Optimal. Leaf size=263 \[ \frac{6 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (3,\frac{d}{d+e \sqrt{x}}\right )}{d^2}+\frac{6 b^2 e^2 n^2 \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}-\frac{3 b e^2 n \log \left (1-\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2}-\frac{3 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x} \]
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Rubi [A] time = 0.594744, antiderivative size = 283, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac{6 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right )}{d^2}+\frac{6 b^2 e^2 n^2 \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}-\frac{3 b e^2 n \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2}+\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{d^2}-\frac{3 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2302
Rule 30
Rule 2317
Rule 2374
Rule 6589
Rule 2318
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}+(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 (d+e x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}+(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}+\frac{(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt{x}\right )}{d}-\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt{x}\right )}{d}\\ &=-\frac{3 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}-\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt{x}\right )}{d^2}+\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e \sqrt{x}\right )}{d^2}+\frac{\left (6 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt{x}\right )}{d^2}\\ &=-\frac{3 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}+\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}-\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}+\frac{\left (6 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{d^2}-\frac{\left (6 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{d^2}\\ &=-\frac{3 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2 \sqrt{x}}+\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{d^2}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}+\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}-\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )}{d^2}-\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{\left (6 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{d^2}\\ &=-\frac{3 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2 \sqrt{x}}+\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{d^2}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{x}+\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}-\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )}{d^2}-\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{6 b^3 e^2 n^3 \text{Li}_3\left (1+\frac{e \sqrt{x}}{d}\right )}{d^2}\\ \end{align*}
Mathematica [B] time = 0.712587, size = 536, normalized size = 2.04 \[ \frac{3 b^2 n^2 \left (-2 e^2 x \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )-2 e^2 x \left (\log \left (d+e \sqrt{x}\right )-1\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )+\left (d+e \sqrt{x}\right ) \log \left (d+e \sqrt{x}\right ) \left (\left (e \sqrt{x}-d\right ) \log \left (d+e \sqrt{x}\right )-2 e \sqrt{x}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )+b^3 n^3 \left (6 e^2 x \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right )-6 e^2 x \left (\log \left (d+e \sqrt{x}\right )-1\right ) \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )-3 e^2 x \left (\log \left (d+e \sqrt{x}\right )-2\right ) \log \left (d+e \sqrt{x}\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )+\left (d+e \sqrt{x}\right ) \log ^2\left (d+e \sqrt{x}\right ) \left (\left (e \sqrt{x}-d\right ) \log \left (d+e \sqrt{x}\right )-3 e \sqrt{x}\right )\right )-3 b d^2 n \log \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^2-d^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^3+3 b e^2 n x \log \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^2-\frac{3}{2} b e^2 n x \log (x) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^2-3 b d e n \sqrt{x} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )-b n \log \left (d+e \sqrt{x}\right )\right )^2}{d^2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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*} -\frac{2 \, b^{3} d^{2} \sqrt{x} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{3} - 3 \,{\left (2 \, b^{3} e^{2} n x^{\frac{3}{2}} \log \left (e \sqrt{x} + d\right ) - 2 \, b^{3} d e n x -{\left (b^{3} e^{2} n x \log \left (x\right ) + 2 \, b^{3} d^{2} \log \left (c\right ) + 2 \, a b^{2} d^{2}\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{2}}{2 \, d^{2} x^{\frac{3}{2}}} - \int -\frac{2 \,{\left (b^{3} d^{2} e \log \left (c\right )^{3} + 3 \, a b^{2} d^{2} e \log \left (c\right )^{2} + 3 \, a^{2} b d^{2} e \log \left (c\right ) + a^{3} d^{2} e\right )} x^{\frac{3}{2}} + 2 \,{\left (b^{3} d^{3} \log \left (c\right )^{3} + 3 \, a b^{2} d^{3} \log \left (c\right )^{2} + 3 \, a^{2} b d^{3} \log \left (c\right ) + a^{3} d^{3}\right )} x - 3 \,{\left (2 \, b^{3} e^{3} n^{2} x^{\frac{5}{2}} \log \left (e \sqrt{x} + d\right ) - 2 \, b^{3} d e^{2} n^{2} x^{2} - 2 \,{\left (b^{3} d^{2} e \log \left (c\right )^{2} + 2 \, a b^{2} d^{2} e \log \left (c\right ) + a^{2} b d^{2} e\right )} x^{\frac{3}{2}} - 2 \,{\left (b^{3} d^{3} \log \left (c\right )^{2} + 2 \, a b^{2} d^{3} \log \left (c\right ) + a^{2} b d^{3}\right )} x -{\left (b^{3} e^{3} n^{2} x^{2} \log \left (x\right ) + 2 \,{\left (b^{3} d^{2} e n \log \left (c\right ) + a b^{2} d^{2} e n\right )} x\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )}{2 \,{\left (d^{2} e x^{\frac{7}{2}} + d^{3} x^{3}\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^{2}}, 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^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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