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

Optimal. Leaf size=83 \[ \frac{f p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{f \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}-\frac{g p x^n}{n} \]

[Out]

-((g*p*x^n)/n) + (g*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e*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 = 0.10986, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2475, 43, 2416, 2389, 2295, 2394, 2315} \[ \frac{f p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{f \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}-\frac{g p x^n}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((g*p*x^n)/n) + (g*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, 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{(f+g x) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (g \log \left (c (d+e x)^p\right )+\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 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\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 \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e 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{g p x^n}{n}+\frac{g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e 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 = 0.0577596, size = 68, normalized size = 0.82 \[ \frac{e f p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+\log \left (c \left (d+e x^n\right )^p\right ) \left (e f \log \left (-\frac{e x^n}{d}\right )+d g+e g x^n\right )-e g p x^n}{e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 4.306, size = 376, normalized size = 4.5 \begin{align*}{\frac{ \left ( f\ln \left ( x \right ) n+g{x}^{n} \right ) \ln \left ( \left ( d+e{x}^{n} \right ) ^{p} \right ) }{n}}+{\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 \right ) +{\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{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 \right ) -{\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{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}f\ln \left ( x \right ) -{\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{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 \right ) +{\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}}+\ln \left ( c \right ) f\ln \left ( x \right ) +{\frac{g\ln \left ( c \right ){x}^{n}}{n}}-{\frac{gp{x}^{n}}{n}}+{\frac{dgp\ln \left ( d+e{x}^{n} \right ) }{en}}-{\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 ) \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+g*x^n)/n*ln((d+e*x^n)^p)+1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*f*ln(x)+1/2*I*Pi*csgn
(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*g*x^n/n-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)*f
*ln(x)-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)*g*x^n/n-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3*f
*ln(x)-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3*g*x^n/n+1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)*f*ln(x)+1/2*I*Pi*cs
gn(I*c*(d+e*x^n)^p)^2*csgn(I*c)*g*x^n/n+ln(c)*f*ln(x)+ln(c)*g*x^n/n-g*p*x^n/n+p/e/n*g*d*ln(d+e*x^n)-p/n*f*dilo
g((d+e*x^n)/d)-p*f*ln(x)*ln((d+e*x^n)/d)

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

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Fricas [A]  time = 2.10273, size = 244, normalized size = 2.94 \begin{align*} -\frac{e f n p \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) - e f n \log \left (c\right ) \log \left (x\right ) + e f p{\rm Li}_2\left (-\frac{e x^{n} + d}{d} + 1\right ) +{\left (e g p - e g \log \left (c\right )\right )} x^{n} -{\left (e f n p \log \left (x\right ) + e g p x^{n} + d g p\right )} \log \left (e x^{n} + d\right )}{e 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]

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

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

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

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