Optimal. Leaf size=149 \[ -\frac {d \log \left (\frac {e x}{d}+1\right ) \left (6 a+6 b \log \left (c x^n\right )+5 b n\right )}{2 e^4}-\frac {x^2 \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{2 e^2 (d+e x)}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}+\frac {x (6 a+5 b n)}{2 e^3}+\frac {3 b x \log \left (c x^n\right )}{e^3}-\frac {3 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {3 b n x}{e^3} \]
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Rubi [A] time = 0.22, antiderivative size = 167, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {43, 2351, 2295, 2319, 44, 2314, 31, 2317, 2391} \[ -\frac {3 b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}-\frac {3 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {a x}{e^3}+\frac {b x \log \left (c x^n\right )}{e^3}-\frac {b d^2 n}{2 e^4 (d+e x)}-\frac {b d n \log (x)}{2 e^4}-\frac {5 b d n \log (d+e x)}{2 e^4}-\frac {b n x}{e^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 44
Rule 2295
Rule 2314
Rule 2317
Rule 2319
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)^3} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{e^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^3}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)^2}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}-\frac {(3 d) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3}+\frac {\left (3 d^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^3}-\frac {d^3 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^3}\\ &=\frac {a x}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}-\frac {3 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {b \int \log \left (c x^n\right ) \, dx}{e^3}+\frac {(3 b d n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}-\frac {\left (b d^3 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 e^4}-\frac {(3 b d n) \int \frac {1}{d+e x} \, dx}{e^3}\\ &=\frac {a x}{e^3}-\frac {b n x}{e^3}+\frac {b x \log \left (c x^n\right )}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}-\frac {3 b d n \log (d+e x)}{e^4}-\frac {3 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {3 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {\left (b d^3 n\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 e^4}\\ &=\frac {a x}{e^3}-\frac {b n x}{e^3}-\frac {b d^2 n}{2 e^4 (d+e x)}-\frac {b d n \log (x)}{2 e^4}+\frac {b x \log \left (c x^n\right )}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}-\frac {5 b d n \log (d+e x)}{2 e^4}-\frac {3 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {3 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 150, normalized size = 1.01 \[ \frac {\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-6 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+2 a e x+2 b e x \log \left (c x^n\right )-6 b d n \text {Li}_2\left (-\frac {e x}{d}\right )+6 b d n (\log (x)-\log (d+e x))-b d n \left (\frac {d}{d+e x}-\log (d+e x)+\log (x)\right )-2 b e n x}{2 e^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \log \left (c x^{n}\right ) + a x^{3}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 764, normalized size = 5.13 \[ \frac {3 b d n \dilog \left (-\frac {e x}{d}\right )}{e^{4}}-\frac {3 b \,d^{2} \ln \left (x^{n}\right )}{\left (e x +d \right ) e^{4}}-\frac {3 b d \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{4}}+\frac {b \,d^{3} \ln \left (x^{n}\right )}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 b \,d^{2} \ln \relax (c )}{\left (e x +d \right ) e^{4}}-\frac {3 b d \ln \relax (c ) \ln \left (e x +d \right )}{e^{4}}+\frac {b \,d^{3} \ln \relax (c )}{2 \left (e x +d \right )^{2} e^{4}}+\frac {5 b d n \ln \left (e x \right )}{2 e^{4}}-\frac {5 b d n \ln \left (e x +d \right )}{2 e^{4}}-\frac {b \,d^{2} n}{2 \left (e x +d \right ) e^{4}}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (e x +d \right )^{2} e^{4}}+\frac {a \,d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 a \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 a d \ln \left (e x +d \right )}{e^{4}}+\frac {b x \ln \relax (c )}{e^{3}}-\frac {b d n}{e^{4}}+\frac {b x \ln \left (x^{n}\right )}{e^{3}}+\frac {3 b d n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{e^{4}}+\frac {a x}{e^{3}}-\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 )}{4 \left (e x +d \right )^{2} 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 )}{2 \left (e x +d \right ) e^{4}}-\frac {3 i \pi b d \,\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 \,\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 x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 e^{3}}-\frac {3 i \pi b \,d^{2} \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 {i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 \left (e x +d \right )^{2} e^{4}}-\frac {3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e x +d \right ) e^{4}}+\frac {3 i \pi b d \,\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 {b n x}{e^{3}}-\frac {i \pi b x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 e^{3}}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 \left (e x +d \right )^{2} e^{4}}+\frac {3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 \left (e x +d \right ) e^{4}}+\frac {3 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 e^{4}}+\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e^{3}}+\frac {i \pi b x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 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 \[ -\frac {1}{2} \, a {\left (\frac {6 \, d^{2} e x + 5 \, d^{3}}{e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}} - \frac {2 \, x}{e^{3}} + \frac {6 \, d \log \left (e x + d\right )}{e^{4}}\right )} + b \int \frac {x^{3} \log \relax (c) + x^{3} \log \left (x^{n}\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}\,{d x} \]
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 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 48.37, size = 372, normalized size = 2.50 \[ - \frac {a d^{3} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 a d^{2} \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 \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 x}{e^{3}} + \frac {b d^{3} n \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 d^{2} e + 2 d e^{2} x} - \frac {\log {\relax (x )}}{2 d^{2} e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {b d^{3} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \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^{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 {3 b d^{2} \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 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 \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 n x}{e^{3}} + \frac {b x \log {\left (c x^{n} \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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