3.386 \(\int x^3 (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\)

Optimal. Leaf size=742 \[ -\frac{b d^4 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{4 e^4}-\frac{b g i^4 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{4 j^4}+\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac{g i^2 m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 j^2}-\frac{g i^4 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 j^4}+\frac{g i m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{12 j}-\frac{1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{a g i^3 m x}{4 j^3}+\frac{b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\frac{b d^2 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}-\frac{b d^4 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e^4}+\frac{b d^3 f n x}{4 e^3}+\frac{b d^3 g n (i+j x) \log \left (h (i+j x)^m\right )}{4 e^3 j}+\frac{b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac{b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac{5 b d^2 g i m n x}{24 e^2 j}+\frac{b d^3 g i m n \log (d+e x)}{12 e^3 j}+\frac{3 b d^2 g m n x^2}{32 e^2}-\frac{5 b d^3 g m n x}{16 e^3}+\frac{b d^4 g m n \log (d+e x)}{16 e^4}+\frac{b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac{5 b d g i^2 m n x}{24 e j^2}+\frac{b d g i^3 m n \log (i+j x)}{12 e j^3}+\frac{b d g i m n x^2}{12 e j}-\frac{7 b d g m n x^3}{144 e}-\frac{1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{3 b g i^2 m n x^2}{32 j^2}-\frac{5 b g i^3 m n x}{16 j^3}+\frac{b g i^4 m n \log (i+j x)}{16 j^4}-\frac{7 b g i m n x^3}{144 j}+\frac{1}{32} b g m n x^4 \]

[Out]

(a*g*i^3*m*x)/(4*j^3) + (b*d^3*f*n*x)/(4*e^3) - (5*b*d^3*g*m*n*x)/(16*e^3) - (5*b*g*i^3*m*n*x)/(16*j^3) - (5*b
*d*g*i^2*m*n*x)/(24*e*j^2) - (5*b*d^2*g*i*m*n*x)/(24*e^2*j) + (3*b*d^2*g*m*n*x^2)/(32*e^2) + (3*b*g*i^2*m*n*x^
2)/(32*j^2) + (b*d*g*i*m*n*x^2)/(12*e*j) - (7*b*d*g*m*n*x^3)/(144*e) - (7*b*g*i*m*n*x^3)/(144*j) + (b*g*m*n*x^
4)/32 + (b*d^4*g*m*n*Log[d + e*x])/(16*e^4) + (b*d^2*g*i^2*m*n*Log[d + e*x])/(8*e^2*j^2) + (b*d^3*g*i*m*n*Log[
d + e*x])/(12*e^3*j) + (b*g*i^3*m*(d + e*x)*Log[c*(d + e*x)^n])/(4*e*j^3) - (g*i^2*m*x^2*(a + b*Log[c*(d + e*x
)^n]))/(8*j^2) + (g*i*m*x^3*(a + b*Log[c*(d + e*x)^n]))/(12*j) - (g*m*x^4*(a + b*Log[c*(d + e*x)^n]))/16 + (b*
g*i^4*m*n*Log[i + j*x])/(16*j^4) + (b*d*g*i^3*m*n*Log[i + j*x])/(12*e*j^3) + (b*d^2*g*i^2*m*n*Log[i + j*x])/(8
*e^2*j^2) - (g*i^4*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(4*j^4) + (b*d^3*g*n*(i + j*x)
*Log[h*(i + j*x)^m])/(4*e^3*j) - (b*d^2*n*x^2*(f + g*Log[h*(i + j*x)^m]))/(8*e^2) + (b*d*n*x^3*(f + g*Log[h*(i
+ j*x)^m]))/(12*e) - (b*n*x^4*(f + g*Log[h*(i + j*x)^m]))/16 - (b*d^4*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f
+ g*Log[h*(i + j*x)^m]))/(4*e^4) + (x^4*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/4 - (b*g*i^4*m*
n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(4*j^4) - (b*d^4*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(4*e
^4)

