3.4.90 \(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m))}{x} \, dx\) [390]

Optimal. Leaf size=637 \[ f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b g m n \log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (i+j x)-b g m \log \left (-\frac {j x}{i}\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \log (i+j x)+\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {e i-d j}{e (i+j x)}\right )-\log \left (-\frac {(e i-d j) x}{d (i+j x)}\right )\right ) \log ^2\left (\frac {d (i+j x)}{i (d+e x)}\right )-\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{i}\right )\right ) \left (\log (d+e x)+\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right )^2-b g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )+a g \log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+b f n \text {Li}_2\left (1+\frac {e x}{d}\right )+b g m n \left (\log (i+j x)-\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-b g n \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+b g m n \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \text {Li}_2\left (\frac {i (d+e x)}{d (i+j x)}\right )-b g m n \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \text {Li}_2\left (\frac {j (d+e x)}{e (i+j x)}\right )+a g m \text {Li}_2\left (1+\frac {j x}{i}\right )-b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {j x}{i}\right )+b g m n \left (\log (d+e x)+\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {j x}{i}\right )-b g m n \text {Li}_3\left (1+\frac {e x}{d}\right )+b g m n \text {Li}_3\left (\frac {i (d+e x)}{d (i+j x)}\right )-b g m n \text {Li}_3\left (\frac {j (d+e x)}{e (i+j x)}\right )-b g m n \text {Li}_3\left (1+\frac {j x}{i}\right ) \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 637, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2488, 2441, 2352, 2487, 2485} \begin {gather*} a g m \text {PolyLog}\left (2,\frac {j x}{i}+1\right )-b g m \text {PolyLog}\left (2,\frac {j x}{i}+1\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+b f n \text {PolyLog}\left (2,\frac {e x}{d}+1\right )-b g n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )+b g m n \text {PolyLog}\left (3,\frac {i (d+e x)}{d (i+j x)}\right )-b g m n \text {PolyLog}\left (3,\frac {j (d+e x)}{e (i+j x)}\right )+b g m n \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \text {PolyLog}\left (2,\frac {i (d+e x)}{d (i+j x)}\right )-b g m n \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \text {PolyLog}\left (2,\frac {j (d+e x)}{e (i+j x)}\right )+b g m n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (\log (i+j x)-\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right )+b g m n \text {PolyLog}\left (2,\frac {j x}{i}+1\right ) \left (\log \left (\frac {d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )-b g m n \text {PolyLog}\left (3,\frac {e x}{d}+1\right )-b g m n \text {PolyLog}\left (3,\frac {j x}{i}+1\right )+f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g \log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )-b g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )-b g m \log \left (-\frac {j x}{i}\right ) \log (i+j x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+\frac {1}{2} b g m n \left (\log \left (\frac {e i-d j}{e (i+j x)}\right )-\log \left (-\frac {x (e i-d j)}{d (i+j x)}\right )+\log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (\frac {d (i+j x)}{i (d+e x)}\right )-\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{i}\right )\right ) \left (\log \left (\frac {d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )^2+b g m n \log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (i+j x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2441

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 2485

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[(-b)*(x/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-(b*c - a*d))*(x/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d
*x))])*Log[a*((c + d*x)/(c*(a + b*x)))]^2, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-d)*(x/c)])*(Log[a + b*x] +
Log[a*((c + d*x)/(c*(a + b*x)))])^2, x] + Simp[(Log[c + d*x] - Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1
+ b*(x/a)], x] + Simp[(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 + d*(x/c)], x] + Simp[Lo
g[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[Log[a*((c + d*x)/(c*(a + b*x
)))]*PolyLog[2, d*((a + b*x)/(b*(c + d*x)))], x] - Simp[PolyLog[3, 1 + b*(x/a)], x] - Simp[PolyLog[3, 1 + d*(x
/c)], x] + Simp[PolyLog[3, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[PolyLog[3, d*((a + b*x)/(b*(c + d*x)))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2487

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[Log[i + j*x]*(Log[c*(d + e*x)^n]/x), x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2488

