Optimal. Leaf size=135 \[ -12 b^2 n^2 \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+6 b n \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+12 b^3 n^3 \text{PolyLog}\left (4,\frac{e \sqrt{x}}{d}+1\right )+2 \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \]
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Rubi [A] time = 0.196425, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2396, 2433, 2374, 2383, 6589} \[ -12 b^2 n^2 \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+6 b n \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+12 b^3 n^3 \text{PolyLog}\left (4,\frac{e \sqrt{x}}{d}+1\right )+2 \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \log \left (-\frac{e \sqrt{x}}{d}\right )-(6 b e n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx,x,\sqrt{x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \log \left (-\frac{e \sqrt{x}}{d}\right )-(6 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \log \left (-\frac{e \sqrt{x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )-\left (12 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \log \left (-\frac{e \sqrt{x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )-12 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \text{Li}_3\left (1+\frac{e \sqrt{x}}{d}\right )+\left (12 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \log \left (-\frac{e \sqrt{x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )-12 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \text{Li}_3\left (1+\frac{e \sqrt{x}}{d}\right )+12 b^3 n^3 \text{Li}_4\left (1+\frac{e \sqrt{x}}{d}\right )\\ \end{align*}
Mathematica [B] time = 0.154776, size = 333, normalized size = 2.47 \[ 6 b^2 n^2 \left (-2 \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right )+2 \log \left (d+e \sqrt{x}\right ) \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )+\log \left (-\frac{e \sqrt{x}}{d}\right ) \log ^2\left (d+e \sqrt{x}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )+3 b n \left (\log (x) \left (\log \left (d+e \sqrt{x}\right )-\log \left (\frac{e \sqrt{x}}{d}+1\right )\right )-2 \text{PolyLog}\left (2,-\frac{e \sqrt{x}}{d}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^2+2 b^3 n^3 \left (6 \text{PolyLog}\left (4,\frac{e \sqrt{x}}{d}+1\right )+3 \log ^2\left (d+e \sqrt{x}\right ) \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )-6 \log \left (d+e \sqrt{x}\right ) \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right )+\log \left (-\frac{e \sqrt{x}}{d}\right ) \log ^3\left (d+e \sqrt{x}\right )\right )+\log (x) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^3 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) ^{3}}\, 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*} b^{3} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{3} \log \left (x\right ) + \int -\frac{3 \,{\left (b^{3} e n x \log \left (x\right ) - 2 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x - 2 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{2} - 2 \,{\left (b^{3} e \log \left (c\right )^{3} + 3 \, a b^{2} e \log \left (c\right )^{2} + 3 \, a^{2} b e \log \left (c\right ) + a^{3} e\right )} x - 6 \,{\left ({\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x +{\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right ) - 2 \,{\left (b^{3} d \log \left (c\right )^{3} + 3 \, a b^{2} d \log \left (c\right )^{2} + 3 \, a^{2} b d \log \left (c\right ) + a^{3} d\right )} \sqrt{x}}{2 \,{\left (e x^{2} + d x^{\frac{3}{2}}\right )}}\,{d x} \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 (\frac{b^{3} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + a^{3}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d + e \sqrt{x}\right )^{n} \right )}\right )^{3}}{x}\, dx \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 \frac{{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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