\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m))}{x^2} \, dx\) [391]

Optimal result

Integrand size = 32, antiderivative size = 270 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\frac {g j m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}-\frac {g j m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{i}+\frac {b e n \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {b e n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}-\frac {b g j m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{i}+\frac {b g j m n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{i}-\frac {b e g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d}+\frac {b e g m n \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2489, 36, 29, 31, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}+\frac {g j m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}-\frac {g j m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}+\frac {b e n \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {b e n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {b g j m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{i}+\frac {b g j m n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{i}-\frac {b e g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d}+\frac {b e g m n \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )}{d} \]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=-\frac {a d f i-b e f i n x \log (x)-a d g j m x \log \left (-\frac {j x}{i}\right )+b e f i n x \log (d+e x)-b d g j m n x \log \left (-\frac {e x}{d}\right ) \log (d+e x)+b d g j m n x \log \left (-\frac {j x}{i}\right ) \log (d+e x)+b d f i \log \left (c (d+e x)^n\right )-b d g j m x \log \left (-\frac {j x}{i}\right ) \log \left (c (d+e x)^n\right )+a d g j m x \log (i+j x)-b e g i m n x \log (d+e x) \log (i+j x)-b d g j m n x \log (d+e x) \log (i+j x)+b e g i m n x \log \left (\frac {j (d+e x)}{-e i+d j}\right ) \log (i+j x)+b d g j m x \log \left (c (d+e x)^n\right ) \log (i+j x)+b d g j m n x \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )+a d g i \log \left (h (i+j x)^m\right )-b e g i n x \log (x) \log \left (h (i+j x)^m\right )+b e g i n x \log (d+e x) \log \left (h (i+j x)^m\right )+b d g i \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+b e g i m n x \log (x) \log \left (1+\frac {j x}{i}\right )+b e g i m n x \operatorname {PolyLog}\left (2,-\frac {j x}{i}\right )+b d g j m n x \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )-b d g j m n x \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+b e g i m n x \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d i x} \]

[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]

\[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x^{2}}d x\]

[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)

Fricas [F]

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

[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)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[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

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x^2} \,d x \]

[In]

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

[Out]

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