Optimal. Leaf size=288 \[ -\frac{b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt{x}}}\right )}{d^4}+\frac{b e^4 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}+\frac{b e^3 n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-\frac{5 b^2 e^3 n^2 \sqrt{x}}{6 d^3}+\frac{b^2 e^2 n^2 x}{6 d^2}+\frac{5 b^2 e^4 n^2 \log \left (d+\frac{e}{\sqrt{x}}\right )}{6 d^4}+\frac{11 b^2 e^4 n^2 \log (x)}{12 d^4} \]
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Rubi [A] time = 0.637296, antiderivative size = 311, normalized size of antiderivative = 1.08, number of steps used = 18, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )}{d^4}-\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}+\frac{b e^4 n \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}+\frac{b e^3 n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-\frac{5 b^2 e^3 n^2 \sqrt{x}}{6 d^3}+\frac{b^2 e^2 n^2 x}{6 d^2}+\frac{5 b^2 e^4 n^2 \log \left (d+\frac{e}{\sqrt{x}}\right )}{6 d^4}+\frac{11 b^2 e^4 n^2 \log (x)}{12 d^4} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^5} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^4 (d+e x)} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-(b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-\frac{(b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d}\\ &=\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2+\frac{(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{3 d}\\ &=-\frac{b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^3}+\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^3}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e^3}{d (d-x)^3}-\frac{e^3}{d^2 (d-x)^2}-\frac{e^3}{d^3 (d-x)}-\frac{e^3}{d^3 x}\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{3 d}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}\\ &=-\frac{b^2 e^3 n^2 \sqrt{x}}{3 d^3}+\frac{b^2 e^2 n^2 x}{6 d^2}+\frac{b^2 e^4 n^2 \log \left (d+\frac{e}{\sqrt{x}}\right )}{3 d^4}+\frac{b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2+\frac{b^2 e^4 n^2 \log (x)}{6 d^4}+\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}-\frac{\left (b e^4 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}\\ &=-\frac{5 b^2 e^3 n^2 \sqrt{x}}{6 d^3}+\frac{b^2 e^2 n^2 x}{6 d^2}+\frac{5 b^2 e^4 n^2 \log \left (d+\frac{e}{\sqrt{x}}\right )}{6 d^4}+\frac{b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}-\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2+\frac{b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^4}+\frac{11 b^2 e^4 n^2 \log (x)}{12 d^4}-\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}\\ &=-\frac{5 b^2 e^3 n^2 \sqrt{x}}{6 d^3}+\frac{b^2 e^2 n^2 x}{6 d^2}+\frac{5 b^2 e^4 n^2 \log \left (d+\frac{e}{\sqrt{x}}\right )}{6 d^4}+\frac{b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 d}-\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2+\frac{b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^4}+\frac{11 b^2 e^4 n^2 \log (x)}{12 d^4}+\frac{b^2 e^4 n^2 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.222446, size = 321, normalized size = 1.11 \[ \frac{1}{6} \left (\frac{b e n \left (-6 b e^3 n \text{PolyLog}\left (2,\frac{d \sqrt{x}}{e}+1\right )-3 a d^2 e x+2 a d^3 x^{3/2}+6 a d e^2 \sqrt{x}-6 a e^3 \log \left (d \sqrt{x}+e\right )+2 b d^3 x^{3/2} \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-3 b d^2 e x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+6 b d e^2 \sqrt{x} \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-6 b e^3 \log \left (d \sqrt{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+b d^2 e n x-5 b d e^2 n \sqrt{x}+3 b e^3 n \log ^2\left (d \sqrt{x}+e\right )+8 b e^3 n \log \left (d+\frac{e}{\sqrt{x}}\right )+3 b e^3 n \log \left (d \sqrt{x}+e\right )-6 b e^3 n \log \left (d \sqrt{x}+e\right ) \log \left (-\frac{d \sqrt{x}}{e}\right )+4 b e^3 n \log (x)\right )}{d^4}+3 x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.381, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{2}\, 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{1}{2} \, b^{2} x^{2} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{2} - \int -\frac{2 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{2} + 2 \,{\left (b^{2} d x^{2} + b^{2} e x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} + 2 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{\frac{3}{2}} -{\left (b^{2} d n x^{2} - 4 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2} - 4 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{3}{2}} + 4 \,{\left (b^{2} d x^{2} + b^{2} e x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right ) - 4 \,{\left ({\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2} +{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )}{2 \,{\left (d x + e \sqrt{x}\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 (b^{2} x \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right )^{2} + 2 \, a b x \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right ) + a^{2} x, 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}{\sqrt{x}}\right )}^{n}\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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