Optimal. Leaf size=148 \[ \frac {a d^2 x}{e^3}-\frac {b d^2 n x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e}+\frac {b d^2 x \log \left (c x^n\right )}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {45, 2393, 2332,
2341, 2354, 2438} \begin {gather*} -\frac {b d^3 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {a d^2 x}{e^3}+\frac {b d^2 x \log \left (c x^n\right )}{e^3}-\frac {b d^2 n x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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+e x} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {d x \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^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d^2 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}-\frac {d^3 \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac {d \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac {\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=\frac {a d^2 x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (b d^2\right ) \int \log \left (c x^n\right ) \, dx}{e^3}+\frac {\left (b d^3 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}\\ &=\frac {a d^2 x}{e^3}-\frac {b d^2 n x}{e^3}+\frac {b d n x^2}{4 e^2}-\frac {b n x^3}{9 e}+\frac {b d^2 x \log \left (c x^n\right )}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 142, normalized size = 0.96 \begin {gather*} \frac {36 a d^2 e x-36 b d^2 e n x-18 a d e^2 x^2+9 b d e^2 n x^2+12 a e^3 x^3-4 b e^3 n x^3-36 a d^3 \log \left (1+\frac {e x}{d}\right )+6 b \log \left (c x^n\right ) \left (e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 d^3 \log \left (1+\frac {e x}{d}\right )\right )-36 b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right )}{36 e^4} \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 = 0.12, size = 693, normalized size = 4.68
method | result | size |
risch | \(-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{3} \ln \left (e x +d \right )}{2 e^{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 ) x^{3}}{6 e}+\frac {a \,x^{3}}{3 e}-\frac {a \,d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {a d \,x^{2}}{2 e^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x \,d^{2}}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \,x^{2}}{4 e^{2}}+\frac {b n \,d^{3} \dilog \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{3}}{6 e}-\frac {49 b n \,d^{3}}{36 e^{4}}+\frac {b \ln \left (c \right ) x^{3}}{3 e}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x \,d^{2}}{2 e^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{3} \ln \left (e x +d \right )}{2 e^{4}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \,x^{2}}{4 e^{2}}+\frac {b \ln \left (x^{n}\right ) x^{3}}{3 e}+\frac {b n \,d^{3} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{3}}{6 e}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x \,d^{2}}{2 e^{3}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{3} \ln \left (e x +d \right )}{2 e^{4}}+\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 {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d \,x^{2}}{4 e^{2}}+\frac {a \,d^{2} x}{e^{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 ) d \,x^{2}}{4 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 \,d^{2}}{2 e^{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 ) d^{3} \ln \left (e x +d \right )}{2 e^{4}}-\frac {b \ln \left (x^{n}\right ) d \,x^{2}}{2 e^{2}}+\frac {b \ln \left (x^{n}\right ) x \,d^{2}}{e^{3}}-\frac {b \ln \left (x^{n}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {b \ln \left (c \right ) d \,x^{2}}{2 e^{2}}+\frac {b \ln \left (c \right ) x \,d^{2}}{e^{3}}-\frac {b \ln \left (c \right ) d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b d n \,x^{2}}{4 e^{2}}-\frac {b n \,x^{3}}{9 e}-\frac {b \,d^{2} n x}{e^{3}}\) | \(693\) |
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 [A]
time = 18.60, size = 267, normalized size = 1.80 \begin {gather*} - \frac {a d^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {a d^{2} x}{e^{3}} - \frac {a d x^{2}}{2 e^{2}} + \frac {a x^{3}}{3 e} + \frac {b d^{3} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\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 (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {b d^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b d^{2} n x}{e^{3}} + \frac {b d^{2} x \log {\left (c x^{n} \right )}}{e^{3}} + \frac {b d n x^{2}}{4 e^{2}} - \frac {b d x^{2} \log {\left (c x^{n} \right )}}{2 e^{2}} - \frac {b n x^{3}}{9 e} + \frac {b x^{3} \log {\left (c x^{n} \right )}}{3 e} \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+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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