Optimal. Leaf size=185 \[ -\frac {a e^3 x}{d^4}+\frac {b e^3 n x}{d^4}-\frac {b e^2 n x^2}{4 d^3}+\frac {b e n x^3}{9 d^2}-\frac {b n x^4}{16 d}-\frac {b e^3 x \log \left (c x^n\right )}{d^4}+\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac {e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac {e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d^5}+\frac {b e^4 n \text {Li}_2\left (-\frac {d x}{e}\right )}{d^5} \]
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Rubi [A]
time = 0.14, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {269, 45, 2393,
2332, 2341, 2354, 2438} \begin {gather*} \frac {b e^4 n \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^5}+\frac {e^4 \log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac {e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}-\frac {a e^3 x}{d^4}-\frac {b e^3 x \log \left (c x^n\right )}{d^4}+\frac {b e^3 n x}{d^4}-\frac {b e^2 n x^2}{4 d^3}+\frac {b e n x^3}{9 d^2}-\frac {b n x^4}{16 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 2332
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+\frac {e}{x}} \, dx &=\int \left (-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac {e x^2 \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {e^4 \left (a+b \log \left (c x^n\right )\right )}{d^4 (e+d x)}\right ) \, dx\\ &=\frac {\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d}-\frac {e \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^2}+\frac {e^2 \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^3}-\frac {e^3 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^4}+\frac {e^4 \int \frac {a+b \log \left (c x^n\right )}{e+d x} \, dx}{d^4}\\ &=-\frac {a e^3 x}{d^4}-\frac {b e^2 n x^2}{4 d^3}+\frac {b e n x^3}{9 d^2}-\frac {b n x^4}{16 d}+\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac {e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac {e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d^5}-\frac {\left (b e^3\right ) \int \log \left (c x^n\right ) \, dx}{d^4}-\frac {\left (b e^4 n\right ) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d^5}\\ &=-\frac {a e^3 x}{d^4}+\frac {b e^3 n x}{d^4}-\frac {b e^2 n x^2}{4 d^3}+\frac {b e n x^3}{9 d^2}-\frac {b n x^4}{16 d}-\frac {b e^3 x \log \left (c x^n\right )}{d^4}+\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac {e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac {e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d^5}+\frac {b e^4 n \text {Li}_2\left (-\frac {d x}{e}\right )}{d^5}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 171, normalized size = 0.92 \begin {gather*} \frac {-144 a d e^3 x+144 b d e^3 n x-36 b d^2 e^2 n x^2+16 b d^3 e n x^3-9 b d^4 n x^4-144 b d e^3 x \log \left (c x^n\right )+72 d^2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-48 d^3 e x^3 \left (a+b \log \left (c x^n\right )\right )+36 d^4 x^4 \left (a+b \log \left (c x^n\right )\right )+144 e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )+144 b e^4 n \text {Li}_2\left (-\frac {d x}{e}\right )}{144 d^5} \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 = 0.10, size = 867, normalized size = 4.69
method | result | size |
risch | \(\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2} e^{2}}{4 d^{3}}-\frac {b n \,e^{4} \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{d^{5}}-\frac {b n \,e^{4} \dilog \left (-\frac {d x}{e}\right )}{d^{5}}+\frac {a \,e^{4} \ln \left (d x +e \right )}{d^{5}}-\frac {a e \,x^{3}}{3 d^{2}}+\frac {a \,x^{2} e^{2}}{2 d^{3}}-\frac {b \ln \left (x^{n}\right ) e \,x^{3}}{3 d^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{4}}{8 d}-\frac {b \,e^{2} n \,x^{2}}{4 d^{3}}+\frac {b e n \,x^{3}}{9 d^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e^{4} \ln \left (d x +e \right )}{2 d^{5}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x \,e^{3}}{2 d^{4}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e^{4} \ln \left (d x +e \right )}{2 d^{5}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x \,e^{3}}{2 d^{4}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2} e^{2}}{4 d^{3}}-\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}}{8 d}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e \,x^{3}}{6 d^{2}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e \,x^{3}}{6 d^{2}}+\frac {b \,e^{3} n x}{d^{4}}+\frac {b \ln \left (c \right ) x^{2} e^{2}}{2 d^{3}}-\frac {b \ln \left (c \right ) x \,e^{3}}{d^{4}}+\frac {b \ln \left (c \right ) e^{4} \ln \left (d x +e \right )}{d^{5}}+\frac {a \,x^{4}}{4 d}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{4}}{8 d}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x \,e^{3}}{2 d^{4}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{4}}{8 d}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{2} e^{2}}{4 d^{3}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} e^{4} \ln \left (d x +e \right )}{2 d^{5}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} e \,x^{3}}{6 d^{2}}+\frac {b \ln \left (x^{n}\right ) x^{4}}{4 d}+\frac {b \ln \left (c \right ) x^{4}}{4 d}+\frac {205 b n \,e^{4}}{144 d^{5}}-\frac {b \ln \left (c \right ) e \,x^{3}}{3 d^{2}}+\frac {b \ln \left (x^{n}\right ) x^{2} e^{2}}{2 d^{3}}-\frac {b \ln \left (x^{n}\right ) x \,e^{3}}{d^{4}}+\frac {b \ln \left (x^{n}\right ) e^{4} \ln \left (d x +e \right )}{d^{5}}-\frac {a \,e^{3} x}{d^{4}}-\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 ) e^{4} \ln \left (d x +e \right )}{2 d^{5}}-\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^{2} e^{2}}{4 d^{3}}-\frac {b n \,x^{4}}{16 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 ) e \,x^{3}}{6 d^{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 \,e^{3}}{2 d^{4}}\) | \(867\) |
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 [A]
time = 115.41, size = 316, normalized size = 1.71 \begin {gather*} \frac {a x^{4}}{4 d} - \frac {a e x^{3}}{3 d^{2}} + \frac {a e^{2} x^{2}}{2 d^{3}} + \frac {a e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}} - \frac {a e^{3} x}{d^{4}} - \frac {b n x^{4}}{16 d} + \frac {b x^{4} \log {\left (c x^{n} \right )}}{4 d} + \frac {b e n x^{3}}{9 d^{2}} - \frac {b e x^{3} \log {\left (c x^{n} \right )}}{3 d^{2}} - \frac {b e^{2} n x^{2}}{4 d^{3}} + \frac {b e^{2} x^{2} \log {\left (c x^{n} \right )}}{2 d^{3}} - \frac {b e^{4} n \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}} + \frac {b e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{4}} + \frac {b e^{3} n x}{d^{4}} - \frac {b e^{3} x \log {\left (c x^{n} \right )}}{d^{4}} \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^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+\frac {e}{x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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