Optimal. Leaf size=152 \[ \frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{e^4}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}+\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e^2}-\frac {d x (3 a+b n)}{e^3}-\frac {3 b d x \log \left (c x^n\right )}{e^3}+\frac {3 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {3 b d n x}{e^3}-\frac {3 b n x^2}{4 e^2} \]
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Rubi [A] time = 0.18, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {43, 2351, 2295, 2304, 2314, 31, 2317, 2391} \[ \frac {3 b d^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac {3 d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 a d x}{e^3}-\frac {2 b d x \log \left (c x^n\right )}{e^3}+\frac {b d^2 n \log (d+e x)}{e^4}+\frac {2 b d n x}{e^3}-\frac {b n x^2}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 2295
Rule 2304
Rule 2314
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx &=\int \left (-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^2}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac {(2 d) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}+\frac {\left (3 d^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}-\frac {d^3 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^3}+\frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}\\ &=-\frac {2 a d x}{e^3}-\frac {b n x^2}{4 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {(2 b d) \int \log \left (c x^n\right ) \, dx}{e^3}-\frac {\left (3 b d^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}+\frac {\left (b d^2 n\right ) \int \frac {1}{d+e x} \, dx}{e^3}\\ &=-\frac {2 a d x}{e^3}+\frac {2 b d n x}{e^3}-\frac {b n x^2}{4 e^2}-\frac {2 b d x \log \left (c x^n\right )}{e^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac {b d^2 n \log (d+e x)}{e^4}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {3 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 141, normalized size = 0.93 \[ \frac {\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+12 d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-8 a d e x-8 b d e x \log \left (c x^n\right )+12 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )-4 b d^2 n (\log (x)-\log (d+e x))+8 b d e n x-b e^2 n x^2}{4 e^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \log \left (c x^{n}\right ) + a x^{3}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 739, normalized size = 4.86 \[ -\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 \left (e x +d \right ) e^{4}}-\frac {3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +d \right )}{2 e^{4}}+\frac {i \pi b d x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{e^{3}}-\frac {3 b \,d^{2} n \dilog \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {b \,d^{2} n \ln \left (e x \right )}{e^{4}}+\frac {b \,d^{2} n \ln \left (e x +d \right )}{e^{4}}-\frac {2 b d x \ln \relax (c )}{e^{3}}+\frac {b \,d^{3} \ln \relax (c )}{\left (e x +d \right ) e^{4}}+\frac {3 b \,d^{2} \ln \relax (c ) \ln \left (e x +d \right )}{e^{4}}+\frac {b \,d^{3} \ln \left (x^{n}\right )}{\left (e x +d \right ) e^{4}}+\frac {3 b \,d^{2} \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{4}}-\frac {2 b d x \ln \left (x^{n}\right )}{e^{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 e^{2}}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e x +d \right ) e^{4}}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e x +d \right ) e^{4}}+\frac {3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{4}}+\frac {3 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{4}}-\frac {i \pi b d x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{e^{3}}-\frac {i \pi b d x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{e^{3}}+\frac {b \,x^{2} \ln \left (x^{n}\right )}{2 e^{2}}-\frac {2 a d x}{e^{3}}+\frac {a \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 a \,d^{2} \ln \left (e x +d \right )}{e^{4}}+\frac {b \,x^{2} \ln \relax (c )}{2 e^{2}}+\frac {9 b \,d^{2} n}{4 e^{4}}+\frac {a \,x^{2}}{2 e^{2}}-\frac {3 b \,d^{2} n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{e^{4}}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 e^{2}}-\frac {b n \,x^{2}}{4 e^{2}}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e^{2}}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e^{2}}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 \left (e x +d \right ) e^{4}}-\frac {3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 e^{4}}+\frac {i \pi b d x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{e^{3}}+\frac {2 b d n x}{e^{3}} \]
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 \[ \frac {1}{2} \, {\left (\frac {2 \, d^{3}}{e^{5} x + d e^{4}} + \frac {6 \, d^{2} \log \left (e x + d\right )}{e^{4}} + \frac {e x^{2} - 4 \, d x}{e^{3}}\right )} a + b \int \frac {x^{3} \log \relax (c) + x^{3} \log \left (x^{n}\right )}{e^{2} x^{2} + 2 \, d e x + d^{2}}\,{d x} \]
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 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.73, size = 304, normalized size = 2.00 \[ - \frac {a d^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 a d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {2 a d x}{e^{3}} + \frac {a x^{2}}{2 e^{2}} + \frac {b d^{3} n \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\relax (x )}}{d e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {b d^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{3}} - \frac {3 b d^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 b d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{3}} + \frac {2 b d n x}{e^{3}} - \frac {2 b d x \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b n x^{2}}{4 e^{2}} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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