________________________________________________________________________________________

Rubi [A]  time = 0.872177, antiderivative size = 742, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {2439, 43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ -\frac{b d^4 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{4 e^4}-\frac{b g i^4 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{4 j^4}+\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac{g i^2 m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 j^2}-\frac{g i^4 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 j^4}+\frac{g i m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{12 j}-\frac{1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{a g i^3 m x}{4 j^3}+\frac{b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\frac{b d^2 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}-\frac{b d^4 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e^4}+\frac{b d^3 f n x}{4 e^3}+\frac{b d^3 g n (i+j x) \log \left (h (i+j x)^m\right )}{4 e^3 j}+\frac{b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac{b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac{5 b d^2 g i m n x}{24 e^2 j}+\frac{b d^3 g i m n \log (d+e x)}{12 e^3 j}+\frac{3 b d^2 g m n x^2}{32 e^2}-\frac{5 b d^3 g m n x}{16 e^3}+\frac{b d^4 g m n \log (d+e x)}{16 e^4}+\frac{b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac{5 b d g i^2 m n x}{24 e j^2}+\frac{b d g i^3 m n \log (i+j x)}{12 e j^3}+\frac{b d g i m n x^2}{12 e j}-\frac{7 b d g m n x^3}{144 e}-\frac{1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{3 b g i^2 m n x^2}{32 j^2}-\frac{5 b g i^3 m n x}{16 j^3}+\frac{b g i^4 m n \log (i+j x)}{16 j^4}-\frac{7 b g i m n x^3}{144 j}+\frac{1}{32} b g m n x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]),x]

[Out]

(a*g*i^3*m*x)/(4*j^3) + (b*d^3*f*n*x)/(4*e^3) - (5*b*d^3*g*m*n*x)/(16*e^3) - (5*b*g*i^3*m*n*x)/(16*j^3) - (5*b
*d*g*i^2*m*n*x)/(24*e*j^2) - (5*b*d^2*g*i*m*n*x)/(24*e^2*j) + (3*b*d^2*g*m*n*x^2)/(32*e^2) + (3*b*g*i^2*m*n*x^
2)/(32*j^2) + (b*d*g*i*m*n*x^2)/(12*e*j) - (7*b*d*g*m*n*x^3)/(144*e) - (7*b*g*i*m*n*x^3)/(144*j) + (b*g*m*n*x^
4)/32 + (b*d^4*g*m*n*Log[d + e*x])/(16*e^4) + (b*d^2*g*i^2*m*n*Log[d + e*x])/(8*e^2*j^2) + (b*d^3*g*i*m*n*Log[
d + e*x])/(12*e^3*j) + (b*g*i^3*m*(d + e*x)*Log[c*(d + e*x)^n])/(4*e*j^3) - (g*i^2*m*x^2*(a + b*Log[c*(d + e*x
)^n]))/(8*j^2) + (g*i*m*x^3*(a + b*Log[c*(d + e*x)^n]))/(12*j) - (g*m*x^4*(a + b*Log[c*(d + e*x)^n]))/16 + (b*
g*i^4*m*n*Log[i + j*x])/(16*j^4) + (b*d*g*i^3*m*n*Log[i + j*x])/(12*e*j^3) + (b*d^2*g*i^2*m*n*Log[i + j*x])/(8
*e^2*j^2) - (g*i^4*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(4*j^4) + (b*d^3*g*n*(i + j*x)
*Log[h*(i + j*x)^m])/(4*e^3*j) - (b*d^2*n*x^2*(f + g*Log[h*(i + j*x)^m]))/(8*e^2) + (b*d*n*x^3*(f + g*Log[h*(i
+ j*x)^m]))/(12*e) - (b*n*x^4*(f + g*Log[h*(i + j*x)^m]))/16 - (b*d^4*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f
+ g*Log[h*(i + j*x)^m]))/(4*e^4) + (x^4*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/4 - (b*g*i^4*m*
n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(4*j^4) - (b*d^4*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(4*e
^4)

