3.392 \(\int \frac{(a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m))}{x^3} \, dx\)

Optimal. Leaf size=421 \[ \frac{b e^2 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{2 d^2}-\frac{b e^2 g m n \text{PolyLog}\left (2,\frac{j x}{i}+1\right )}{2 d^2}+\frac{b g j^2 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{2 i^2}-\frac{b g j^2 m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{2 i^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}-\frac{g j^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}+\frac{g j^2 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}-\frac{g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i x}-\frac{b e^2 n \log \left (-\frac{j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}+\frac{b e^2 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}-\frac{b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d x}+\frac{b e g j m n \log (x)}{d i}-\frac{b e g j m n \log (d+e x)}{2 d i}-\frac{b e g j m n \log (i+j x)}{2 d i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.455179, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2439, 44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ \frac{b e^2 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{2 d^2}-\frac{b e^2 g m n \text{PolyLog}\left (2,\frac{j x}{i}+1\right )}{2 d^2}+\frac{b g j^2 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{2 i^2}-\frac{b g j^2 m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{2 i^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}-\frac{g j^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}+\frac{g j^2 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}-\frac{g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i x}-\frac{b e^2 n \log \left (-\frac{j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}+\frac{b e^2 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}-\frac{b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d x}+\frac{b e g j m n \log (x)}{d i}-\frac{b e g j m n \log (d+e x)}{2 d i}-\frac{b e g j m n \log (i+j x)}{2 d i} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x^3,x]

[Out]

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

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 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]

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 (392+j x)^m\right )\right )}{x^3} \, dx &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 x^2}+\frac{1}{2} (g j m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2 (392+j x)} \, dx+\frac{1}{2} (b e n) \int \frac{f+g \log \left (h (392+j x)^m\right )}{x^2 (d+e x)} \, dx\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 x^2}+\frac{1}{2} (g j m) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{392 x^2}-\frac{j \left (a+b \log \left (c (d+e x)^n\right )\right )}{153664 x}+\frac{j^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{153664 (392+j x)}\right ) \, dx+\frac{1}{2} (b e n) \int \left (\frac{f+g \log \left (h (392+j x)^m\right )}{d x^2}-\frac{e \left (f+g \log \left (h (392+j x)^m\right )\right )}{d^2 x}+\frac{e^2 \left (f+g \log \left (h (392+j x)^m\right )\right )}{d^2 (d+e x)}\right ) \, dx\\ &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 x^2}+\frac{1}{784} (g j m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx-\frac{\left (g j^2 m\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{307328}+\frac{\left (g j^3 m\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{392+j x} \, dx}{307328}+\frac{(b e n) \int \frac{f+g \log \left (h (392+j x)^m\right )}{x^2} \, dx}{2 d}-\frac{\left (b e^2 n\right ) \int \frac{f+g \log \left (h (392+j x)^m\right )}{x} \, dx}{2 d^2}+\frac{\left (b e^3 n\right ) \int \frac{f+g \log \left (h (392+j x)^m\right )}{d+e x} \, dx}{2 d^2}\\ &=-\frac{g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{784 x}-\frac{g j^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{307328}+\frac{g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (392+j x)}{392 e-d j}\right )}{307328}-\frac{b e n \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d x}-\frac{b e^2 n \log \left (-\frac{j x}{392}\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d^2}+\frac{b e^2 n \log \left (-\frac{j (d+e x)}{392 e-d j}\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 x^2}+\frac{1}{784} (b e g j m n) \int \frac{1}{x (d+e x)} \, dx+\frac{(b e g j m n) \int \frac{1}{x (392+j x)} \, dx}{2 d}+\frac{\left (b e^2 g j m n\right ) \int \frac{\log \left (-\frac{j x}{392}\right )}{392+j x} \, dx}{2 d^2}-\frac{\left (b e^2 g j m n\right ) \int \frac{\log \left (\frac{j (d+e x)}{-392 e+d j}\right )}{392+j x} \, dx}{2 d^2}+\frac{\left (b e g j^2 m n\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{307328}-\frac{\left (b e g j^2 m n\right ) \int \frac{\log \left (\frac{e (392+j x)}{392 e-d j}\right )}{d+e x} \, dx}{307328}\\ &=-\frac{g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{784 x}-\frac{g j^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{307328}+\frac{g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (392+j x)}{392 e-d j}\right )}{307328}-\frac{b e n \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d x}-\frac{b e^2 n \log \left (-\frac{j x}{392}\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d^2}+\frac{b e^2 n \log \left (-\frac{j (d+e x)}{392 e-d j}\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 x^2}-\frac{b g j^2 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{307328}-\frac{b e^2 g m n \text{Li}_2\left (1+\frac{j x}{392}\right )}{2 d^2}-\frac{\left (b e^2 g m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-392 e+d j}\right )}{x} \, dx,x,392+j x\right )}{2 d^2}+2 \frac{(b e g j m n) \int \frac{1}{x} \, dx}{784 d}-\frac{\left (b e^2 g j m n\right ) \int \frac{1}{d+e x} \, dx}{784 d}-\frac{\left (b g j^2 m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{392 e-d j}\right )}{x} \, dx,x,d+e x\right )}{307328}-\frac{\left (b e g j^2 m n\right ) \int \frac{1}{392+j x} \, dx}{784 d}\\ &=\frac{b e g j m n \log (x)}{392 d}-\frac{b e g j m n \log (d+e x)}{784 d}-\frac{g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{784 x}-\frac{g j^2 m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{307328}-\frac{b e g j m n \log (392+j x)}{784 d}+\frac{g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (392+j x)}{392 e-d j}\right )}{307328}-\frac{b e n \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d x}-\frac{b e^2 n \log \left (-\frac{j x}{392}\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d^2}+\frac{b e^2 n \log \left (-\frac{j (d+e x)}{392 e-d j}\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 d^2}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (392+j x)^m\right )\right )}{2 x^2}+\frac{b g j^2 m n \text{Li}_2\left (-\frac{j (d+e x)}{392 e-d j}\right )}{307328}-\frac{b g j^2 m n \text{Li}_2\left (1+\frac{e x}{d}\right )}{307328}-\frac{b e^2 g m n \text{Li}_2\left (1+\frac{j x}{392}\right )}{2 d^2}+\frac{b e^2 g m n \text{Li}_2\left (\frac{e (392+j x)}{392 e-d j}\right )}{2 d^2}\\ \end{align*}

