\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m))}{x^3} \, dx\) [392]

Optimal result

Integrand size = 32, antiderivative size = 421 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\frac {b e g j m n \log (x)}{d i}-\frac {b e g j m n \log (d+e x)}{2 d i}-\frac {g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i x}-\frac {g j^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}-\frac {b e g j m n \log (i+j x)}{2 d i}+\frac {g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 i^2}-\frac {b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d x}-\frac {b e^2 n \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}+\frac {b e^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}+\frac {b g j^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 i^2}-\frac {b g j^2 m n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 i^2}+\frac {b e^2 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 d^2}-\frac {b e^2 g m n \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2489, 46, 2463, 2442, 36, 29, 31, 2441, 2352, 2440, 2438} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}-\frac {g j^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}+\frac {g j^2 m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}-\frac {g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i x}-\frac {b e^2 n \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}+\frac {b e^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}+\frac {b e^2 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 d^2}-\frac {b e^2 g m n \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )}{2 d^2}-\frac {b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d x}+\frac {b g j^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 i^2}-\frac {b g j^2 m n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{2 i^2}+\frac {b e g j m n \log (x)}{d i}-\frac {b e g j m n \log (d+e x)}{2 d i}-\frac {b e g j m n \log (i+j x)}{2 d i} \]

[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 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 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 46

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] && Lt
Q[m + n + 2, 0])

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 2438

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]

Rule 2440

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 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 2442

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 2463

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 2489

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]

Rubi steps \begin{align*} \text {integral}& = -\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 {1}{2} (g j m) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (i+j x)} \, dx+\frac {1}{2} (b e n) \int \frac {f+g \log \left (h (i+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 (i+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 )}{i x^2}-\frac {j \left (a+b \log \left (c (d+e x)^n\right )\right )}{i^2 x}+\frac {j^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i^2 (i+j x)}\right ) \, dx+\frac {1}{2} (b e n) \int \left (\frac {f+g \log \left (h (i+j x)^m\right )}{d x^2}-\frac {e \left (f+g \log \left (h (i+j x)^m\right )\right )}{d^2 x}+\frac {e^2 \left (f+g \log \left (h (i+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 (i+j x)^m\right )\right )}{2 x^2}+\frac {(g j m) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{2 i}-\frac {\left (g j^2 m\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{2 i^2}+\frac {\left (g j^3 m\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{i+j x} \, dx}{2 i^2}+\frac {(b e n) \int \frac {f+g \log \left (h (i+j x)^m\right )}{x^2} \, dx}{2 d}-\frac {\left (b e^2 n\right ) \int \frac {f+g \log \left (h (i+j x)^m\right )}{x} \, dx}{2 d^2}+\frac {\left (b e^3 n\right ) \int \frac {f+g \log \left (h (i+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 )}{2 i 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 )}{2 i^2}+\frac {g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 i^2}-\frac {b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d 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 {\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 {(b e g j m n) \int \frac {1}{x (i+j x)} \, dx}{2 d}+\frac {\left (b e^2 g j m n\right ) \int \frac {\log \left (-\frac {j x}{i}\right )}{i+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)}{-e i+d j}\right )}{i+j x} \, dx}{2 d^2}+\frac {(b e g j m n) \int \frac {1}{x (d+e x)} \, dx}{2 i}+\frac {\left (b e g j^2 m n\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{2 i^2}-\frac {\left (b e g j^2 m n\right ) \int \frac {\log \left (\frac {e (i+j x)}{e i-d j}\right )}{d+e x} \, dx}{2 i^2} \\ & = -\frac {g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i 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 )}{2 i^2}+\frac {g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 i^2}-\frac {b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d 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 {\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 {b g j^2 m n \text {Li}_2\left (1+\frac {e x}{d}\right )}{2 i^2}-\frac {b e^2 g m n \text {Li}_2\left (1+\frac {j x}{i}\right )}{2 d^2}-\frac {\left (b e^2 g m n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-e i+d j}\right )}{x} \, dx,x,i+j x\right )}{2 d^2}+2 \frac {(b e g j m n) \int \frac {1}{x} \, dx}{2 d i}-\frac {\left (b e^2 g j m n\right ) \int \frac {1}{d+e x} \, dx}{2 d i}-\frac {\left (b g j^2 m n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{e i-d j}\right )}{x} \, dx,x,d+e x\right )}{2 i^2}-\frac {\left (b e g j^2 m n\right ) \int \frac {1}{i+j x} \, dx}{2 d i} \\ & = \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 {g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i 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 )}{2 i^2}-\frac {b e g j m n \log (i+j x)}{2 d i}+\frac {g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 i^2}-\frac {b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d 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 {\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 {b g j^2 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{2 i^2}-\frac {b g j^2 m n \text {Li}_2\left (1+\frac {e x}{d}\right )}{2 i^2}+\frac {b e^2 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{2 d^2}-\frac {b e^2 g m n \text {Li}_2\left (1+\frac {j x}{i}\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.82 \[ \int \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^3} \, dx=-\frac {b e^2 n \log (x) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 d^2}+\frac {b e^2 n \log (d+e x) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 d^2}-\frac {b n \log (d+e x) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 x^2}-\frac {\left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right ) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 x^2}-\frac {e \left (b f n+b g n \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 d x}+\frac {1}{2} a g m \left (\frac {j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}-\left (\frac {j^2 (i+j x)^2}{i^4 \left (1-\frac {i+j x}{i}\right )^2}+\frac {2 j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}\right ) \log (i+j x)-\frac {j^2 \log \left (1-\frac {i+j x}{i}\right )}{i^2}\right )+\frac {1}{2} b g m \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\frac {j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}-\left (\frac {j^2 (i+j x)^2}{i^4 \left (1-\frac {i+j x}{i}\right )^2}+\frac {2 j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}\right ) \log (i+j x)-\frac {j^2 \log \left (1-\frac {i+j x}{i}\right )}{i^2}\right )+\frac {1}{2} b g m n \left (-\frac {\log (d+e x) \log (i+j x)}{x^2}+j \left (\frac {\frac {e \log (x)}{d}-\frac {e \log (d+e x)}{d}-\frac {\log (d+e x)}{x}}{i}-\frac {j \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x)+\operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )\right )}{i^2}+\frac {j^2 \left (\frac {\log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}+\frac {\operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{j}\right )}{i^2}\right )+e \left (\frac {\frac {j \log (x)}{i}-\frac {j \log (i+j x)}{i}-\frac {\log (i+j x)}{x}}{d}-\frac {e \left (\log (x) \left (\log (i+j x)-\log \left (1+\frac {j x}{i}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {j x}{i}\right )\right )}{d^2}+\frac {e^2 \left (\frac {\log \left (\frac {j (d+e x)}{-e i+d j}\right ) \log (i+j x)}{e}+\frac {\operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e}\right )}{d^2}\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x^{3}}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[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

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x^3} \,d x \]

[In]

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

[Out]

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