3.512 \(\int \frac{a+b \log (c (d+\frac{e}{x^{2/3}})^n)}{x} \, dx\)

Optimal. Leaf size=55 \[ -\frac{3}{2} b n \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )-\frac{3}{2} \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \]

[Out]

(-3*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))])/2 - (3*b*n*PolyLog[2, 1 + e/(d*x^(2/3))])/2

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Rubi [A]  time = 0.0514524, antiderivative size = 55, 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} \[ -\frac{3}{2} b n \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )-\frac{3}{2} \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/x^(2/3))^n])/x,x]

[Out]

(-3*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))])/2 - (3*b*n*PolyLog[2, 1 + e/(d*x^(2/3))])/2

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x} \, dx &=-\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\frac{1}{x^{2/3}}\right )\right )\\ &=-\frac{3}{2} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )+\frac{1}{2} (3 b e n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=-\frac{3}{2} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )-\frac{3}{2} b n \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0114116, size = 55, normalized size = 1. \[ a \log (x)-\frac{3}{2} b \left (n \text{PolyLog}\left (2,\frac{d+\frac{e}{x^{2/3}}}{d}\right )+\log \left (-\frac{e}{d x^{2/3}}\right ) \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])/x,x]

[Out]

a*Log[x] - (3*b*(Log[c*(d + e/x^(2/3))^n]*Log[-(e/(d*x^(2/3)))] + n*PolyLog[2, (d + e/x^(2/3))/d]))/2

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Maple [F]  time = 0.417, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(2/3))^n))/x,x)

[Out]

int((a+b*ln(c*(d+e/x^(2/3))^n))/x,x)

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Maxima [B]  time = 2.41227, size = 171, normalized size = 3.11 \begin{align*} -\frac{3}{2} \,{\left (2 \, \log \left (\frac{d x^{\frac{2}{3}}}{e} + 1\right ) \log \left (x^{\frac{1}{3}}\right ) +{\rm Li}_2\left (-\frac{d x^{\frac{2}{3}}}{e}\right )\right )} b n + \frac{2 \, b e n \log \left (x\right )^{2} + 6 \, b d n x^{\frac{2}{3}} \log \left (x\right ) + 6 \, b e \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right ) \log \left (x\right ) - 12 \, b e \log \left (x\right ) \log \left (x^{\frac{1}{3} \, n}\right ) - 9 \, b d n x^{\frac{2}{3}} + 6 \,{\left (b e \log \left (c\right ) + a e\right )} \log \left (x\right ) - \frac{3 \,{\left (2 \, b d n x \log \left (x\right ) - 3 \, b d n x\right )}}{x^{\frac{1}{3}}}}{6 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))/x,x, algorithm="maxima")

[Out]

-3/2*(2*log(d*x^(2/3)/e + 1)*log(x^(1/3)) + dilog(-d*x^(2/3)/e))*b*n + 1/6*(2*b*e*n*log(x)^2 + 6*b*d*n*x^(2/3)
*log(x) + 6*b*e*log((d*x^(2/3) + e)^n)*log(x) - 12*b*e*log(x)*log(x^(1/3*n)) - 9*b*d*n*x^(2/3) + 6*(b*e*log(c)
+ a*e)*log(x) - 3*(2*b*d*n*x*log(x) - 3*b*d*n*x)/x^(1/3))/e

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))/x,x, algorithm="fricas")

[Out]

integral((b*log(c*((d*x + e*x^(1/3))/x)^n) + a)/x, x)

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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.

[In]

integrate((a+b*ln(c*(d+e/x**(2/3))**n))/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + a}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))/x,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(2/3))^n) + a)/x, x)