Optimal. Leaf size=107 \[ \frac {e^2 \log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d}-\frac {a e x}{d^2}-\frac {b e x \log \left (c x^n\right )}{d^2}+\frac {b e^2 n \text {Li}_2\left (-\frac {d x}{e}\right )}{d^3}+\frac {b e n x}{d^2}-\frac {b n x^2}{4 d} \]
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Rubi [A] time = 0.12, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {263, 43, 2351, 2295, 2304, 2317, 2391} \[ \frac {b e^2 n \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^3}+\frac {e^2 \log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d}-\frac {a e x}{d^2}-\frac {b e x \log \left (c x^n\right )}{d^2}+\frac {b e n x}{d^2}-\frac {b n x^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 2295
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+\frac {e}{x}} \, dx &=\int \left (-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (e+d x)}\right ) \, dx\\ &=\frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{d}-\frac {e \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{e+d x} \, dx}{d^2}\\ &=-\frac {a e x}{d^2}-\frac {b n x^2}{4 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d^3}-\frac {(b e) \int \log \left (c x^n\right ) \, dx}{d^2}-\frac {\left (b e^2 n\right ) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d^3}\\ &=-\frac {a e x}{d^2}+\frac {b e n x}{d^2}-\frac {b n x^2}{4 d}-\frac {b e x \log \left (c x^n\right )}{d^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d^3}+\frac {b e^2 n \text {Li}_2\left (-\frac {d x}{e}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 105, normalized size = 0.98 \[ \frac {2 a d^2 x^2+4 a e^2 \log \left (\frac {d x}{e}+1\right )-4 a d e x+2 b \log \left (c x^n\right ) \left (2 e^2 \log \left (\frac {d x}{e}+1\right )+d x (d x-2 e)\right )-b d^2 n x^2+4 b e^2 n \text {Li}_2\left (-\frac {d x}{e}\right )+4 b d e n x}{4 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left (c x^{n}\right ) + a x^{2}}{d x + e}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{d + \frac {e}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 521, normalized size = 4.87 \[ -\frac {b \,e^{2} n \dilog \left (-\frac {d x}{e}\right )}{d^{3}}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 d}-\frac {i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d^{2}}-\frac {b \,e^{2} n \ln \left (-\frac {d x}{e}\right ) \ln \left (d x +e \right )}{d^{3}}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 d}-\frac {b e x \ln \relax (c )}{d^{2}}+\frac {b \,e^{2} \ln \relax (c ) \ln \left (d x +e \right )}{d^{3}}+\frac {b \,e^{2} \ln \left (x^{n}\right ) \ln \left (d x +e \right )}{d^{3}}-\frac {b e x \ln \left (x^{n}\right )}{d^{2}}+\frac {a \,x^{2}}{2 d}+\frac {i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 d^{2}}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (d x +e \right )}{2 d^{3}}+\frac {b \,x^{2} \ln \left (x^{n}\right )}{2 d}+\frac {a \,e^{2} \ln \left (d x +e \right )}{d^{3}}+\frac {b \,x^{2} \ln \relax (c )}{2 d}+\frac {5 b \,e^{2} n}{4 d^{3}}-\frac {i \pi b e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d^{2}}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d x +e \right )}{2 d^{3}}+\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d x +e \right )}{2 d^{3}}-\frac {a e x}{d^{2}}+\frac {b e n x}{d^{2}}+\frac {i \pi b e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 d^{2}}-\frac {i \pi b \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d x +e \right )}{2 d^{3}}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 d}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 d}-\frac {b n \,x^{2}}{4 d} \]
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 \[ \frac {1}{2} \, a {\left (\frac {2 \, e^{2} \log \left (d x + e\right )}{d^{3}} + \frac {d x^{2} - 2 \, e x}{d^{2}}\right )} + b \int \frac {x^{2} \log \relax (c) + x^{2} \log \left (x^{n}\right )}{d x + e}\,{d x} \]
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 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+\frac {e}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 114.15, size = 199, normalized size = 1.86 \[ \frac {a x^{2}}{2 d} + \frac {a e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e x}{d^{2}} - \frac {b n x^{2}}{4 d} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 d} - \frac {b e^{2} n \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} \log {\relax (e )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (e )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (e )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (e )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b e n x}{d^{2}} - \frac {b e x \log {\left (c x^{n} \right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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