Optimal. Leaf size=150 \[ \frac {e^3 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {a+b \log \left (c x^n\right )}{3 d x^3}-\frac {b e^3 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}-\frac {b e^2 n}{d^3 x}+\frac {b e n}{4 d^2 x^2}-\frac {b n}{9 d x^3} \]
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Rubi [A] time = 0.21, antiderivative size = 173, normalized size of antiderivative = 1.15, 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 = {44, 2351, 2304, 2301, 2317, 2391} \[ \frac {b e^3 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac {e^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {a+b \log \left (c x^n\right )}{3 d x^3}-\frac {b e^2 n}{d^3 x}+\frac {b e n}{4 d^2 x^2}-\frac {b n}{9 d x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2301
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^4 (d+e x)} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d x^4}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^2}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {e^4 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^4} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}-\frac {e^3 \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^4}+\frac {e^4 \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4}\\ &=-\frac {b n}{9 d x^3}+\frac {b e n}{4 d^2 x^2}-\frac {b e^2 n}{d^3 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac {e^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {\left (b e^3 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^4}\\ &=-\frac {b n}{9 d x^3}+\frac {b e n}{4 d^2 x^2}-\frac {b e^2 n}{d^3 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}+\frac {e^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {b e^3 n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 159, normalized size = 1.06 \[ \frac {-\frac {12 d^3 \left (a+b \log \left (c x^n\right )\right )}{x^3}+\frac {18 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^2}+36 e^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {36 d e^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {18 e^3 \left (a+b \log \left (c x^n\right )\right )^2}{b n}-\frac {4 b d^3 n}{x^3}+\frac {9 b d^2 e n}{x^2}+36 b e^3 n \text {Li}_2\left (-\frac {e x}{d}\right )-\frac {36 b d e^2 n}{x}}{36 d^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{5} + d x^{4}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 868, normalized size = 5.79 \[ \text {result too large to display} \]
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 \[ \frac {1}{6} \, a {\left (\frac {6 \, e^{3} \log \left (e x + d\right )}{d^{4}} - \frac {6 \, e^{3} \log \relax (x)}{d^{4}} - \frac {6 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}}{d^{3} x^{3}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x^{5} + d x^{4}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 91.86, size = 296, normalized size = 1.97 \[ - \frac {a}{3 d x^{3}} + \frac {a e}{2 d^{2} x^{2}} - \frac {a e^{2}}{d^{3} x} + \frac {a e^{4} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{4}} - \frac {a e^{3} \log {\relax (x )}}{d^{4}} - \frac {b n}{9 d x^{3}} - \frac {b \log {\left (c x^{n} \right )}}{3 d x^{3}} + \frac {b e n}{4 d^{2} x^{2}} + \frac {b e \log {\left (c x^{n} \right )}}{2 d^{2} x^{2}} - \frac {b e^{2} n}{d^{3} x} - \frac {b e^{2} \log {\left (c x^{n} \right )}}{d^{3} x} - \frac {b e^{4} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{4}} + \frac {b e^{4} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{4}} + \frac {b e^{3} n \log {\relax (x )}^{2}}{2 d^{4}} - \frac {b e^{3} \log {\relax (x )} \log {\left (c x^{n} \right )}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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