\(\int (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [389]

Optimal result

Integrand size = 29, antiderivative size = 232 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-a g m x-b f n x+2 b g m n x-\frac {b g m (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {g i m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}-\frac {b g n (i+j x) \log \left (h (i+j x)^m\right )}{j}+\frac {b d n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j}+\frac {b d g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2479, 45, 2463, 2436, 2332, 2441, 2440, 2438} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {g i m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-a g m x-\frac {b g m (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {b d n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}+\frac {b g i m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j}+\frac {b d g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e}-b f n x-\frac {b g n (i+j x) \log \left (h (i+j x)^m\right )}{j}+2 b g m n x \]

[In]

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

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2479

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.42 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {-b d f j n+b d g j m n+a e f j x-a e g j m x-b e f j n x+2 b e g j m n x+b e f j x \log \left (c (d+e x)^n\right )-b e g j m x \log \left (c (d+e x)^n\right )+a e g i m \log (i+j x)-b e g i m n \log (i+j x)+b d g j m n \log (i+j x)+b e g i m \log \left (c (d+e x)^n\right ) \log (i+j x)-b d g j n \log \left (h (i+j x)^m\right )+a e g j x \log \left (h (i+j x)^m\right )-b e g j n x \log \left (h (i+j x)^m\right )+b e g j x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+b n \log (d+e x) \left (-e g i m \log (i+j x)+g (e i-d j) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (f-g m+g \log \left (h (i+j x)^m\right )\right )\right )+b g (e i-d j) m n \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{e j} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 49.44 (sec) , antiderivative size = 1012, normalized size of antiderivative = 4.36

method	result	size
risch	\(\text {Expression too large to display}\)	\(1012\)

[In]

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

[Out]

(x*b*g*ln((j*x+i)^m)+1/2*b*(I*Pi*g*j*x*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-I*Pi*g*j*x*csgn(I*(j*x+i)^m)*cs
gn(I*h*(j*x+i)^m)*csgn(I*h)-I*Pi*g*j*x*csgn(I*h*(j*x+i)^m)^3+I*Pi*g*j*x*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)+2*j*x*
ln(h)*g+2*g*i*m*ln(j*x+i)-2*x*g*m*j+2*f*j*x)/j)*ln((e*x+d)^n)+(-1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*
c*(e*x+d)^n)+1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/4
*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+1/2*b*ln(c)+1/2*a)*(I*Pi*g*x*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+I*Pi*g*x*cs
gn(I*h*(j*x+i)^m)^2*csgn(I*h)+2*f*x+2*ln(h)*g*x+2*g*ln((j*x+i)^m)*x-2*g*m*x+2*g*m/j*i*ln(j*x+i)-I*Pi*g*x*csgn(
I*h*(j*x+i)^m)^3-I*Pi*g*x*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h))+1/2*I*n*b*x*g*Pi*csgn(I*(j*x+i)^m)*
csgn(I*h*(j*x+i)^m)*csgn(I*h)+1/2*I*n*b*x*g*Pi*csgn(I*h*(j*x+i)^m)^3-1/2*I*n*b*x*g*Pi*csgn(I*h*(j*x+i)^m)^2*cs
gn(I*h)+1/2*I/e*n*b*d*ln(e*x+d)*g*Pi*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-ln(h)*x*b*g*n+2*b*g*m*n*x-b*f*n*x+1/2*I/e
*n*b*d*ln(e*x+d)*g*Pi*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/2*I*n*b*x*g*Pi*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x
+i)^m)^2-1/2*I/e*n*b*d*ln(e*x+d)*g*Pi*csgn(I*h*(j*x+i)^m)^3-1/2*I/e*n*b*d*ln(e*x+d)*g*Pi*csgn(I*(j*x+i)^m)*csg
n(I*h*(j*x+i)^m)*csgn(I*h)+1/e*n*b*d*ln(e*x+d)*g*ln(h)-1/e*n*b*d*ln(e*x+d)*g*m+b*f/e*n*d*ln(e*x+d)-n*b*g*ln((j
*x+i)^m)*x+1/e*n*b*g*ln((j*x+i)^m)*d*ln(e*x+d)+b*d*g*m*n/e-n*b*g*m/j*i*ln((e*x+d)*j-d*j+e*i)-1/e*n*b*g*m*d*dil
og(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))-1/e*n*b*g*m*d*ln(e*x+d)*ln(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))-n*b*g*i*m/j*dilo
g(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-n*b*g*i*m/j*ln(j*x+i)*ln(((j*x+i)*e+d*j-e*i)/(d*j-e*i))

Fricas [F]

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

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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