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.) + (f_))
)/(x_), x_Symbol] :> Dist[f, Int[(a + b*Log[c*(d + e*x)^n])/x, x], x] + Dist[g, Int[Log[h*(i + j*x)^m]*((a + b
*Log[c*(d + e*x)^n])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (390+j x)^m\right )\right )}{x} \, dx &=f \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx+g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (390+j x)^m\right )}{x} \, dx\\ &=f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+(a g) \int \frac {\log \left (h (390+j x)^m\right )}{x} \, dx+(b g) \int \frac {\log \left (c (d+e x)^n\right ) \log \left (h (390+j x)^m\right )}{x} \, dx-(b e f n) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx\\ &=f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g \log \left (-\frac {j x}{390}\right ) \log \left (h (390+j x)^m\right )+b f n \text {Li}_2\left (1+\frac {e x}{d}\right )+(b g m) \int \frac {\log \left (c (d+e x)^n\right ) \log (390+j x)}{x} \, dx-(a g j m) \int \frac {\log \left (-\frac {j x}{390}\right )}{390+j x} \, dx-\left (b g \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )\right ) \int \frac {\log \left (c (d+e x)^n\right )}{x} \, dx\\ &=f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )+a g \log \left (-\frac {j x}{390}\right ) \log \left (h (390+j x)^m\right )+b f n \text {Li}_2\left (1+\frac {e x}{d}\right )+a g m \text {Li}_2\left (1+\frac {j x}{390}\right )+(b g m n) \int \frac {\log (d+e x) \log (390+j x)}{x} \, dx-\left (b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )\right ) \int \frac {\log (390+j x)}{x} \, dx+\left (b e g n \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx\\ &=-b g m \log (390) \log (x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b g m n \log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (390+j x)+\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {390 e-d j}{e (390+j x)}\right )-\log \left (-\frac {(390 e-d j) x}{d (390+j x)}\right )\right ) \log ^2\left (\frac {d (390+j x)}{390 (d+e x)}\right )-\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{390}\right )\right ) \left (\log (d+e x)+\log \left (\frac {d (390+j x)}{390 (d+e x)}\right )\right )^2-b g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )+a g \log \left (-\frac {j x}{390}\right ) \log \left (h (390+j x)^m\right )+b f n \text {Li}_2\left (1+\frac {e x}{d}\right )+b g m n \left (\log (390+j x)-\log \left (\frac {d (390+j x)}{390 (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-b g n \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+a g m \text {Li}_2\left (1+\frac {j x}{390}\right )+b g m n \left (\log (d+e x)+\log \left (\frac {d (390+j x)}{390 (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {j x}{390}\right )+b g m n \log \left (\frac {d (390+j x)}{390 (d+e x)}\right ) \text {Li}_2\left (\frac {390 (d+e x)}{d (390+j x)}\right )-b g m n \log \left (\frac {d (390+j x)}{390 (d+e x)}\right ) \text {Li}_2\left (\frac {j (d+e x)}{e (390+j x)}\right )-b g m n \text {Li}_3\left (1+\frac {e x}{d}\right )-b g m n \text {Li}_3\left (1+\frac {j x}{390}\right )+b g m n \text {Li}_3\left (\frac {390 (d+e x)}{d (390+j x)}\right )-b g m n \text {Li}_3\left (\frac {j (d+e x)}{e (390+j x)}\right )-\left (b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )\right ) \int \frac {\log \left (1+\frac {j x}{390}\right )}{x} \, dx\\ &=-b g m \log (390) \log (x) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )+f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b g m n \log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (390+j x)+\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {390 e-d j}{e (390+j x)}\right )-\log \left (-\frac {(390 e-d j) x}{d (390+j x)}\right )\right ) \log ^2\left (\frac {d (390+j x)}{390 (d+e x)}\right )-\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{390}\right )\right ) \left (\log (d+e x)+\log \left (\frac {d (390+j x)}{390 (d+e x)}\right )\right )^2-b g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right )+a g \log \left (-\frac {j x}{390}\right ) \log \left (h (390+j x)^m\right )+b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j x}{390}\right )+b f n \text {Li}_2\left (1+\frac {e x}{d}\right )+b g m n \left (\log (390+j x)-\log \left (\frac {d (390+j x)}{390 (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-b g n \left (m \log (390+j x)-\log \left (h (390+j x)^m\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+a g m \text {Li}_2\left (1+\frac {j x}{390}\right )+b g m n \left (\log (d+e x)+\log \left (\frac {d (390+j x)}{390 (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {j x}{390}\right )+b g m n \log \left (\frac {d (390+j x)}{390 (d+e x)}\right ) \text {Li}_2\left (\frac {390 (d+e x)}{d (390+j x)}\right )-b g m n \log \left (\frac {d (390+j x)}{390 (d+e x)}\right ) \text {Li}_2\left (\frac {j (d+e x)}{e (390+j x)}\right )-b g m n \text {Li}_3\left (1+\frac {e x}{d}\right )-b g m n \text {Li}_3\left (1+\frac {j x}{390}\right )+b g m n \text {Li}_3\left (\frac {390 (d+e x)}{d (390+j x)}\right )-b g m n \text {Li}_3\left (\frac {j (d+e x)}{e (390+j x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 605, normalized size = 0.95 \begin {gather*} \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (f-g m \log (i+j x)+g \log \left (h (i+j x)^m\right )\right )+b n \left (f-g m \log (i+j x)+g \log \left (h (i+j x)^m\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\text {Li}_2\left (-\frac {e x}{d}\right )\right )+a g m \left (\log (x) \left (\log (i+j x)-\log \left (1+\frac {j x}{i}\right )\right )-\text {Li}_2\left (-\frac {j x}{i}\right )\right )+b g m \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (i+j x)-\log \left (1+\frac {j x}{i}\right )\right )-\text {Li}_2\left (-\frac {j x}{i}\right )\right )+b g m n \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (i+j x)+\frac {1}{2} \log ^2\left (\frac {d (i+j x)}{i (d+e x)}\right ) \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {-e i+d j}{j (d+e x)}\right )-\log \left (\frac {e i x-d j x}{d i+e i x}\right )\right )+\left (-\log \left (-\frac {e x}{d}\right )+\log \left (-\frac {j x}{i}\right )\right ) \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \log \left (1+\frac {j x}{i}\right )+\frac {1}{2} \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{i}\right )\right ) \log \left (1+\frac {j x}{i}\right ) \left (-2 \log (d+e x)+\log \left (1+\frac {j x}{i}\right )\right )+\left (\log (i+j x)-\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+\log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \left (-\text {Li}_2\left (\frac {d (i+j x)}{i (d+e x)}\right )+\text {Li}_2\left (\frac {e (i+j x)}{j (d+e x)}\right )\right )+\left (\log (d+e x)+\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \text {Li}_2\left (1+\frac {j x}{i}\right )-\text {Li}_3\left (1+\frac {e x}{d}\right )+\text {Li}_3\left (\frac {d (i+j x)}{i (d+e x)}\right )-\text {Li}_3\left (\frac {e (i+j x)}{j (d+e x)}\right )-\text {Li}_3\left (1+\frac {j x}{i}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x}\, dx\]

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,x)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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