Optimal. Leaf size=389 \[ 4 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b^2 d^2 f^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b d^2 f^2 n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}-\frac{2 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{1}{2} b^2 d^2 f^2 n^2 \log ^2(x)+2 b^2 d^2 f^2 n^2 \log \left (d f \sqrt{x}+1\right )-b^2 d^2 f^2 n^2 \log (x)-\frac{14 b^2 d f n^2}{\sqrt{x}}-\frac{2 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{x} \]
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Rubi [A] time = 0.408889, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2454, 2395, 44, 2377, 2304, 2376, 2391, 2301, 2374, 6589, 2366, 12, 2302, 30} \[ 4 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b^2 d^2 f^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b d^2 f^2 n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}-\frac{2 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+\frac{1}{2} b^2 d^2 f^2 n^2 \log ^2(x)+2 b^2 d^2 f^2 n^2 \log \left (d f \sqrt{x}+1\right )-b^2 d^2 f^2 n^2 \log (x)-\frac{14 b^2 d f n^2}{\sqrt{x}}-\frac{2 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 44
Rule 2377
Rule 2304
Rule 2376
Rule 2391
Rule 2301
Rule 2374
Rule 6589
Rule 2366
Rule 12
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac{d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 x}\right ) \, dx\\ &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2+(2 b n) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+(2 b d f n) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\left (b d^2 f^2 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (2 b d^2 f^2 n\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=-\frac{8 b^2 d f n^2}{\sqrt{x}}-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-\left (b d^2 f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (2 b^2 n^2\right ) \int \left (-\frac{d f}{x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right )}{x^2}+\frac{d^2 f^2 \log \left (1+d f \sqrt{x}\right )}{x}-\frac{d^2 f^2 \log (x)}{2 x}\right ) \, dx-\left (4 b^2 d^2 f^2 n^2\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{12 b^2 d f n^2}{\sqrt{x}}-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )-\frac{1}{2} \left (d^2 f^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (2 b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x^2} \, dx+\left (b^2 d^2 f^2 n^2\right ) \int \frac{\log (x)}{x} \, dx-\left (2 b^2 d^2 f^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{12 b^2 d f n^2}{\sqrt{x}}+\frac{1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b^2 d^2 f^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )-\frac{\left (d^2 f^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}+\left (4 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+d f x)}{x^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{12 b^2 d f n^2}{\sqrt{x}}-\frac{2 b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{x}+\frac{1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )+\left (2 b^2 d f n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+d f x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{12 b^2 d f n^2}{\sqrt{x}}-\frac{2 b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{x}+\frac{1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )+\left (2 b^2 d f n^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{d f}{x}+\frac{d^2 f^2}{1+d f x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{14 b^2 d f n^2}{\sqrt{x}}+2 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt{x}\right )-\frac{2 b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{x}-b^2 d^2 f^2 n^2 \log (x)+\frac{1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac{6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}}+d^2 f^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-8 b^2 d^2 f^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.400513, size = 627, normalized size = 1.61 \[ -\frac{-24 b d^2 f^2 n x \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )+b n\right )+48 b^2 d^2 f^2 n^2 x \text{PolyLog}\left (3,-d f \sqrt{x}\right )+3 a^2 d^2 f^2 x \log (x)-6 a^2 d^2 f^2 x \log \left (d f \sqrt{x}+1\right )+6 a^2 d f \sqrt{x}+6 a^2 \log \left (d f \sqrt{x}+1\right )+6 a b d^2 f^2 x \log (x) \log \left (c x^n\right )-12 a b d^2 f^2 x \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+12 a b \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+12 a b d f \sqrt{x} \log \left (c x^n\right )-3 a b d^2 f^2 n x \log ^2(x)+6 a b d^2 f^2 n x \log (x)-12 a b d^2 f^2 n x \log \left (d f \sqrt{x}+1\right )+36 a b d f n \sqrt{x}+12 a b n \log \left (d f \sqrt{x}+1\right )-3 b^2 d^2 f^2 n x \log ^2(x) \log \left (c x^n\right )+3 b^2 d^2 f^2 x \log (x) \log ^2\left (c x^n\right )-6 b^2 d^2 f^2 x \log ^2\left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+6 b^2 d^2 f^2 n x \log (x) \log \left (c x^n\right )-12 b^2 d^2 f^2 n x \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+6 b^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+6 b^2 d f \sqrt{x} \log ^2\left (c x^n\right )+12 b^2 n \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+36 b^2 d f n \sqrt{x} \log \left (c x^n\right )+b^2 d^2 f^2 n^2 x \log ^3(x)-3 b^2 d^2 f^2 n^2 x \log ^2(x)+6 b^2 d^2 f^2 n^2 x \log (x)-12 b^2 d^2 f^2 n^2 x \log \left (d f \sqrt{x}+1\right )+84 b^2 d f n^2 \sqrt{x}+12 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{6 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{2}}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) }\, 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*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{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{{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left (d f \sqrt{x} + 1\right )}{x^{2}}, 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 (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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