### 3.65 $$\int \frac{\log (d (b x+c x^2)^n)}{x} \, dx$$

Optimal. Leaf size=53 $-n \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \log (x) \log \left (\frac{c x}{b}+1\right )-\frac{1}{2} n \log ^2(x)$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.125706, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {2524, 1593, 2357, 2301, 2317, 2391} $-n \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \log (x) \log \left (\frac{c x}{b}+1\right )-\frac{1}{2} n \log ^2(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[d*(b*x + c*x^2)^n]/x,x]

[Out]

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

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2391

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

Rubi steps

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

Mathematica [A]  time = 0.0203252, size = 50, normalized size = 0.94 $\log (x) \log \left (d (x (b+c x))^n\right )-n \left (\text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (\frac{b+c x}{b}\right )+\frac{\log ^2(x)}{2}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[d*(b*x + c*x^2)^n]/x,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) }{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0547, size = 108, normalized size = 2.04 \begin{align*} -n \log \left (c x^{2} + b x\right ) \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, \log \left (c x^{2} + b x\right ) \log \left (x\right ) - 2 \, \log \left (\frac{c x}{b} + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \,{\rm Li}_2\left (-\frac{c x}{b}\right )\right )} n + \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-n*log(c*x^2 + b*x)*log(x) + 1/2*(2*log(c*x^2 + b*x)*log(x) - 2*log(c*x/b + 1)*log(x) - log(x)^2 - 2*dilog(-c*
x/b))*n + log((c*x^2 + b*x)^n*d)*log(x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((c*x^2 + b*x)^n*d)/x, x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(d*(b*x + c*x**2)**n)/x, x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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