Optimal. Leaf size=51 \[ 3 b n \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )+3 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0505669, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2394, 2315} \[ 3 b n \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )+3 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx &=3 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )-(3 b e n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )+3 b n \text{Li}_2\left (1+\frac{e \sqrt [3]{x}}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.0029209, size = 53, normalized size = 1.04 \[ 3 b n \text{PolyLog}\left (2,\frac{d+e \sqrt [3]{x}}{d}\right )+a \log (x)+3 b \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.59889, size = 224, normalized size = 4.39 \begin{align*} -3 \,{\left (\log \left (\frac{e x^{\frac{1}{3}}}{d} + 1\right ) \log \left (x^{\frac{1}{3}}\right ) +{\rm Li}_2\left (-\frac{e x^{\frac{1}{3}}}{d}\right )\right )} b n + \frac{4 \, b d^{2} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right ) \log \left (x\right ) + 4 \,{\left (b d^{2} \log \left (c\right ) + a d^{2}\right )} \log \left (x\right ) + \frac{2 \, b e^{2} n x \log \left (x\right ) - 3 \, b e^{2} n x}{x^{\frac{1}{3}}} - \frac{4 \,{\left (b d e n x \log \left (x\right ) - 3 \, b d e n x\right )}}{x^{\frac{2}{3}}}}{4 \, d^{2}} + \frac{3 \,{\left (b e^{2} n x^{\frac{2}{3}} - 4 \, b d e n x^{\frac{1}{3}} - 2 \,{\left (b e^{2} n x^{\frac{2}{3}} - 2 \, b d e n x^{\frac{1}{3}}\right )} \log \left (x^{\frac{1}{3}}\right )\right )}}{4 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]