3.391 \(\int \frac{(a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m))}{x^2} \, dx\)

Optimal. Leaf size=270 \[ -\frac{b g j m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{i}+\frac{b g j m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{i}-\frac{b e g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{d}+\frac{b e g m n \text{PolyLog}\left (2,\frac{j x}{i}+1\right )}{d}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}+\frac{g j m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}-\frac{g j m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}+\frac{b e n \log \left (-\frac{j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac{b e n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d} \]

[Out]

(g*j*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/i - (g*j*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*
i - d*j)])/i + (b*e*n*Log[-((j*x)/i)]*(f + g*Log[h*(i + j*x)^m]))/d - (b*e*n*Log[-((j*(d + e*x))/(e*i - d*j))]
*(f + g*Log[h*(i + j*x)^m]))/d - ((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x - (b*g*j*m*n*PolyLo
g[2, -((j*(d + e*x))/(e*i - d*j))])/i + (b*g*j*m*n*PolyLog[2, 1 + (e*x)/d])/i - (b*e*g*m*n*PolyLog[2, (e*(i +
j*x))/(e*i - d*j)])/d + (b*e*g*m*n*PolyLog[2, 1 + (j*x)/i])/d

________________________________________________________________________________________

Rubi [A]  time = 0.332609, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {2439, 36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac{b g j m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{i}+\frac{b g j m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{i}-\frac{b e g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{d}+\frac{b e g m n \text{PolyLog}\left (2,\frac{j x}{i}+1\right )}{d}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}+\frac{g j m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}-\frac{g j m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}+\frac{b e n \log \left (-\frac{j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac{b e n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x^2,x]

[Out]

(g*j*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/i - (g*j*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*
i - d*j)])/i + (b*e*n*Log[-((j*x)/i)]*(f + g*Log[h*(i + j*x)^m]))/d - (b*e*n*Log[-((j*(d + e*x))/(e*i - d*j))]
*(f + g*Log[h*(i + j*x)^m]))/d - ((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x - (b*g*j*m*n*PolyLo
g[2, -((j*(d + e*x))/(e*i - d*j))])/i + (b*g*j*m*n*PolyLog[2, 1 + (e*x)/d])/i - (b*e*g*m*n*PolyLog[2, (e*(i +
j*x))/(e*i - d*j)])/d + (b*e*g*m*n*PolyLog[2, 1 + (j*x)/i])/d

Rule 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*(x_)^(r_.), x_Symbol] :> Simp[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]))/(r +
1), x] + (-Dist[(g*j*m)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[(b*e*n*
p)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /
; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && N
eQ[r, -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 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 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]