Rule 2439

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]

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 2395

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 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 x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx &=\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{1}{4} (g j m) \int \frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{386+j x} \, dx-\frac{1}{4} (b e n) \int \frac{x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{1}{4} (g j m) \int \left (-\frac{57512456 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^4}+\frac{148996 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac{386 x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac{22199808016 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^4 (386+j x)}\right ) \, dx-\frac{1}{4} (b e n) \int \left (-\frac{d^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^4}+\frac{d^2 x \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^3}-\frac{d x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^2}+\frac{x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e}+\frac{d^4 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{1}{4} (g m) \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+\frac{(14378114 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j^3}-\frac{(5549952004 g m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{386+j x} \, dx}{j^3}-\frac{(37249 g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j^2}+\frac{(193 g m) \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{2 j}-\frac{1}{4} (b n) \int x^3 \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx+\frac{\left (b d^3 n\right ) \int \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx}{4 e^3}-\frac{\left (b d^4 n\right ) \int \frac{f+g \log \left (h (386+j x)^m\right )}{d+e x} \, dx}{4 e^3}-\frac{\left (b d^2 n\right ) \int x \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx}{4 e^2}+\frac{(b d n) \int x^2 \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx}{4 e}\\ &=\frac{14378114 a g m x}{j^3}+\frac{b d^3 f n x}{4 e^3}-\frac{37249 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}+\frac{193 g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac{1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{5549952004 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (386+j x)}{386 e-d j}\right )}{j^4}-\frac{b d^2 n x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{8 e^2}+\frac{b d n x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{12 e}-\frac{1}{16} b n x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{b d^4 n \log \left (-\frac{j (d+e x)}{386 e-d j}\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )+\frac{(14378114 b g m) \int \log \left (c (d+e x)^n\right ) \, dx}{j^3}+\frac{\left (b d^3 g n\right ) \int \log \left (h (386+j x)^m\right ) \, dx}{4 e^3}+\frac{1}{16} (b e g m n) \int \frac{x^4}{d+e x} \, dx+\frac{(5549952004 b e g m n) \int \frac{\log \left (\frac{e (386+j x)}{386 e-d j}\right )}{d+e x} \, dx}{j^4}+\frac{(37249 b e g m n) \int \frac{x^2}{d+e x} \, dx}{2 j^2}-\frac{(193 b e g m n) \int \frac{x^3}{d+e x} \, dx}{6 j}+\frac{1}{16} (b g j m n) \int \frac{x^4}{386+j x} \, dx+\frac{\left (b d^4 g j m n\right ) \int \frac{\log \left (\frac{j (d+e x)}{-386 e+d j}\right )}{386+j x} \, dx}{4 e^4}+\frac{\left (b d^2 g j m n\right ) \int \frac{x^2}{386+j x} \, dx}{8 e^2}-\frac{(b d g j m n) \int \frac{x^3}{386+j x} \, dx}{12 e}\\ &=\frac{14378114 a g m x}{j^3}+\frac{b d^3 f n x}{4 e^3}-\frac{37249 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}+\frac{193 g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac{1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{5549952004 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (386+j x)}{386 e-d j}\right )}{j^4}-\frac{b d^2 n x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{8 e^2}+\frac{b d n x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{12 e}-\frac{1}{16} b n x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{b d^4 n \log \left (-\frac{j (d+e x)}{386 e-d j}\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )+\frac{(14378114 b g m) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j^3}+\frac{\left (b d^3 g n\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,386+j x\right )}{4 e^3 j}+\frac{\left (b d^4 g m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-386 e+d j}\right )}{x} \, dx,x,386+j x\right )}{4 e^4}+\frac{1}{16} (b e g m n) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x}{e^3}-\frac{d x^2}{e^2}+\frac{x^3}{e}+\frac{d^4}{e^4 (d+e x)}\right ) \, dx+\frac{(5549952004 b g m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{386 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^4}+\frac{(37249 b e g m n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx}{2 j^2}-\frac{(193 b e g m n) \int \left (\frac{d^2}{e^3}-\frac{d x}{e^2}+\frac{x^2}{e}-\frac{d^3}{e^3 (d+e x)}\right ) \, dx}{6 j}+\frac{1}{16} (b g j m n) \int \left (-\frac{57512456}{j^4}+\frac{148996 x}{j^3}-\frac{386 x^2}{j^2}+\frac{x^3}{j}+\frac{22199808016}{j^4 (386+j x)}\right ) \, dx+\frac{\left (b d^2 g j m n\right ) \int \left (-\frac{386}{j^2}+\frac{x}{j}+\frac{148996}{j^2 (386+j x)}\right ) \, dx}{8 e^2}-\frac{(b d g j m n) \int \left (\frac{148996}{j^3}-\frac{386 x}{j^2}+\frac{x^2}{j}-\frac{57512456}{j^3 (386+j x)}\right ) \, dx}{12 e}\\ &=\frac{14378114 a g m x}{j^3}+\frac{b d^3 f n x}{4 e^3}-\frac{5 b d^3 g m n x}{16 e^3}-\frac{35945285 b g m n x}{2 j^3}-\frac{186245 b d g m n x}{6 e j^2}-\frac{965 b d^2 g m n x}{12 e^2 j}+\frac{3 b d^2 g m n x^2}{32 e^2}+\frac{111747 b g m n x^2}{8 j^2}+\frac{193 b d g m n x^2}{6 e j}-\frac{7 b d g m n x^3}{144 e}-\frac{1351 b g m n x^3}{72 j}+\frac{1}{32} b g m n x^4+\frac{b d^4 g m n \log (d+e x)}{16 e^4}+\frac{37249 b d^2 g m n \log (d+e x)}{2 e^2 j^2}+\frac{193 b d^3 g m n \log (d+e x)}{6 e^3 j}+\frac{14378114 b g m (d+e x) \log \left (c (d+e x)^n\right )}{e j^3}-\frac{37249 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}+\frac{193 g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac{1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1387488001 b g m n \log (386+j x)}{j^4}+\frac{14378114 b d g m n \log (386+j x)}{3 e j^3}+\frac{37249 b d^2 g m n \log (386+j x)}{2 e^2 j^2}-\frac{5549952004 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (386+j x)}{386 e-d j}\right )}{j^4}+\frac{b d^3 g n (386+j x) \log \left (h (386+j x)^m\right )}{4 e^3 j}-\frac{b d^2 n x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{8 e^2}+\frac{b d n x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{12 e}-\frac{1}{16} b n x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{b d^4 n \log \left (-\frac{j (d+e x)}{386 e-d j}\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac{5549952004 b g m n \text{Li}_2\left (-\frac{j (d+e x)}{386 e-d j}\right )}{j^4}-\frac{b d^4 g m n \text{Li}_2\left (\frac{e (386+j x)}{386 e-d j}\right )}{4 e^4}\\ \end{align*}

