Optimal. Leaf size=221 \[ \frac{24 b^3 e p^3 \text{PolyLog}\left (3,\frac{e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )}{d g}-\frac{12 b^2 e p^2 \text{PolyLog}\left (2,\frac{e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac{24 b^4 e p^4 \text{PolyLog}\left (4,\frac{e}{d (f+g x)}+1\right )}{d g}-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g} \]
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Rubi [A] time = 0.27778, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2483, 2449, 2454, 2396, 2433, 2374, 2383, 6589} \[ \frac{24 b^3 e p^3 \text{PolyLog}\left (3,\frac{e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )}{d g}-\frac{12 b^2 e p^2 \text{PolyLog}\left (2,\frac{e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac{24 b^4 e p^4 \text{PolyLog}\left (4,\frac{e}{d (f+g x)}+1\right )}{d g}-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g} \]
Antiderivative was successfully verified.
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Rule 2483
Rule 2449
Rule 2454
Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x}\right )^p\right )\right )^4 \, dx,x,f+g x\right )}{g}\\ &=\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}+\frac{(4 b e p) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x}\right )^p\right )\right )^3}{x} \, dx,x,f+g x\right )}{d g}\\ &=\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}-\frac{(4 b e p) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^p\right )\right )^3}{x} \, dx,x,\frac{1}{f+g x}\right )}{d g}\\ &=-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}+\frac{\left (12 b^2 e^2 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^p\right )\right )^2}{d+e x} \, dx,x,\frac{1}{f+g x}\right )}{d g}\\ &=-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}+\frac{\left (12 b^2 e p^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^p\right )\right )^2 \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac{e}{f+g x}\right )}{d g}\\ &=-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}-\frac{12 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2 \text{Li}_2\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}+\frac{\left (24 b^3 e p^3\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^p\right )\right ) \text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{f+g x}\right )}{d g}\\ &=-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}-\frac{12 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2 \text{Li}_2\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}+\frac{24 b^3 e p^3 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right ) \text{Li}_3\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}-\frac{\left (24 b^4 e p^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{f+g x}\right )}{d g}\\ &=-\frac{4 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^4}{d g}-\frac{12 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2 \text{Li}_2\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}+\frac{24 b^3 e p^3 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right ) \text{Li}_3\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}-\frac{24 b^4 e p^4 \text{Li}_4\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}\\ \end{align*}
Mathematica [B] time = 1.36468, size = 739, normalized size = 3.34 \[ \frac{4 b^3 p^3 \left (6 e \text{PolyLog}\left (3,\frac{e}{d f+d g x}+1\right )-6 e \log \left (d+\frac{e}{f+g x}\right ) \text{PolyLog}\left (2,\frac{e}{d f+d g x}+1\right )+\left ((d f+d g x+e) \log \left (d+\frac{e}{f+g x}\right )-3 e \log \left (-\frac{e}{d f+d g x}\right )\right ) \log ^2\left (d+\frac{e}{f+g x}\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )+6 b^2 p^2 \left (-d f \left (\log \left (-\frac{e}{d f+d g x}\right ) \left (\log \left (-\frac{e}{d f+d g x}\right )+2 \log \left (\frac{d f+d g x+e}{e}\right )\right )-2 \text{PolyLog}\left (2,-\frac{d (f+g x)}{e}\right )\right )+(d f+e) \left (2 \text{PolyLog}\left (2,\frac{d f+d g x+e}{e}\right )+\left (2 \log \left (-\frac{d (f+g x)}{e}\right )-\log (d f+d g x+e)\right ) \log (d f+d g x+e)\right )+d g x \log ^2\left (\frac{d f+d g x+e}{f+g x}\right )+2 d f \log \left (-\frac{e}{d f+d g x}\right ) \log \left (\frac{d f+d g x+e}{f+g x}\right )+2 (d f+e) \log (d f+d g x+e) \log \left (\frac{d f+d g x+e}{f+g x}\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )^2-b^4 p^4 \left (24 e \text{PolyLog}\left (4,\frac{e}{d f+d g x}+1\right )+12 e \log ^2\left (d+\frac{e}{f+g x}\right ) \text{PolyLog}\left (2,\frac{e}{d f+d g x}+1\right )-24 e \log \left (d+\frac{e}{f+g x}\right ) \text{PolyLog}\left (3,\frac{e}{d f+d g x}+1\right )-e \log ^4\left (d+\frac{e}{f+g x}\right )-d f \log ^4\left (d+\frac{e}{f+g x}\right )-d g x \log ^4\left (d+\frac{e}{f+g x}\right )+4 e \log \left (-\frac{e}{d f+d g x}\right ) \log ^3\left (d+\frac{e}{f+g x}\right )\right )-4 b p \left (-(d f+e) \log (d f+d g x+e)-d g x \log \left (\frac{d f+d g x+e}{f+g x}\right )+d f \log (f+g x)\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )^3+d g x \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )^4}{d g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.599, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{\frac{e}{gx+f}} \right ) ^{p} \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{4} \log \left (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right )^{4} + 4 \, a b^{3} \log \left (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right )^{3} + 6 \, a^{2} b^{2} \log \left (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right )^{2} + 4 \, a^{3} b \log \left (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right ) + a^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c{\left (d + \frac{e}{g x + f}\right )}^{p}\right ) + a\right )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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