\(\int x (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [388]

Optimal result

Integrand size = 30, antiderivative size = 397 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {a g i m x}{2 j}+\frac {b d f n x}{2 e}-\frac {3 b d g m n x}{4 e}-\frac {3 b g i m n x}{4 j}+\frac {1}{4} b g m n x^2+\frac {b d^2 g m n \log (d+e x)}{4 e^2}+\frac {b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g i^2 m n \log (i+j x)}{4 j^2}-\frac {g i^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}+\frac {b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}-\frac {b d^2 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 e^2} \]

[Out]

1/2*a*g*i*m*x/j+1/2*b*d*f*n*x/e-3/4*b*d*g*m*n*x/e-3/4*b*g*i*m*n*x/j+1/4*b*g*m*n*x^2+1/4*b*d^2*g*m*n*ln(e*x+d)/
e^2+1/2*b*g*i*m*(e*x+d)*ln(c*(e*x+d)^n)/e/j-1/4*g*m*x^2*(a+b*ln(c*(e*x+d)^n))+1/4*b*g*i^2*m*n*ln(j*x+i)/j^2-1/
2*g*i^2*m*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j^2+1/2*b*d*g*n*(j*x+i)*ln(h*(j*x+i)^m)/e/j-1/4*b*n*x
^2*(f+g*ln(h*(j*x+i)^m))-1/2*b*d^2*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^2+1/2*x^2*(a+b*ln(c*(e*
x+d)^n))*(f+g*ln(h*(j*x+i)^m))-1/2*b*g*i^2*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2-1/2*b*d^2*g*m*n*polylog(2,
e*(j*x+i)/(-d*j+e*i))/e^2

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2489, 45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {g i^2 m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {a g i m x}{2 j}+\frac {b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac {b d^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2}-\frac {b d^2 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 e^2}+\frac {b d^2 g m n \log (d+e x)}{4 e^2}+\frac {b d f n x}{2 e}+\frac {b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac {b g i^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}-\frac {3 b d g m n x}{4 e}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i^2 m n \log (i+j x)}{4 j^2}-\frac {3 b g i m n x}{4 j}+\frac {1}{4} b g m n x^2 \]

[In]

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

[Out]

(a*g*i*m*x)/(2*j) + (b*d*f*n*x)/(2*e) - (3*b*d*g*m*n*x)/(4*e) - (3*b*g*i*m*n*x)/(4*j) + (b*g*m*n*x^2)/4 + (b*d
^2*g*m*n*Log[d + e*x])/(4*e^2) + (b*g*i*m*(d + e*x)*Log[c*(d + e*x)^n])/(2*e*j) - (g*m*x^2*(a + b*Log[c*(d + e
*x)^n]))/4 + (b*g*i^2*m*n*Log[i + j*x])/(4*j^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i -
d*j)])/(2*j^2) + (b*d*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(2*e*j) - (b*n*x^2*(f + g*Log[h*(i + j*x)^m]))/4 - (b
*d^2*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(2*e^2) + (x^2*(a + b*Log[c*(d + e*x)^n])
*(f + g*Log[h*(i + j*x)^m]))/2 - (b*g*i^2*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) - (b*d^2*g*m*n
*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*e^2)