Mathematica [A]  time = 1.21918, size = 605, normalized size = 0.82 \[ \frac{-72 b g m n \left (e^4 i^4-d^4 j^4\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )+e \left (j \left (-6 g j^3 x \left (b n \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )-12 a e^3 x^3\right ) \log \left (h (i+j x)^m\right )+6 a e^3 x \left (12 f j^3 x^3+g m \left (-6 i^2 j x+12 i^3+4 i j^2 x^2-3 j^3 x^3\right )\right )-b n \left (3 d^2 e j^2 x (12 f j x+g m (20 i-9 j x))+18 d^3 j^3 x (5 g m-4 f)+2 d e^2 \left (g m \left (30 i^2 j x+36 i^3-12 i j^2 x^2+7 j^3 x^3\right )-12 f j^3 x^3\right )+e^3 x \left (18 f j^3 x^3+g m \left (-27 i^2 j x+90 i^3+14 i j^2 x^2-9 j^3 x^3\right )\right )\right )\right )+6 g i m \log (i+j x) \left (b n \left (6 d^2 e i j^2+12 d^3 j^3+4 d e^2 i^2 j+3 e^3 i^3\right )-12 a e^3 i^3\right )-6 b e^3 \log \left (c (d+e x)^n\right ) \left (-12 f j^4 x^4-12 g j^4 x^4 \log \left (h (i+j x)^m\right )+g j m x \left (6 i^2 j x-12 i^3-4 i j^2 x^2+3 j^3 x^3\right )+12 g i^4 m \log (i+j x)\right )\right )+6 b n \log (d+e x) \left (d j \left (4 d^2 e g i j^2 m+3 d^3 j^3 (g m-4 f)-12 d^3 g j^3 \log \left (h (i+j x)^m\right )+6 d e^2 g i^2 j m+12 e^3 g i^3 m\right )-12 g m \left (e^4 i^4-d^4 j^4\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )+12 e^4 g i^4 m \log (i+j x)\right )}{288 e^4 j^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]),x]

