Optimal. Leaf size=372 \[ \frac{2}{3} b d^6 f^6 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+\frac{1}{3} d^6 f^6 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{b d^3 f^3 n}{9 x^{3/2}}+\frac{5 b d^2 f^2 n}{72 x^2}-\frac{7 b d^5 f^5 n}{9 \sqrt{x}}+\frac{2 b d^4 f^4 n}{9 x}+\frac{1}{12} b d^6 f^6 n \log ^2(x)+\frac{1}{9} b d^6 f^6 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{18} b d^6 f^6 n \log (x)-\frac{11 b d f n}{225 x^{5/2}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{9 x^3} \]
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Rubi [A] time = 0.251879, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2454, 2395, 44, 2376, 2391, 2301} \[ \frac{2}{3} b d^6 f^6 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+\frac{1}{3} d^6 f^6 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{b d^3 f^3 n}{9 x^{3/2}}+\frac{5 b d^2 f^2 n}{72 x^2}-\frac{7 b d^5 f^5 n}{9 \sqrt{x}}+\frac{2 b d^4 f^4 n}{9 x}+\frac{1}{12} b d^6 f^6 n \log ^2(x)+\frac{1}{9} b d^6 f^6 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{18} b d^6 f^6 n \log (x)-\frac{11 b d f n}{225 x^{5/2}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{9 x^3} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 44
Rule 2376
Rule 2391
Rule 2301
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 )}{x^4} \, dx &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{1}{3} d^6 f^6 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d f}{15 x^{7/2}}+\frac{d^2 f^2}{12 x^3}-\frac{d^3 f^3}{9 x^{5/2}}+\frac{d^4 f^4}{6 x^2}-\frac{d^5 f^5}{3 x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right )}{3 x^4}+\frac{d^6 f^6 \log \left (1+d f \sqrt{x}\right )}{3 x}-\frac{d^6 f^6 \log (x)}{6 x}\right ) \, dx\\ &=-\frac{2 b d f n}{75 x^{5/2}}+\frac{b d^2 f^2 n}{24 x^2}-\frac{2 b d^3 f^3 n}{27 x^{3/2}}+\frac{b d^4 f^4 n}{6 x}-\frac{2 b d^5 f^5 n}{3 \sqrt{x}}-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{1}{3} d^6 f^6 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} (b n) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x^4} \, dx+\frac{1}{6} \left (b d^6 f^6 n\right ) \int \frac{\log (x)}{x} \, dx-\frac{1}{3} \left (b d^6 f^6 n\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{2 b d f n}{75 x^{5/2}}+\frac{b d^2 f^2 n}{24 x^2}-\frac{2 b d^3 f^3 n}{27 x^{3/2}}+\frac{b d^4 f^4 n}{6 x}-\frac{2 b d^5 f^5 n}{3 \sqrt{x}}+\frac{1}{12} b d^6 f^6 n \log ^2(x)-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{1}{3} d^6 f^6 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} b d^6 f^6 n \text{Li}_2\left (-d f \sqrt{x}\right )+\frac{1}{3} (2 b n) \operatorname{Subst}\left (\int \frac{\log (1+d f x)}{x^7} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b d f n}{75 x^{5/2}}+\frac{b d^2 f^2 n}{24 x^2}-\frac{2 b d^3 f^3 n}{27 x^{3/2}}+\frac{b d^4 f^4 n}{6 x}-\frac{2 b d^5 f^5 n}{3 \sqrt{x}}-\frac{b n \log \left (1+d f \sqrt{x}\right )}{9 x^3}+\frac{1}{12} b d^6 f^6 n \log ^2(x)-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{1}{3} d^6 f^6 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} b d^6 f^6 n \text{Li}_2\left (-d f \sqrt{x}\right )+\frac{1}{9} (b d f n) \operatorname{Subst}\left (\int \frac{1}{x^6 (1+d f x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b d f n}{75 x^{5/2}}+\frac{b d^2 f^2 n}{24 x^2}-\frac{2 b d^3 f^3 n}{27 x^{3/2}}+\frac{b d^4 f^4 n}{6 x}-\frac{2 b d^5 f^5 n}{3 \sqrt{x}}-\frac{b n \log \left (1+d f \sqrt{x}\right )}{9 x^3}+\frac{1}{12} b d^6 f^6 n \log ^2(x)-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{1}{3} d^6 f^6 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} b d^6 f^6 n \text{Li}_2\left (-d f \sqrt{x}\right )+\frac{1}{9} (b d f n) \operatorname{Subst}\left (\int \left (\frac{1}{x^6}-\frac{d f}{x^5}+\frac{d^2 f^2}{x^4}-\frac{d^3 f^3}{x^3}+\frac{d^4 f^4}{x^2}-\frac{d^5 f^5}{x}+\frac{d^6 f^6}{1+d f x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{11 b d f n}{225 x^{5/2}}+\frac{5 b d^2 f^2 n}{72 x^2}-\frac{b d^3 f^3 n}{9 x^{3/2}}+\frac{2 b d^4 f^4 n}{9 x}-\frac{7 b d^5 f^5 n}{9 \sqrt{x}}+\frac{1}{9} b d^6 f^6 n \log \left (1+d f \sqrt{x}\right )-\frac{b n \log \left (1+d f \sqrt{x}\right )}{9 x^3}-\frac{1}{18} b d^6 f^6 n \log (x)+\frac{1}{12} b d^6 f^6 n \log ^2(x)-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac{d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt{x}}+\frac{1}{3} d^6 f^6 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} b d^6 f^6 n \text{Li}_2\left (-d f \sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.361322, size = 288, normalized size = 0.77 \[ \frac{2}{3} b d^6 f^6 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )-\frac{d f \left (100 d^5 f^5 x^{5/2} \log (x) \left (3 a+3 b \log \left (c x^n\right )+b n\right )+600 a d^4 f^4 x^2-300 a d^3 f^3 x^{3/2}+200 a d^2 f^2 x-150 a d f \sqrt{x}+120 a+10 b \left (60 d^4 f^4 x^2-30 d^3 f^3 x^{3/2}+20 d^2 f^2 x-15 d f \sqrt{x}+12\right ) \log \left (c x^n\right )+1400 b d^4 f^4 n x^2-400 b d^3 f^3 n x^{3/2}-150 b d^5 f^5 n x^{5/2} \log ^2(x)+200 b d^2 f^2 n x-125 b d f n \sqrt{x}+88 b n\right )}{1800 x^{5/2}}+\frac{\left (d^6 f^6 x^3-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{9 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{4}}\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 )} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{4}}\,{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 \log \left (c x^{n}\right ) + a\right )} \log \left (d f \sqrt{x} + 1\right )}{x^{4}}, 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 )} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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