Optimal. Leaf size=167 \[ -\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {i b d^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {i b d^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {308, 211,
2393, 2332, 2341, 2361, 12, 4940, 2438} \begin {gather*} -\frac {i b d^{3/2} n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {i b d^{3/2} n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {a d x}{e^2}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 308
Rule 2332
Rule 2341
Rule 2361
Rule 2393
Rule 2438
Rule 4940
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac {d \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e^2}+\frac {\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac {a d x}{e^2}-\frac {b n x^3}{9 e}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {(b d) \int \log \left (c x^n\right ) \, dx}{e^2}-\frac {\left (b d^2 n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{e^2}\\ &=-\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {\left (b d^{3/2} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2}}\\ &=-\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {\left (i b d^{3/2} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{5/2}}+\frac {\left (i b d^{3/2} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{5/2}}\\ &=-\frac {a d x}{e^2}+\frac {b d n x}{e^2}-\frac {b n x^3}{9 e}-\frac {b d x \log \left (c x^n\right )}{e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac {i b d^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {i b d^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 208, normalized size = 1.25 \begin {gather*} \frac {-18 a d \sqrt {e} x+18 b d \sqrt {e} n x-2 b e^{3/2} n x^3-18 b d \sqrt {e} x \log \left (c x^n\right )+6 e^{3/2} x^3 \left (a+b \log \left (c x^n\right )\right )+9 \sqrt {-d} d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+9 (-d)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+9 b (-d)^{3/2} n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )-9 b (-d)^{3/2} n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{18 e^{5/2}} \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 = 0.13, size = 693, normalized size = 4.15
method | result | size |
risch | \(\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d x}{2 e^{2}}+\frac {a \,x^{3}}{3 e}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3}}{6 e}-\frac {b \,d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right ) n \ln \left (x \right )}{e^{2} \sqrt {e d}}+\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 e^{2} \sqrt {-e d}}-\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 e^{2} \sqrt {-e d}}+\frac {b \ln \left (c \right ) d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{e^{2} \sqrt {e d}}+\frac {b \ln \left (c \right ) x^{3}}{3 e}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3}}{6 e}+\frac {b \,d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right ) \ln \left (x^{n}\right )}{e^{2} \sqrt {e d}}-\frac {b n \,d^{2} \dilog \left (\frac {e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 e^{2} \sqrt {-e d}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{3}}{6 e}+\frac {b n \,d^{2} \dilog \left (\frac {-e x +\sqrt {-e d}}{\sqrt {-e d}}\right )}{2 e^{2} \sqrt {-e d}}-\frac {b \ln \left (x^{n}\right ) d x}{e^{2}}+\frac {a \,d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{e^{2} \sqrt {e d}}-\frac {b \ln \left (c \right ) d x}{e^{2}}-\frac {a d x}{e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d x}{2 e^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d x}{2 e^{2}}-\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^{3}}{6 e}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e^{2} \sqrt {e d}}+\frac {b \ln \left (x^{n}\right ) x^{3}}{3 e}-\frac {b n \,x^{3}}{9 e}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e^{2} \sqrt {e d}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e^{2} \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 ) d x}{2 e^{2}}-\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 ) d^{2} \arctan \left (\frac {x e}{\sqrt {e d}}\right )}{2 e^{2} \sqrt {e d}}+\frac {b d n x}{e^{2}}\) | \(693\) |
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^{4} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{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^4\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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