[Out]

(6*b*n*Log[d + e*x]*(12*e^4*g*i^4*m*Log[i + j*x] - 12*g*(e^4*i^4 - d^4*j^4)*m*Log[(e*(i + j*x))/(e*i - d*j)] +
d*j*(12*e^3*g*i^3*m + 6*d*e^2*g*i^2*j*m + 4*d^2*e*g*i*j^2*m + 3*d^3*j^3*(-4*f + g*m) - 12*d^3*g*j^3*Log[h*(i
+ j*x)^m])) + e*(6*g*i*m*(-12*a*e^3*i^3 + b*(3*e^3*i^3 + 4*d*e^2*i^2*j + 6*d^2*e*i*j^2 + 12*d^3*j^3)*n)*Log[i
+ j*x] - 6*b*e^3*Log[c*(d + e*x)^n]*(-12*f*j^4*x^4 + g*j*m*x*(-12*i^3 + 6*i^2*j*x - 4*i*j^2*x^2 + 3*j^3*x^3) +
12*g*i^4*m*Log[i + j*x] - 12*g*j^4*x^4*Log[h*(i + j*x)^m]) + j*(6*a*e^3*x*(12*f*j^3*x^3 + g*m*(12*i^3 - 6*i^2
*j*x + 4*i*j^2*x^2 - 3*j^3*x^3)) - b*n*(18*d^3*j^3*(-4*f + 5*g*m)*x + 3*d^2*e*j^2*x*(12*f*j*x + g*m*(20*i - 9*
j*x)) + e^3*x*(18*f*j^3*x^3 + g*m*(90*i^3 - 27*i^2*j*x + 14*i*j^2*x^2 - 9*j^3*x^3)) + 2*d*e^2*(-12*f*j^3*x^3 +
g*m*(36*i^3 + 30*i^2*j*x - 12*i*j^2*x^2 + 7*j^3*x^3))) - 6*g*j^3*x*(-12*a*e^3*x^3 + b*n*(-12*d^3 + 6*d^2*e*x
- 4*d*e^2*x^2 + 3*e^3*x^3))*Log[h*(i + j*x)^m])) - 72*b*g*(e^4*i^4 - d^4*j^4)*m*n*PolyLog[2, (j*(d + e*x))/(-(
e*i) + d*j)])/(288*e^4*j^4)

________________________________________________________________________________________

Maple [C]  time = 2.398, size = 4217, normalized size = 5.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m)),x)

[Out]

