g^{2}}-\frac {b n \,f^{2} \ln \left (e x +d \right ) \ln \left (g \,x^{2}+f \right )}{2 g^{3}}+\frac {b n \,f^{2} \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 g^{3}}+\frac {b n \,f^{2} \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 g^{3}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{4}}{4 g}-\frac {a f \,x^{2}}{2 g^{2}}+\frac {a \,f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}+\frac {b n \,f^{2} \dilog \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 g^{3}}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f \,x^{2}}{2 g^{2}}+\frac {a \,x^{4}}{4 g}-\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} f \,x^{2}}{4 g^{2}}+\frac {b n \,f^{2} \dilog \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 g^{3}}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} f^{2} \ln \left (g \,x^{2}+f \right )}{4 g^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} x^{4}}{8 g}+\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} x^{4}}{8 g}-\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 ) f^{2} \ln \left (g \,x^{2}+f \right )}{4 g^{3}}+\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} f^{2} \ln \left (g \,x^{2}+f \right )}{4 g^{3}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}+\frac {b \ln \left (c \right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}+\frac {b d n \,x^{3}}{12 e g}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} x^{4}}{8 g}-\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 ) x^{4}}{8 g}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} f \,x^{2}}{4 g^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} f^{2} \ln \left (g \,x^{2}+f \right )}{4 g^{3}}+\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} f \,x^{2}}{4 g^{2}}-\frac {b d f n x}{2 e \,g^{2}}+\frac {b \,d^{2} f n \ln \left (e x +d \right )}{2 e^{2} g^{2}}-\frac {b \,d^{4} n \ln \left (e x +d \right )}{4 e^{4} g}+\frac {b f n \,x^{2}}{4 g^{2}}+\frac {b \,d^{3} n x}{4 e^{3} g}-\frac {b \,d^{2} n \,x^{2}}{8 e^{2} g}-\frac {b n \,x^{4}}{16 g}\) | \(905\) |
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 {x^5\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{g\,x^2+f} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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