3.624 \(\int \frac{a+b \log (c (d+e x^m)^n)}{x} \, dx\)

Optimal. Leaf size=49 \[ \frac{b n \text{PolyLog}\left (2,\frac{e x^m}{d}+1\right )}{m}+\frac{\log \left (-\frac{e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m} \]

[Out]

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

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Rubi [A]  time = 0.0528, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2454, 2394, 2315} \[ \frac{b n \text{PolyLog}\left (2,\frac{e x^m}{d}+1\right )}{m}+\frac{\log \left (-\frac{e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^m)^n])/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0147763, size = 49, normalized size = 1. \[ \frac{b \left (n \text{PolyLog}\left (2,\frac{d+e x^m}{d}\right )+\log \left (-\frac{e x^m}{d}\right ) \log \left (c \left (d+e x^m\right )^n\right )\right )}{m}+a \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 3.542, size = 189, normalized size = 3.9 \begin{align*} b\ln \left ( x \right ) \ln \left ( \left ( d+e{x}^{m} \right ) ^{n} \right ) -{\frac{i}{2}}\ln \left ( x \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( d+e{x}^{m} \right ) ^{n} \right ){\it csgn} \left ( ic \left ( d+e{x}^{m} \right ) ^{n} \right ) +{\frac{i}{2}}\ln \left ( x \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{m} \right ) ^{n} \right ) \right ) ^{2}+{\frac{i}{2}}\ln \left ( x \right ) b\pi \,{\it csgn} \left ( i \left ( d+e{x}^{m} \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{m} \right ) ^{n} \right ) \right ) ^{2}-{\frac{i}{2}}\ln \left ( x \right ) b\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{m} \right ) ^{n} \right ) \right ) ^{3}+\ln \left ( c \right ) \ln \left ( x \right ) b+\ln \left ( x \right ) a-{\frac{bn}{m}{\it dilog} \left ({\frac{d+e{x}^{m}}{d}} \right ) }-bn\ln \left ( x \right ) \ln \left ({\frac{d+e{x}^{m}}{d}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*ln(x)*ln((d+e*x^m)^n)-1/2*I*ln(x)*b*Pi*csgn(I*c)*csgn(I*(d+e*x^m)^n)*csgn(I*c*(d+e*x^m)^n)+1/2*I*ln(x)*b*Pi*
csgn(I*c)*csgn(I*c*(d+e*x^m)^n)^2+1/2*I*ln(x)*b*Pi*csgn(I*(d+e*x^m)^n)*csgn(I*c*(d+e*x^m)^n)^2-1/2*I*ln(x)*b*P
i*csgn(I*c*(d+e*x^m)^n)^3+ln(c)*ln(x)*b+ln(x)*a-b/m*n*dilog((d+e*x^m)/d)-b*n*ln(x)*ln((d+e*x^m)/d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*d*m*n*integrate(log(x)/(e*x*x^m + d*x), x) - m*n*log(x)^2 + 2*log((e*x^m + d)^n)*log(x) + 2*log(c)*log(
x))*b + a*log(x)

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Fricas [A]  time = 1.66569, size = 171, normalized size = 3.49 \begin{align*} \frac{b m n \log \left (e x^{m} + d\right ) \log \left (x\right ) - b m n \log \left (x\right ) \log \left (\frac{e x^{m} + d}{d}\right ) - b n{\rm Li}_2\left (-\frac{e x^{m} + d}{d} + 1\right ) +{\left (b m \log \left (c\right ) + a m\right )} \log \left (x\right )}{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*m*n*log(e*x^m + d)*log(x) - b*m*n*log(x)*log((e*x^m + d)/d) - b*n*dilog(-(e*x^m + d)/d + 1) + (b*m*log(c) +
a*m)*log(x))/m

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*(d + e*x**m)**n))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x^{m} + d\right )}^{n} c\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^m)^n))/x,x, algorithm="giac")

[Out]

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