1/4*ln(c)*x^4*b*f+1/4*ln(h)*x^4*a*g-1/16/e/j^3*g*i^3*m*b*d*n+1/4/j^4*b*g*i^4*m*n*ln(j*x+i)*ln(((j*x+i)*e+d*j-e
*i)/(d*j-e*i))+1/8*I*ln(c)*Pi*x^4*b*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/8*I*ln(c)*Pi*x^4*b*g*csgn(I*(j*x+i)^m)
*csgn(I*h*(j*x+i)^m)^2-1/16*x^4*a*g*m-1/16*x^4*b*f*n-1/16*n*b*g*ln((j*x+i)^m)*x^4+1/4*b*ln(c)*g*x^4*ln((j*x+i)
^m)+1/16/j^4*g*i^4*m*ln((e*x+d)*j-d*j+e*i)*b*n+1/4/e^4*b*d^4*g*m*n*dilog(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))-1/4*a
*g*m/j^4*i^4*ln(j*x+i)+1/8*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*g*x^4*ln((j*x+i)^m)-1/32*I*Pi*x^4*b*
g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-5/16*b*d^3*g*m*n*x/e^3-5/16*b*g*i^3*m*n*x/j^3+3/32*b*d^2*g*m*n*x^2/e^2+3/3
2*b*g*i^2*m*n*x^2/j^2-7/144*b*d*g*m*n*x^3/e-7/144*b*g*i*m*n*x^3/j-1/8*I*Pi*x^4*a*g*csgn(I*h*(j*x+i)^m)^3-1/8*I
*Pi*x^4*b*f*csgn(I*c*(e*x+d)^n)^3-1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^4*g*csgn(I*h*(j*x+i)^m)^3-1/8*I*Pi*x^4*a
*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^4*g*csgn(I*h)
*csgn(I*h*(j*x+i)^m)^2-1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^4*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)
^2-1/8*I*Pi*x^4*b*f*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4*x^4*a*f+1/4*a*g*x^4*ln((j*x+i)^m)-1/16
*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^4*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/32*b*g*m*n*x^4+1/4/j^4
*b*g*i^4*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))-1/4/e^4*ln(e*x+d)*b*d^4*f*n+1/12/e*x^3*b*d*f*n-1/8/e^2*x^2*b
*d^2*f*n+1/12/j*x^3*a*g*i*m-1/8/j^2*x^2*a*g*i^2*m-1/32*I*Pi*x^4*b*g*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+
1/8*I/e^3*Pi*x*b*d^3*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/16*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x
+d)^n)*x^4*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/8*I/j^3*Pi*x*b*g*i^3*m*csgn(I*c)*csgn(I*c*(e*x+
d)^n)^2+1/24*I/e*Pi*x^3*b*d*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/8*I/e^3*Pi*x*b*d^3*g*n*csgn(I*h)*csgn(I*(j*x
+i)^m)*csgn(I*h*(j*x+i)^m)+1/8*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g*m/j^4*i^4*ln(j*x+i)+1/
8*b*d^2*g*i^2*m*n*ln(e*x+d)/e^2/j^2+1/12*b*d^3*g*i*m*n*ln(e*x+d)/e^3/j-1/8*I/e^3*Pi*x*b*d^3*g*n*csgn(I*h*(j*x+
i)^m)^3+1/16*I/j^2*Pi*x^2*b*g*i^2*m*csgn(I*c*(e*x+d)^n)^3+1/8*I/e^4*ln(e*x+d)*Pi*b*d^4*g*n*csgn(I*h*(j*x+i)^m)
^3-1/8*I/j^3*Pi*x*b*g*i^3*m*csgn(I*c*(e*x+d)^n)^3+1/24*I/j*Pi*x^3*b*g*i*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)^2+1/8*I/j^3*Pi*x*b*g*i^3*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/8*I/e^4*ln(e*x+d)*Pi*b*d^4*g*n*csgn(I*(
j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+(1/4*g*b*x^4*ln((j*x+i)^m)-1/48*b*(6*I*Pi*g*j^4*x^4*csgn(I*h)*csgn(I*(j*x+i)^m