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 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 {1}{2} x^2 \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 {1}{2} (g j m) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x} \, dx-\frac {1}{2} (b e n) \int \frac {x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x} \, dx \\ & = \frac {1}{2} x^2 \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 {1}{2} (g j m) \int \left (-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac {i^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2 (i+j x)}\right ) \, dx-\frac {1}{2} (b e n) \int \left (-\frac {d \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^2}+\frac {x \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}+\frac {d^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^2 (d+e x)}\right ) \, dx \\ & = \frac {1}{2} x^2 \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 {1}{2} (g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+\frac {(g i m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{2 j}-\frac {\left (g i^2 m\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{i+j x} \, dx}{2 j}-\frac {1}{2} (b n) \int x \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx+\frac {(b d n) \int \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx}{2 e}-\frac {\left (b d^2 n\right ) \int \frac {f+g \log \left (h (i+j x)^m\right )}{d+e x} \, dx}{2 e} \\ & = \frac {a g i m x}{2 j}+\frac {b d f n x}{2 e}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {g i^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 j^2}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^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 e^2}+\frac {1}{2} x^2 \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) \int \log \left (c (d+e x)^n\right ) \, dx}{2 j}+\frac {(b d g n) \int \log \left (h (i+j x)^m\right ) \, dx}{2 e}+\frac {1}{4} (b e g m n) \int \frac {x^2}{d+e x} \, dx+\frac {\left (b e g i^2 m n\right ) \int \frac {\log \left (\frac {e (i+j x)}{e i-d j}\right )}{d+e x} \, dx}{2 j^2}+\frac {1}{4} (b g j m n) \int \frac {x^2}{i+j x} \, dx+\frac {\left (b d^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 e^2} \\ & = \frac {a g i m x}{2 j}+\frac {b d f n x}{2 e}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {g i^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 j^2}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^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 e^2}+\frac {1}{2} x^2 \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) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{2 e j}+\frac {(b d g n) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,i+j x\right )}{2 e j}+\frac {\left (b d^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 e^2}+\frac {1}{4} (b e g m n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx+\frac {\left (b g i^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 j^2}+\frac {1}{4} (b g j m n) \int \left (-\frac {i}{j^2}+\frac {x}{j}+\frac {i^2}{j^2 (i+j x)}\right ) \, dx \\ & = \frac {a g i m x}{2 j}+\frac {b d f n x}{2 e}-\frac {3 b d g m n x}{4 e}-\frac {3 b g i m n x}{4 j}+\frac {1}{4} b g m n x^2+\frac {b d^2 g m n \log (d+e x)}{4 e^2}+\frac {b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g i^2 m n \log (i+j x)}{4 j^2}-\frac {g i^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 j^2}+\frac {b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^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 e^2}+\frac {1}{2} x^2 \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^2 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}-\frac {b d^2 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{2 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.86 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {b n \log (d+e x) \left (2 e^2 g i^2 m \log (i+j x)+2 g \left (-e^2 i^2+d^2 j^2\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (-2 d f j+2 e g i m+d g j m-2 d g j \log \left (h (i+j x)^m\right )\right )\right )+e \left (g i m (-2 a e i+b (e i+2 d j) n) \log (i+j x)+j \left (a e x (2 f j x+g m (2 i-j x))-b n (e x (3 g i m+f j x-g j m x)+d (2 g i m-2 f j x+3 g j m x))+g j x (2 a e x+b n (2 d-e x)) \log \left (h (i+j x)^m\right )\right )+b e \log \left (c (d+e x)^n\right ) \left (-2 g i^2 m \log (i+j x)+j x \left (2 g i m+2 f j x-g j m x+2 g j x \log \left (h (i+j x)^m\right )\right )\right )\right )+2 b g \left (-e^2 i^2+d^2 j^2\right ) m n \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{4 e^2 j^2} \]

[In]

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

[Out]

(b*n*Log[d + e*x]*(2*e^2*g*i^2*m*Log[i + j*x] + 2*g*(-(e^2*i^2) + d^2*j^2)*m*Log[(e*(i + j*x))/(e*i - d*j)] +
d*j*(-2*d*f*j + 2*e*g*i*m + d*g*j*m - 2*d*g*j*Log[h*(i + j*x)^m])) + e*(g*i*m*(-2*a*e*i + b*(e*i + 2*d*j)*n)*L
og[i + j*x] + j*(a*e*x*(2*f*j*x + g*m*(2*i - j*x)) - b*n*(e*x*(3*g*i*m + f*j*x - g*j*m*x) + d*(2*g*i*m - 2*f*j
*x + 3*g*j*m*x)) + g*j*x*(2*a*e*x + b*n*(2*d - e*x))*Log[h*(i + j*x)^m]) + b*e*Log[c*(d + e*x)^n]*(-2*g*i^2*m*
Log[i + j*x] + j*x*(2*g*i*m + 2*f*j*x - g*j*m*x + 2*g*j*x*Log[h*(i + j*x)^m]))) + 2*b*g*(-(e^2*i^2) + d^2*j^2)
*m*n*PolyLog[2, (j*(d + e*x))/(-(e*i) + d*j)])/(4*e^2*j^2)

Maple [C] (warning: unable to verify)

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

Time = 111.26 (sec) , antiderivative size = 1372, normalized size of antiderivative = 3.46

method	result	size
risch	\(\text {Expression too large to display}\)	\(1372\)

[In]

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

[Out]

