Optimal. Leaf size=217 \[ -\frac {3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac {6 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}-\frac {a+b \log \left (c x^n\right )}{2 d^3 x^2}+\frac {6 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^5}-\frac {b e^2 n \log (x)}{2 d^5}+\frac {7 b e^2 n \log (d+e x)}{2 d^5}-\frac {b e^2 n}{2 d^4 (d+e x)}+\frac {3 b e n}{d^4 x}-\frac {b n}{4 d^3 x^2} \]
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Rubi [A] time = 0.27, antiderivative size = 239, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {44, 2351, 2304, 2301, 2319, 2314, 31, 2317, 2391} \[ -\frac {6 b e^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^5}-\frac {3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}-\frac {6 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{2 d^3 x^2}-\frac {b e^2 n}{2 d^4 (d+e x)}-\frac {b e^2 n \log (x)}{2 d^5}+\frac {7 b e^2 n \log (d+e x)}{2 d^5}+\frac {3 b e n}{d^4 x}-\frac {b n}{4 d^3 x^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2304
Rule 2314
Rule 2317
Rule 2319
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^3} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d^3 x^3}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x^2}+\frac {6 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^5 x}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^3}-\frac {3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^2}-\frac {6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^4}+\frac {\left (6 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^5}-\frac {\left (6 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^5}-\frac {\left (3 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^4}-\frac {e^3 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^3}\\ &=-\frac {b n}{4 d^3 x^2}+\frac {3 b e n}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{2 d^3 x^2}+\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}-\frac {3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}-\frac {6 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {\left (6 b e^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^5}-\frac {\left (b e^2 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^3}+\frac {\left (3 b e^3 n\right ) \int \frac {1}{d+e x} \, dx}{d^5}\\ &=-\frac {b n}{4 d^3 x^2}+\frac {3 b e n}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{2 d^3 x^2}+\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}-\frac {3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}+\frac {3 b e^2 n \log (d+e x)}{d^5}-\frac {6 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {6 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}-\frac {\left (b e^2 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 d^3}\\ &=-\frac {b n}{4 d^3 x^2}+\frac {3 b e n}{d^4 x}-\frac {b e^2 n}{2 d^4 (d+e x)}-\frac {b e^2 n \log (x)}{2 d^5}-\frac {a+b \log \left (c x^n\right )}{2 d^3 x^2}+\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}-\frac {3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}+\frac {7 b e^2 n \log (d+e x)}{2 d^5}-\frac {6 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {6 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 227, normalized size = 1.05 \[ -\frac {-\frac {2 d^2 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {12 d e^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+24 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {12 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {12 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac {b d^2 n}{x^2}+24 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )+12 b e^2 n (\log (x)-\log (d+e x))+\frac {2 b e^2 n (\log (x) (d+e x)-(d+e x) \log (d+e x)+d)}{d+e x}-\frac {12 b d e n}{x}}{4 d^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 1119, normalized size = 5.16 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, a {\left (\frac {12 \, e^{3} x^{3} + 18 \, d e^{2} x^{2} + 4 \, d^{2} e x - d^{3}}{d^{4} e^{2} x^{4} + 2 \, d^{5} e x^{3} + d^{6} x^{2}} - \frac {12 \, e^{2} \log \left (e x + d\right )}{d^{5}} + \frac {12 \, e^{2} \log \relax (x)}{d^{5}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 110.20, size = 478, normalized size = 2.20 \[ - \frac {a e^{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 )}{d^{3}} - \frac {a}{2 d^{3} x^{2}} - \frac {3 a e^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{d^{4}} + \frac {3 a e}{d^{4} x} - \frac {6 a e^{3} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{5}} + \frac {6 a e^{2} \log {\relax (x )}}{d^{5}} + \frac {b e^{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 )}{d^{3}} - \frac {b e^{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 )}}{d^{3}} - \frac {b n}{4 d^{3} x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 d^{3} x^{2}} + \frac {3 b e^{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 )}{d^{4}} - \frac {3 b e^{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 )}}{d^{4}} + \frac {3 b e n}{d^{4} x} + \frac {3 b e \log {\left (c x^{n} \right )}}{d^{4} x} + \frac {6 b e^{3} 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 )}{d^{5}} - \frac {6 b e^{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 )}}{d^{5}} - \frac {3 b e^{2} n \log {\relax (x )}^{2}}{d^{5}} + \frac {6 b e^{2} \log {\relax (x )} \log {\left (c x^{n} \right )}}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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