Optimal. Leaf size=205 \[ \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {24 b^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {24 b^4 n^4 \text {Li}_5\left (-\frac {g (d+e x)}{e f-d g}\right )}{g} \]
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Rubi [A]
time = 0.16, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2443, 2481,
2421, 2430, 6724} \begin {gather*} \frac {24 b^3 n^3 \text {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {12 b^2 n^2 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac {4 b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac {24 b^4 n^4 \text {PolyLog}\left (5,-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g} \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2430
Rule 2443
Rule 2481
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx &=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}-\frac {(4 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g}\\ &=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}-\frac {(4 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {\left (12 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {\left (24 b^3 n^3\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {24 b^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {\left (24 b^4 n^4\right ) \text {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {24 b^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {24 b^4 n^4 \text {Li}_5\left (-\frac {g (d+e x)}{e f-d g}\right )}{g}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(503\) vs. \(2(205)=410\).
time = 0.10, size = 503, normalized size = 2.45 \begin {gather*} \frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^4 \log (f+g x)+4 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )\right )+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-2 \text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )\right )-4 b^3 n^3 \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )+6 \text {Li}_4\left (\frac {g (d+e x)}{-e f+d g}\right )\right )+b^4 n^4 \left (\log ^4(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+4 \log ^3(d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-12 \log ^2(d+e x) \text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )+24 \log (d+e x) \text {Li}_4\left (\frac {g (d+e x)}{-e f+d g}\right )-24 \text {Li}_5\left (\frac {g (d+e x)}{-e f+d g}\right )\right )}{g} \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 = 1.42, size = 33189, normalized size = 161.90
method | result | size |
risch | \(\text {Expression too large to display}\) | \(33189\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{4}}{f + g x}\, dx \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^4}{f+g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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