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

Optimal. Leaf size=232 \[ -a g m x-b f n x+2 b g m n x-\frac {b g m (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {g i m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}-\frac {b g n (i+j x) \log \left (h (i+j x)^m\right )}{j}+\frac {b d n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}+\frac {b d g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{e} \]

[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]
time = 0.20, antiderivative size = 232, normalized size of antiderivative = 1.00, 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 = {2479, 45, 2463, 2436, 2332, 2441, 2440, 2438} \begin {gather*} \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 \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]

[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*} \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) \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,389+j x\right )}{j}-\frac {(b d g m n) \text {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) \text {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*}

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Mathematica [A]
time = 0.13, size = 329, normalized size = 1.42 \begin {gather*} \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 \text {Li}_2\left (\frac {j (d+e x)}{-e i+d j}\right )}{e j} \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]

[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.07, size = 2544, normalized size = 10.97

method	result	size
risch	\(\text {Expression too large to display}\)	\(2544\)

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,method=_RETURNVERBOSE)

[Out]

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

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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, algorithm="maxima")

[Out]

-a*g*j*m*(x/j - I*log(j*x + I)/j^2) + (d*e^(-2)*log(x*e + d) - x*e^(-1))*b*f*n*e + b*f*x*log((x*e + d)^n*c) +
a*g*x*log((j*x + I)^m*h) + a*f*x + b*g*((-I*m*n*e*log(j*x + I)*log(x*e + d) + (d*j*n*log(x*e + d) + j*x*e*log(
(x*e + d)^n) - (j*n - j*log(c))*x*e)*log((j*x + I)^m) - ((j*m - j*log(h))*x*e - I*m*e*log(j*x + I))*log((x*e +
d)^n))*e^(-1)/j + integrate(((2*j*m*n - j*n*log(h) - (j*m - j*log(h))*log(c))*x^2*e^2 + I*d*e*log(c)*log(h) +
((I*m*n - I*n*log(h) + I*log(c)*log(h))*e^2 + (d*j*m*n - (j*m - j*log(h))*d*log(c))*e)*x - (d^2*j*m*n - I*d*m
*n*e + (d*j*m*n*e - I*m*n*e^2)*x)*log(x*e + d))/(j*x^2*e^2 + (d*j*e + I*e^2)*x + I*d*e), 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, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (h \left (i + j x\right )^{m} \right )}\right )\, dx \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)

[Out]

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

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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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \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)

[Out]

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

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