Optimal. Leaf size=293 \[ \frac{b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt{x}}\right )}{d^4}-\frac{b e^4 n \log \left (1-\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4}-\frac{b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4 \sqrt{x}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^2}+\frac{5 b^2 e^3 n^2}{6 d^3 \sqrt{x}}-\frac{b^2 e^2 n^2}{6 d^2 x}-\frac{5 b^2 e^4 n^2 \log \left (d+e \sqrt{x}\right )}{6 d^4}+\frac{11 b^2 e^4 n^2 \log (x)}{12 d^4} \]
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Rubi [A] time = 0.667209, antiderivative size = 318, normalized size of antiderivative = 1.09, number of steps used = 18, 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, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{d^4}+\frac{e^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4}-\frac{b e^4 n \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4}-\frac{b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4 \sqrt{x}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^2}+\frac{5 b^2 e^3 n^2}{6 d^3 \sqrt{x}}-\frac{b^2 e^2 n^2}{6 d^2 x}-\frac{5 b^2 e^4 n^2 \log \left (d+e \sqrt{x}\right )}{6 d^4}+\frac{11 b^2 e^4 n^2 \log (x)}{12 d^4} \]
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 \frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^5} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^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,\sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^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+e \sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^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+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+e \sqrt{x}\right )}{d}\\ &=-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^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+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+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+e \sqrt{x}\right )}{3 d}\\ &=-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^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+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+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+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+e \sqrt{x}\right )}{2 d^2}\\ &=-\frac{b^2 e^2 n^2}{6 d^2 x}+\frac{b^2 e^3 n^2}{3 d^3 \sqrt{x}}-\frac{b^2 e^4 n^2 \log \left (d+e \sqrt{x}\right )}{3 d^4}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^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+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+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+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+e \sqrt{x}\right )}{d^4}\\ &=-\frac{b^2 e^2 n^2}{6 d^2 x}+\frac{5 b^2 e^3 n^2}{6 d^3 \sqrt{x}}-\frac{5 b^2 e^4 n^2 \log \left (d+e \sqrt{x}\right )}{6 d^4}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4 \sqrt{x}}+\frac{e^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^2}-\frac{b e^4 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\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+e \sqrt{x}\right )}{d^4}\\ &=-\frac{b^2 e^2 n^2}{6 d^2 x}+\frac{5 b^2 e^3 n^2}{6 d^3 \sqrt{x}}-\frac{5 b^2 e^4 n^2 \log \left (d+e \sqrt{x}\right )}{6 d^4}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4 \sqrt{x}}+\frac{e^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x^2}-\frac{b e^4 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\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 \sqrt{x}}{d}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.32548, size = 353, normalized size = 1.2 \[ -\frac{\frac{e \sqrt{x} \left (12 b^2 e^3 n^2 x^{3/2} \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )+4 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-6 b d^2 e n \sqrt{x} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-6 e^3 x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+12 b e^3 n x^{3/2} \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+12 b d e^2 n x \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+6 b^2 e^3 n^2 x^{3/2} \left (2 \log \left (d+e \sqrt{x}\right )-\log (x)\right )-3 b^2 e^2 n^2 x \left (-2 e \sqrt{x} \log \left (d+e \sqrt{x}\right )+2 d+e \sqrt{x} \log (x)\right )+2 b^2 e n^2 \sqrt{x} \left (2 e^2 x \log \left (d+e \sqrt{x}\right )+d \left (d-2 e \sqrt{x}\right )-e^2 x \log (x)\right )\right )}{d^4}+6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{12 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+e\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{b^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{2}}{2 \, x^{2}} + \int \frac{2 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x +{\left (b^{2} e n x + 4 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x + 4 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right ) + 2 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} \sqrt{x}}{2 \,{\left (e x^{4} + d x^{\frac{7}{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^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + a^{2}}{x^{3}}, 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 \frac{{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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