)*csgn(I*h*(j*x+i)^m)-6*I*Pi*g*j^4*x^4*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-6*I*Pi*g*j^4*x^4*csgn(I*(j*x+i)^m)*csgn
(I*h*(j*x+i)^m)^2+6*I*Pi*g*j^4*x^4*csgn(I*h*(j*x+i)^m)^3-12*ln(h)*g*j^4*x^4+3*g*j^4*m*x^4-12*f*j^4*x^4-4*g*i*j
^3*m*x^3+6*g*i^2*j^2*m*x^2+12*g*i^4*m*ln(j*x+i)-12*g*i^3*j*m*x)/j^4)*ln((e*x+d)^n)-1/16*b*Pi^2*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^2*x^4*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^4*g
*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/8*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g*x^4*ln((j*x+i)^m)+1/16*b*P
i^2*csgn(I*c*(e*x+d)^n)^3*x^4*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)
^n)^2*x^4*g*csgn(I*h*(j*x+i)^m)^3+1/16*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^4*g*csgn(I*h*(j*x+i)^m
)^3+1/16*I/j^2*Pi*x^2*b*g*i^2*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/8*I/e^4*ln(e*x+d)*Pi*b*d^4*g
*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/8*I*Pi*x^4*b*f*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/8*I*Pi*x
^4*b*f*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/32*I*Pi*x^4*b*g*n*csgn(I*h*(j*x+i)^m)^3+1/8*I*Pi*x^4*a*g*csgn
(I*h)*csgn(I*h*(j*x+i)^m)^2-1/32*I*Pi*x^4*b*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/8*I*Pi*ln(h)*x^4*b*g
*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/8*I*Pi*ln(h)*x^4*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/32*I*Pi*x^4*
b*g*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4/j^3*ln(c)*x*b*g*i^3*m-1/4/e^4*ln(e*x+d)*ln(h)*b*d^4*g*n+1/4/e^3*ln(h
)*x*b*d^3*g*n+1/12/e*ln(h)*x^3*b*d*g*n-1/8/e^2*ln(h)*x^2*b*d^2*g*n+1/12/j*ln(c)*x^3*b*g*i*m-1/8/j^2*ln(c)*x^2*
b*g*i^2*m-1/4*b*ln(c)*g*m/j^4*i^4*ln(j*x+i)-1/16*ln(h)*x^4*b*g*n+1/4*ln(c)*ln(h)*x^4*b*g-1/16*ln(c)*x^4*b*g*m-
1/16*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^4*g*csgn(I*h*(j*x+i)^m)^3+1/8*I*b*Pi*csgn(I*c)*c
sgn(I*c*(e*x+d)^n)^2*g*x^4*ln((j*x+i)^m)-1/24*I/j*Pi*x^3*b*g*i*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^
n)-3/16/e^3/j*b*d^3*g*i*m*n-11/96/e^2/j^2*b*d^2*g*i^2*m*n-1/8*I/j^3*Pi*x*b*g*i^3*m*csgn(I*c)*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)-1/24*I/e*Pi*x^3*b*d*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/16*I/e^2*Pi*x^2
*b*d^2*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/4/e^4*b*d^4*g*m*n*ln(e*x+d)*ln(((e*x+d)*j-d*j+e*i
)/(-d*j+e*i))-205/576/e^4*b*d^4*g*m*n+1/4*a*g*i^3*m*x/j^3+1/4*b*d^3*f*n*x/e^3+1/8*I/e^3*Pi*x*b*d^3*g*n*csgn(I*
(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/8*I/e^4*ln(e*x+d)*Pi*b*d^4*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/16*I/e^2*P
i*x^2*b*d^2*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/24*I/e*Pi*x^3*b*d*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^
