Optimal. Leaf size=39 \[ 4 b n \text{PolyLog}\left (3,-d f \sqrt{x}\right )-2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A] time = 0.0321699, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2374, 6589} \[ 4 b n \text{PolyLog}\left (3,-d f \sqrt{x}\right )-2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 6589
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} \, dx &=-2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+(2 b n) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+4 b n \text{Li}_3\left (-d f \sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0071839, size = 50, normalized size = 1.28 \[ -2 a \text{PolyLog}\left (2,-d f \sqrt{x}\right )-2 b \log \left (c x^n\right ) \text{PolyLog}\left (2,-d f \sqrt{x}\right )+4 b n \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x}\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}\,{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}, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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