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

Optimal. Leaf size=232 \[ \frac{b g i m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{j}+\frac{b d g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{g i m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-a g m x-\frac{b g m (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{b d n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}-b f n x-\frac{b g n (i+j x) \log \left (h (i+j x)^m\right )}{j}+2 b g m n x \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.284739, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2430, 43, 2416, 2389, 2295, 2394, 2393, 2391} \[ \frac{b g i m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{j}+\frac{b d g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{g i m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-a g m x-\frac{b g m (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{b d n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}-b f n x-\frac{b g n (i+j x) \log \left (h (i+j x)^m\right )}{j}+2 b g m n x \]

Antiderivative was successfully verified.

[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 2430

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]

Rule 43

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 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 2389

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, 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 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 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (389+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 (389+j x)^m\right )\right )-(g j m) \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{389+j x} \, dx-(b e n) \int \frac{x \left (f+g \log \left (h (389+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 (389+j x)^m\right )\right )-(g j m) \int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{j}-\frac{389 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (389+j x)}\right ) \, dx-(b e n) \int \left (\frac{f+g \log \left (h (389+j x)^m\right )}{e}-\frac{d \left (f+g \log \left (h (389+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 (389+j x)^m\right )\right )-(g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+(389 g m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{389+j x} \, dx-(b n) \int \left (f+g \log \left (h (389+j x)^m\right )\right ) \, dx+(b d n) \int \frac{f+g \log \left (h (389+j x)^m\right )}{d+e x} \, dx\\ &=-a g m x-b f n x+\frac{389 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (389+j x)}{389 e-d j}\right )}{j}+\frac{b d n \log \left (-\frac{j (d+e x)}{389 e-d j}\right ) \left (f+g \log \left (h (389+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 (389+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 (389+j x)^m\right ) \, dx-\frac{(389 b e g m n) \int \frac{\log \left (\frac{e (389+j x)}{389 e-d j}\right )}{d+e x} \, dx}{j}-\frac{(b d g j m n) \int \frac{\log \left (\frac{j (d+e x)}{-389 e+d j}\right )}{389+j x} \, dx}{e}\\ &=-a g m x-b f n x+\frac{389 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (389+j x)}{389 e-d j}\right )}{j}+\frac{b d n \log \left (-\frac{j (d+e x)}{389 e-d j}\right ) \left (f+g \log \left (h (389+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 (389+j x)^m\right )\right )-\frac{(b g m) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac{(b g n) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,389+j x\right )}{j}-\frac{(b d g m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-389 e+d j}\right )}{x} \, dx,x,389+j x\right )}{e}-\frac{(389 b g m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{389 e-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{389 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (389+j x)}{389 e-d j}\right )}{j}-\frac{b g n (389+j x) \log \left (h (389+j x)^m\right )}{j}+\frac{b d n \log \left (-\frac{j (d+e x)}{389 e-d j}\right ) \left (f+g \log \left (h (389+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 (389+j x)^m\right )\right )+\frac{389 b g m n \text{Li}_2\left (-\frac{j (d+e x)}{389 e-d j}\right )}{j}+\frac{b d g m n \text{Li}_2\left (\frac{e (389+j x)}{389 e-d j}\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.226182, size = 329, normalized size = 1.42 \[ \frac{b g m n (e i-d j) \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )+a e f j x+a e g j x \log \left (h (i+j x)^m\right )+a e g i m \log (i+j x)-a e g j m x+b e f j x \log \left (c (d+e x)^n\right )+b e g j x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+b e g i m \log (i+j x) \log \left (c (d+e x)^n\right )-b e g j m x \log \left (c (d+e x)^n\right )+b n \log (d+e x) \left (g m (e i-d j) \log \left (\frac{e (i+j x)}{e i-d j}\right )+d j \left (f+g \log \left (h (i+j x)^m\right )-g m\right )-e g i m \log (i+j x)\right )-b d f j n-b d g j n \log \left (h (i+j x)^m\right )+b d g j m n \log (i+j x)+b d g j m n-b e f j n x-b e g j n x \log \left (h (i+j x)^m\right )-b e g i m n \log (i+j x)+2 b e g j m n x}{e j} \]

Antiderivative was successfully verified.

[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]  time = 1.346, size = 2544, normalized size = 11. \begin{align*} \text{result too large to display} \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)

[Out]

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

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (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\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, 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)

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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)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \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, algorithm="giac")

[Out]

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