Mathematica [A]  time = 0.418051, size = 765, normalized size = 1.82 \[ \frac{1}{2} b g m n \left (e \left (\frac{e^2 \left (\frac{\text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+\frac{\log (i+j x) \log \left (\frac{j (d+e x)}{d j-e i}\right )}{e}\right )}{d^2}-\frac{e \left (\log (x) \left (\log (i+j x)-\log \left (\frac{j x}{i}+1\right )\right )-\text{PolyLog}\left (2,-\frac{j x}{i}\right )\right )}{d^2}+\frac{\frac{j \log (x)}{i}-\frac{j \log (i+j x)}{i}-\frac{\log (i+j x)}{x}}{d}\right )+j \left (\frac{j^2 \left (\frac{\text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )}{j}+\frac{\log (d+e x) \log \left (\frac{e (i+j x)}{e i-d j}\right )}{j}\right )}{i^2}-\frac{j \left (\text{PolyLog}\left (2,\frac{d+e x}{d}\right )+\log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )}{i^2}+\frac{\frac{e \log (x)}{d}-\frac{e \log (d+e x)}{d}-\frac{\log (d+e x)}{x}}{i}\right )-\frac{\log (d+e x) \log (i+j x)}{x^2}\right )-\frac{\left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right ) \left (f+g \left (\log \left (h (i+j x)^m\right )-m \log (i+j x)\right )\right )}{2 x^2}+\frac{1}{2} a g m \left (\frac{j^2 (i+j x)}{i^3 \left (1-\frac{i+j x}{i}\right )}-\frac{j^2 \log \left (1-\frac{i+j x}{i}\right )}{i^2}-\left (\frac{2 j^2 (i+j x)}{i^3 \left (1-\frac{i+j x}{i}\right )}+\frac{j^2 (i+j x)^2}{i^4 \left (1-\frac{i+j x}{i}\right )^2}\right ) \log (i+j x)\right )+\frac{1}{2} b g m \left (\frac{j^2 (i+j x)}{i^3 \left (1-\frac{i+j x}{i}\right )}-\frac{j^2 \log \left (1-\frac{i+j x}{i}\right )}{i^2}-\left (\frac{2 j^2 (i+j x)}{i^3 \left (1-\frac{i+j x}{i}\right )}+\frac{j^2 (i+j x)^2}{i^4 \left (1-\frac{i+j x}{i}\right )^2}\right ) \log (i+j x)\right ) \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )-\frac{b e^2 n \log (x) \left (f+g \left (\log \left (h (i+j x)^m\right )-m \log (i+j x)\right )\right )}{2 d^2}+\frac{b e^2 n \log (d+e x) \left (f+g \left (\log \left (h (i+j x)^m\right )-m \log (i+j x)\right )\right )}{2 d^2}-\frac{b n \log (d+e x) \left (f+g \left (\log \left (h (i+j x)^m\right )-m \log (i+j x)\right )\right )}{2 x^2}-\frac{e \left (b f n+b g n \left (\log \left (h (i+j x)^m\right )-m \log (i+j x)\right )\right )}{2 d x} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x^3,x]