1/2/e*n*b*g*ln((j*x+i)^m)*x*d-1/2*n*b*d^2*f/e^2*ln(e*x+d)-1/4*b*f*n*x^2-1/4*ln(h)*x^2*b*g*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)*(1/2*(I*g*Pi*csgn(I*(j*x+i)^m)*
csgn(I*h*(j*x+i)^m)^2-I*g*Pi*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)-I*g*Pi*csgn(I*h*(j*x+i)^m)^3+I*g*
Pi*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)+2*g*ln(h)+2*f)*x^2+g*ln((j*x+i)^m)*x^2-1/2*g*m*x^2+g*m/j*x*i-g*m/j^2*i^2*ln
(j*x+i))+1/2/e*n*b/j*d*ln(e*x+d)*g*i*m+1/2/e*n*b*g*m/j*i*ln((e*x+d)*j-d*j+e*i)*d-1/4*I/e*n*b*x*Pi*d*g*csgn(I*h
*(j*x+i)^m)^3+1/4*I/e^2*n*b*d^2*ln(e*x+d)*Pi*g*csgn(I*h*(j*x+i)^m)^3+1/8*I*n*b*Pi*x^2*g*csgn(I*(j*x+i)^m)*csgn
(I*h*(j*x+i)^m)*csgn(I*h)+(1/2*x^2*b*g*ln((j*x+i)^m)-1/4*b*(-I*Pi*g*j^2*x^2*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)
^m)^2+I*Pi*g*j^2*x^2*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+I*Pi*g*j^2*x^2*csgn(I*h*(j*x+i)^m)^3-I*Pi
*g*j^2*x^2*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-2*j^2*x^2*ln(h)*g+g*j^2*m*x^2+2*g*i^2*m*ln(j*x+i)-2*f*j^2*x^2-2*g*i
*j*m*x)/j^2)*ln((e*x+d)^n)-1/4*b*d*g*i*m*n/e/j+1/2/e^2*n*b*g*m*d^2*ln(e*x+d)*ln(((e*x+d)*j-d*j+e*i)/(-d*j+e*i)
)+1/2*n*b*g*i^2*m/j^2*ln(j*x+i)*ln(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/8*I*n*b*Pi*x^2*g*csgn(I*h*(j*x+i)^m)^2*csg
n(I*h)-1/8*I*n*b*Pi*x^2*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-5/8*b*d^2*g*m*n/e^2-1/4*I/e*n*b*x*Pi*d*g*csg
n(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+1/4*I/e^2*n*b*d^2*ln(e*x+d)*Pi*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+
i)^m)*csgn(I*h)+1/2/e*ln(h)*x*b*d*g*n+1/4*b*g*m*n*x^2+1/4*n*b*g*m/j^2*i^2*ln((e*x+d)*j-d*j+e*i)+1/2*n*b*g*i^2*
m/j^2*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/2/e^2*n*b*g*ln((j*x+i)^m)*d^2*ln(e*x+d)+1/2/e^2*n*b*g*m*d^2*dilog
(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))-1/2/e^2*n*b*d^2*ln(e*x+d)*ln(h)*g+1/8*I*n*b*Pi*x^2*g*csgn(I*h*(j*x+i)^m)^3-1/
4*n*b*g*ln((j*x+i)^m)*x^2-3/4*b*d*g*m*n*x/e-3/4*b*g*i*m*n*x/j+1/4*b*d^2*g*m*n*ln(e*x+d)/e^2+1/4*I/e*n*b*x*Pi*d
*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/4*I/e*n*b*x*Pi*d*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-1/4*I/e^2*n*b*
d^2*ln(e*x+d)*Pi*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/4*I/e^2*n*b*d^2*ln(e*x+d)*Pi*g*csgn(I*h*(j*x+i)^m
)^2*csgn(I*h)+1/2*b*d*f*n*x/e

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/4*b*e*f*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) - 1/4*a*g*j*m*(2*i^2*log(j*x + i)/j^3 + (j*x^2 - 2
*i*x)/j^2) + 1/2*b*f*x^2*log((e*x + d)^n*c) + 1/2*a*g*x^2*log((j*x + i)^m*h) + 1/2*a*f*x^2 + 1/4*b*g*((2*e^2*i
^2*m*n*log(e*x + d)*log(j*x + i) + (2*e^2*i*j*m*x - 2*e^2*i^2*m*log(j*x + i) - (j^2*m - 2*j^2*log(h))*e^2*x^2)
*log((e*x + d)^n) + (2*e^2*j^2*x^2*log((e*x + d)^n) + 2*d*e*j^2*n*x - 2*d^2*j^2*n*log(e*x + d) - (e^2*j^2*n -
2*e^2*j^2*log(c))*x^2)*log((j*x + i)^m))/(e^2*j^2) + 4*integrate(-1/4*(2*((j^2*m - 2*j^2*log(h))*e^3*log(c) -
(j^2*m*n - j^2*n*log(h))*e^3)*x^3 + (d*e^2*j^2*m*n + (i*j*m*n + 2*i*j*n*log(h))*e^3 - 2*(2*e^3*i*j*log(h) - (j
^2*m - 2*j^2*log(h))*d*e^2)*log(c))*x^2 + 2*(e^3*i^2*m*n + d^2*e*j^2*m*n - 2*d*e^2*i*j*log(c)*log(h))*x + 2*(d
*e^2*i^2*m*n - d^3*j^2*m*n + (e^3*i^2*m*n - d^2*e*j^2*m*n)*x)*log(e*x + d))/(e^3*j^2*x^2 + d*e^2*i*j + (e^3*i*
j + d*e^2*j^2)*x), x))

Giac [F]

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

[In]

integrate(x*(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, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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