Optimal. Leaf size=149 \[ -\frac {3 b n x}{e^3}+\frac {(6 a+5 b n) x}{2 e^3}+\frac {3 b x \log \left (c x^n\right )}{e^3}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{2 e (d+e x)^2}-\frac {x^2 \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{2 e^2 (d+e x)}-\frac {d \left (6 a+5 b n+6 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{2 e^4}-\frac {3 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4} \]
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Rubi [A]
time = 0.17, antiderivative size = 149, 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 = {2384, 45, 2393,
2332, 2354, 2438} \begin {gather*} -\frac {3 b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\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 n x}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2354
Rule 2384
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)^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 = 0.10, size = 150, normalized size = 1.01 \begin {gather*} \frac {2 a e x-2 b e n x+2 b e x \log \left (c x^n\right )+\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 b d n (\log (x)-\log (d+e x))-b d n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )-6 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{2 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 = 764, normalized size = 5.13
method | result | size |
risch | \(\frac {3 b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{4}}+\frac {3 b n d \dilog \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}{2 e^{3}}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2 e^{3}}+\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d \ln \left (e x +d \right )}{2 e^{4}}+\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2}}{2 e^{4} \left (e x +d \right )}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{3}}{4 e^{4} \left (e x +d \right )^{2}}+\frac {a x}{e^{3}}-\frac {3 i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \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}{2 e^{3}}-\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d \ln \left (e x +d \right )}{2 e^{4}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{2 e^{3}}-\frac {3 b \ln \left (x^{n}\right ) d^{2}}{e^{4} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right ) d^{3}}{2 e^{4} \left (e x +d \right )^{2}}-\frac {3 b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{4}}-\frac {3 b \ln \left (c \right ) d^{2}}{e^{4} \left (e x +d \right )}+\frac {b \ln \left (c \right ) d^{3}}{2 e^{4} \left (e x +d \right )^{2}}-\frac {3 b \ln \left (c \right ) d \ln \left (e x +d \right )}{e^{4}}-\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2}}{2 e^{4} \left (e x +d \right )}-\frac {3 a d \ln \left (e x +d \right )}{e^{4}}-\frac {3 a \,d^{2}}{e^{4} \left (e x +d \right )}+\frac {a \,d^{3}}{2 e^{4} \left (e x +d \right )^{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 ) d^{3}}{4 e^{4} \left (e x +d \right )^{2}}+\frac {3 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 \ln \left (e x +d \right )}{2 e^{4}}+\frac {3 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^{2}}{2 e^{4} \left (e x +d \right )}+\frac {5 b n d \ln \left (e x \right )}{2 e^{4}}-\frac {5 b n d \ln \left (e x +d \right )}{2 e^{4}}-\frac {b n \,d^{2}}{2 e^{4} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right ) x}{e^{3}}+\frac {b \ln \left (c \right ) x}{e^{3}}-\frac {b n d}{e^{4}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{3}}{4 e^{4} \left (e x +d \right )^{2}}-\frac {3 i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2}}{2 e^{4} \left (e x +d \right )}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{3}}{4 e^{4} \left (e x +d \right )^{2}}-\frac {b n x}{e^{3}}\) | \(764\) |
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 = 29.62, size = 391, normalized size = 2.62 \begin {gather*} - \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 {\left (x \right )}}{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 {\left (x \right )}}{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} - \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 {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}} \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 )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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