[Out]

-(b*e^2*n*Log[x]*(f + g*(-(m*Log[i + j*x]) + Log[h*(i + j*x)^m])))/(2*d^2) + (b*e^2*n*Log[d + e*x]*(f + g*(-(m
*Log[i + j*x]) + Log[h*(i + j*x)^m])))/(2*d^2) - (b*n*Log[d + e*x]*(f + g*(-(m*Log[i + j*x]) + Log[h*(i + j*x)
^m])))/(2*x^2) - ((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))*(f + g*(-(m*Log[i + j*x]) + Log[h*(i + j*x)
^m])))/(2*x^2) - (e*(b*f*n + b*g*n*(-(m*Log[i + j*x]) + Log[h*(i + j*x)^m])))/(2*d*x) + (a*g*m*((j^2*(i + j*x)
)/(i^3*(1 - (i + j*x)/i)) - ((j^2*(i + j*x)^2)/(i^4*(1 - (i + j*x)/i)^2) + (2*j^2*(i + j*x))/(i^3*(1 - (i + j*
x)/i)))*Log[i + j*x] - (j^2*Log[1 - (i + j*x)/i])/i^2))/2 + (b*g*m*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])*((
j^2*(i + j*x))/(i^3*(1 - (i + j*x)/i)) - ((j^2*(i + j*x)^2)/(i^4*(1 - (i + j*x)/i)^2) + (2*j^2*(i + j*x))/(i^3
*(1 - (i + j*x)/i)))*Log[i + j*x] - (j^2*Log[1 - (i + j*x)/i])/i^2))/2 + (b*g*m*n*(-((Log[d + e*x]*Log[i + j*x
])/x^2) + j*(((e*Log[x])/d - (e*Log[d + e*x])/d - Log[d + e*x]/x)/i - (j*(Log[-((e*x)/d)]*Log[d + e*x] + PolyL
og[2, (d + e*x)/d]))/i^2 + (j^2*((Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)])/j + PolyLog[2, (j*(d + e*x))/(-
(e*i) + d*j)]/j))/i^2) + e*(((j*Log[x])/i - (j*Log[i + j*x])/i - Log[i + j*x]/x)/d - (e*(Log[x]*(Log[i + j*x]
- Log[1 + (j*x)/i]) - PolyLog[2, -((j*x)/i)]))/d^2 + (e^2*((Log[(j*(d + e*x))/(-(e*i) + d*j)]*Log[i + j*x])/e
+ PolyLog[2, (e*(i + j*x))/(e*i - d*j)]/e))/d^2)))/2

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Maple [F]  time = 1.299, 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}^{3}}}\, 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^3,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b e f n{\left (\frac{e \log \left (e x + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{d^{2}} - \frac{1}{d x}\right )} + \frac{1}{2} \, a g j m{\left (\frac{j \log \left (j x + i\right )}{i^{2}} - \frac{j \log \left (x\right )}{i^{2}} - \frac{1}{i x}\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^{3}}\,{d x} - \frac{b f \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac{a g \log \left ({\left (j x + i\right )}^{m} h\right )}{2 \, x^{2}} - \frac{a f}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

1/2*b*e*f*n*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + 1/2*a*g*j*m*(j*log(j*x + i)/i^2 - j*log(x)/i^2 - 1
/(i*x)) + 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^3, x) - 1/2*b*f*log((e*x + d)^n*c)/x^2 - 1/2*a*g*log((j*x + i)^m*h)/x^2 - 1/2*a*f/x^2

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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^{3}}, 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^3,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^3, 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((a+b*ln(c*(e*x+d)**n))*(f+g*ln(h*(j*x+i)**m))/x**3,x)

[Out]

Timed out

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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^{3}}\,{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^3,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)*(g*log((j*x + i)^m*h) + f)/x^3, x)