3.363 \(\int \frac{(f+g x^{-n}) \log (c (d+e x^n)^p)}{x} \, dx\)

Optimal. Leaf size=97 \[ \frac{f p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{f \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac{g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac{e g p \log \left (d+e x^n\right )}{d n}+\frac{e g p \log (x)}{d} \]

[Out]

(e*g*p*Log[x])/d - (e*g*p*Log[d + e*x^n])/(d*n) - (g*Log[c*(d + e*x^n)^p])/(n*x^n) + (f*Log[-((e*x^n)/d)]*Log[
c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n

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Rubi [A]  time = 0.140534, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2475, 14, 2416, 2395, 36, 29, 31, 2394, 2315} \[ \frac{f p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{f \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac{g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac{e g p \log \left (d+e x^n\right )}{d n}+\frac{e g p \log (x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((f + g/x^n)*Log[c*(d + e*x^n)^p])/x,x]

[Out]

(e*g*p*Log[x])/d - (e*g*p*Log[d + e*x^n])/(d*n) - (g*Log[c*(d + e*x^n)^p])/(n*x^n) + (f*Log[-((e*x^n)/d)]*Log[
c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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

Mathematica [A]  time = 0.0934804, size = 87, normalized size = 0.9 \[ \frac{f \left (p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )\right )-g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )+\frac{e g p \left (n \log (x)-\log \left (d+e x^n\right )\right )}{d}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g/x^n)*Log[c*(d + e*x^n)^p])/x,x]

[Out]

((e*g*p*(n*Log[x] - Log[d + e*x^n]))/d - (g*Log[c*(d + e*x^n)^p])/x^n + f*(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)
^p] + p*PolyLog[2, 1 + (e*x^n)/d]))/n

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Maple [C]  time = 4.279, size = 423, normalized size = 4.4 \begin{align*}{\frac{ \left ( f\ln \left ( x \right ) n{x}^{n}-g \right ) \ln \left ( \left ( d+e{x}^{n} \right ) ^{p} \right ) }{{x}^{n}n}}+{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}f\ln \left ({x}^{n} \right ) }{n}}-{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}g}{{x}^{n}n}}-{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \right ) f\ln \left ({x}^{n} \right ) }{n}}+{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \right ) g}{{x}^{n}n}}-{\frac{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}f\ln \left ({x}^{n} \right ) }{n}}+{\frac{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}g}{{x}^{n}n}}+{\frac{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) f\ln \left ({x}^{n} \right ) }{n}}-{\frac{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) g}{{x}^{n}n}}+{\frac{\ln \left ( c \right ) f\ln \left ({x}^{n} \right ) }{n}}-{\frac{g\ln \left ( c \right ) }{{x}^{n}n}}-{\frac{pf}{n}{\it dilog} \left ({\frac{d+e{x}^{n}}{d}} \right ) }-pf\ln \left ( x \right ) \ln \left ({\frac{d+e{x}^{n}}{d}} \right ) -{\frac{egp\ln \left ( d+e{x}^{n} \right ) }{dn}}+{\frac{egp\ln \left ({x}^{n} \right ) }{dn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f+g/(x^n))*ln(c*(d+e*x^n)^p)/x,x)

[Out]

(f*ln(x)*n*x^n-g)/n/(x^n)*ln((d+e*x^n)^p)+1/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*f*ln(x^n)-1/2
*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*g/(x^n)-1/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^
p)*csgn(I*c)*f*ln(x^n)+1/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)*g/(x^n)-1/2*I/n*Pi*csgn(
I*c*(d+e*x^n)^p)^3*f*ln(x^n)+1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^3*g/(x^n)+1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csg
n(I*c)*f*ln(x^n)-1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)*g/(x^n)+1/n*ln(c)*f*ln(x^n)-1/n*ln(c)*g/(x^n)-p/
n*f*dilog((d+e*x^n)/d)-p*f*ln(x)*ln((d+e*x^n)/d)-e*g*p*ln(d+e*x^n)/d/n+p*e/n*g/d*ln(x^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f+g/(x^n))*log(c*(d+e*x^n)^p)/x,x, algorithm="maxima")

[Out]

-1/2*((f*n^2*p*log(x)^2 - 2*f*n*log(c)*log(x))*x^n - 2*(f*n*x^n*log(x) - g)*log((e*x^n + d)^p) + 2*g*log(c))/(
n*x^n) + integrate((d*f*n*p*log(x) + e*g*p)/(e*x*x^n + d*x), x)

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Fricas [A]  time = 2.1328, size = 266, normalized size = 2.74 \begin{align*} -\frac{d f n p x^{n} \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) + d f p x^{n}{\rm Li}_2\left (-\frac{e x^{n} + d}{d} + 1\right ) + d g \log \left (c\right ) -{\left (e g n p + d f n \log \left (c\right )\right )} x^{n} \log \left (x\right ) +{\left (d g p -{\left (d f n p \log \left (x\right ) - e g p\right )} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f+g/(x^n))*log(c*(d+e*x^n)^p)/x,x, algorithm="fricas")

[Out]

-(d*f*n*p*x^n*log(x)*log((e*x^n + d)/d) + d*f*p*x^n*dilog(-(e*x^n + d)/d + 1) + d*g*log(c) - (e*g*n*p + d*f*n*
log(c))*x^n*log(x) + (d*g*p - (d*f*n*p*log(x) - e*g*p)*x^n)*log(e*x^n + d))/(d*n*x^n)

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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((f+g/(x**n))*ln(c*(d+e*x**n)**p)/x,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f+g/(x^n))*log(c*(d+e*x^n)^p)/x,x, algorithm="giac")

[Out]

integrate((f + g/x^n)*log((e*x^n + d)^p*c)/x, x)