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{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x^2} \, dx &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x}+(g j m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x (391+j x)} \, dx+(b e n) \int \frac{f+g \log \left (h (391+j x)^m\right )}{x (d+e x)} \, dx\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x}+(g j m) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{391 x}-\frac{j \left (a+b \log \left (c (d+e x)^n\right )\right )}{391 (391+j x)}\right ) \, dx+(b e n) \int \left (\frac{f+g \log \left (h (391+j x)^m\right )}{d x}-\frac{e \left (f+g \log \left (h (391+j x)^m\right )\right )}{d (d+e x)}\right ) \, dx\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x}+\frac{1}{391} (g j m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx-\frac{1}{391} \left (g j^2 m\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{391+j x} \, dx+\frac{(b e n) \int \frac{f+g \log \left (h (391+j x)^m\right )}{x} \, dx}{d}-\frac{\left (b e^2 n\right ) \int \frac{f+g \log \left (h (391+j x)^m\right )}{d+e x} \, dx}{d}\\ &=\frac{1}{391} g j m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{391} g j m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (391+j x)}{391 e-d j}\right )+\frac{b e n \log \left (-\frac{j x}{391}\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{d}-\frac{b e n \log \left (-\frac{j (d+e x)}{391 e-d j}\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{d}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x}-\frac{1}{391} (b e g j m n) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx+\frac{1}{391} (b e g j m n) \int \frac{\log \left (\frac{e (391+j x)}{391 e-d j}\right )}{d+e x} \, dx-\frac{(b e g j m n) \int \frac{\log \left (-\frac{j x}{391}\right )}{391+j x} \, dx}{d}+\frac{(b e g j m n) \int \frac{\log \left (\frac{j (d+e x)}{-391 e+d j}\right )}{391+j x} \, dx}{d}\\ &=\frac{1}{391} g j m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{391} g j m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (391+j x)}{391 e-d j}\right )+\frac{b e n \log \left (-\frac{j x}{391}\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{d}-\frac{b e n \log \left (-\frac{j (d+e x)}{391 e-d j}\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{d}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x}+\frac{1}{391} b g j m n \text{Li}_2\left (1+\frac{e x}{d}\right )+\frac{b e g m n \text{Li}_2\left (1+\frac{j x}{391}\right )}{d}+\frac{(b e g m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-391 e+d j}\right )}{x} \, dx,x,391+j x\right )}{d}+\frac{1}{391} (b g j m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{391 e-d j}\right )}{x} \, dx,x,d+e x\right )\\ &=\frac{1}{391} g j m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{1}{391} g j m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (391+j x)}{391 e-d j}\right )+\frac{b e n \log \left (-\frac{j x}{391}\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{d}-\frac{b e n \log \left (-\frac{j (d+e x)}{391 e-d j}\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{d}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (391+j x)^m\right )\right )}{x}-\frac{1}{391} b g j m n \text{Li}_2\left (-\frac{j (d+e x)}{391 e-d j}\right )+\frac{1}{391} b g j m n \text{Li}_2\left (1+\frac{e x}{d}\right )+\frac{b e g m n \text{Li}_2\left (1+\frac{j x}{391}\right )}{d}-\frac{b e g m n \text{Li}_2\left (\frac{e (391+j x)}{391 e-d j}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.230228, size = 476, normalized size = 1.76 \[ -\frac{b e g i m n x \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )+b d g j m n x \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )-b d g j m n x \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+b e g i m n x \text{PolyLog}\left (2,-\frac{j x}{i}\right )+a d f i+a d g i \log \left (h (i+j x)^m\right )-a d g j m x \log \left (-\frac{j x}{i}\right )+a d g j m x \log (i+j x)+b d f i \log \left (c (d+e x)^n\right )+b d g i \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )-b d g j m x \log \left (-\frac{j x}{i}\right ) \log \left (c (d+e x)^n\right )+b d g j m x \log (i+j x) \log \left (c (d+e x)^n\right )+b e f i n x \log (d+e x)+b e g i n x \log (d+e x) \log \left (h (i+j x)^m\right )-b e g i m n x \log (d+e x) \log (i+j x)+b e g i m n x \log (i+j x) \log \left (\frac{j (d+e x)}{d j-e i}\right )+b d g j m n x \log (d+e x) \log \left (-\frac{j x}{i}\right )-b d g j m n x \log (d+e x) \log (i+j x)+b d g j m n x \log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )-b d g j m n x \log \left (-\frac{e x}{d}\right ) \log (d+e x)-b e f i n x \log (x)-b e g i n x \log (x) \log \left (h (i+j x)^m\right )+b e g i m n x \log (x) \log \left (\frac{j x}{i}+1\right )}{d i x} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 1.977, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) \left ( f+g\ln \left ( h \left ( jx+i \right ) ^{m} \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))/x^2,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))/x^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="maxima")

[Out]

-b*e*f*n*(log(e*x + d)/d - log(x)/d) - a*g*j*m*(log(j*x + i)/i - log(x)/i) + b*g*integrate(((log((e*x + d)^n)
+ log(c))*log((j*x + i)^m) + log((e*x + d)^n)*log(h) + log(c)*log(h))/x^2, x) - b*f*log((e*x + d)^n*c)/x - a*g
*log((j*x + i)^m*h)/x - a*f/x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="fricas")

[Out]

integral((b*f*log((e*x + d)^n*c) + a*f + (b*g*log((e*x + d)^n*c) + a*g)*log((j*x + i)^m*h))/x^2, x)

________________________________________________________________________________________

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((a+b*ln(c*(e*x+d)**n))*(f+g*ln(h*(j*x+i)**m))/x**2,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)*(g*log((j*x + i)^m*h) + f)/x^2, x)