### 3.267 $$\int \frac{x \log (c+d x)}{a+b x} \, dx$$

Optimal. Leaf size=81 $-\frac{a \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2}-\frac{a \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{x}{b}$

[Out]

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

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Rubi [A]  time = 0.0994792, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.467, Rules used = {43, 2416, 2389, 2295, 2394, 2393, 2391} $-\frac{a \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2}-\frac{a \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{x}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x*Log[c + d*x])/(a + b*x),x]

[Out]

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

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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{x \log (c+d x)}{a+b x} \, dx &=\int \left (\frac{\log (c+d x)}{b}-\frac{a \log (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac{\int \log (c+d x) \, dx}{b}-\frac{a \int \frac{\log (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac{a \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,c+d x)}{b d}+\frac{(a d) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2}\\ &=-\frac{x}{b}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{a \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}+\frac{a \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2}\\ &=-\frac{x}{b}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{a \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}-\frac{a \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.0291286, size = 73, normalized size = 0.9 $\frac{-a d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (-a d \log \left (\frac{d (a+b x)}{a d-b c}\right )+b c+b d x\right )-b d x}{b^2 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x*Log[c + d*x])/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.026, size = 114, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) x}{b}}+{\frac{\ln \left ( dx+c \right ) c}{db}}-{\frac{x}{b}}-{\frac{c}{db}}-{\frac{a}{{b}^{2}}{\it dilog} \left ({\frac{ \left ( dx+c \right ) b+ad-bc}{ad-bc}} \right ) }-{\frac{a\ln \left ( dx+c \right ) }{{b}^{2}}\ln \left ({\frac{ \left ( dx+c \right ) b+ad-bc}{ad-bc}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(d*x+c)/(b*x+a),x)

[Out]

1/b*ln(d*x+c)*x+1/d/b*ln(d*x+c)*c-x/b-1/d/b*c-a/b^2*dilog(((d*x+c)*b+a*d-b*c)/(a*d-b*c))-a/b^2*ln(d*x+c)*ln(((
d*x+c)*b+a*d-b*c)/(a*d-b*c))

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Maxima [A]  time = 1.01102, size = 150, normalized size = 1.85 \begin{align*} d{\left (\frac{{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )} a}{b^{2} d} - \frac{x}{b d} + \frac{c \log \left (d x + c\right )}{b d^{2}}\right )} +{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} \log \left (d x + c\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*log(d*x + c)/(b*x + a), 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(x*ln(d*x+c)/(b*x+a),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*log(d*x + c)/(b*x + a), x)