2-1/16*I/e^2*Pi*x^2*b*d^2*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/16*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(
e*x+d)^n)^2*x^4*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/24*I/j*Pi*x^3*b*g*i*m*csgn(I*c)*csgn(I*c*(
e*x+d)^n)^2-1/8*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g*m/j^4*i^4*ln(j*x+i)-1/8*I*b*Pi*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^2*g*m/j^4*i^4*ln(j*x+i)-5/24*b*d*g*i^2*m*n*x/e/j^2-5/24*b*d^2*g*i*m*n*x/e^2/j+1/12*b*d*g*i*m*n
*x^2/e/j+1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^4*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/8*I*Pi*x^4*a*g*csgn(I*(j*x+
i)^m)*csgn(I*h*(j*x+i)^m)^2-1/8*I*ln(c)*Pi*x^4*b*g*csgn(I*h*(j*x+i)^m)^3-1/8*I*Pi*ln(h)*x^4*b*g*csgn(I*c*(e*x+
d)^n)^3+1/32*I*Pi*x^4*b*g*m*csgn(I*c*(e*x+d)^n)^3+1/12/e/j^3*g*i^3*m*ln((e*x+d)*j-d*j+e*i)*b*d*n+1/4/e/j^3*ln(
e*x+d)*b*d*g*i^3*m*n+1/4/e^3/j*g*i*m*ln((e*x+d)*j-d*j+e*i)*b*d^3*n+1/8/e^2/j^2*g*i^2*m*ln((e*x+d)*j-d*j+e*i)*b
*d^2*n-1/16*I/j^2*Pi*x^2*b*g*i^2*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/16*I/j^2*Pi*x^2*b*g*i^2*m*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^2+1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^4*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*
h*(j*x+i)^m)-1/8*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g*x^4*ln((j*x+i)^m)+1/8*I*b*Pi*csgn(I*
c*(e*x+d)^n)^3*g*m/j^4*i^4*ln(j*x+i)+1/16*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^4*g*csgn(I*
h)*csgn(I*h*(j*x+i)^m)^2+1/16*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^4*g*csgn(I*(j*x+i)^m)*c
sgn(I*h*(j*x+i)^m)^2-1/8*I*Pi*ln(h)*x^4*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/8*I*ln(c)*Pi*x^4
*b*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/32*I*Pi*x^4*b*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*
(j*x+i)^m)+1/32*I*Pi*x^4*b*g*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/24*I/j*Pi*x^3*b*g*i*m*csgn(I*
c*(e*x+d)^n)^3-1/24*I/e*Pi*x^3*b*d*g*n*csgn(I*h*(j*x+i)^m)^3+1/16*I/e^2*Pi*x^2*b*d^2*g*n*csgn(I*h*(j*x+i)^m)^3
+1/16*b*d^4*g*m*n*ln(e*x+d)/e^4-1/4/e^4*n*b*g*ln((j*x+i)^m)*d^4*ln(e*x+d)+1/12/e*n*b*g*ln((j*x+i)^m)*d*x^3-1/8
/e^2*n*b*g*ln((j*x+i)^m)*x^2*d^2+1/4/e^3*n*b*g*ln((j*x+i)^m)*x*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

1/4*b*f*x^4*log((e*x + d)^n*c) + 1/4*a*g*x^4*log((j*x + i)^m*h) + 1/4*a*f*x^4 - 1/48*b*e*f*n*(12*d^4*log(e*x +
d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^4) - 1/48*a*g*j*m*(12*i^4*log(j*x + i)/j^5 + (3
*j^3*x^4 - 4*i*j^2*x^3 + 6*i^2*j*x^2 - 12*i^3*x)/j^4) + 1/48*b*g*((12*e^4*i^4*m*n*log(e*x + d)*log(j*x + i) +
(4*e^4*i*j^3*m*x^3 - 6*e^4*i^2*j^2*m*x^2 + 12*e^4*i^3*j*m*x - 12*e^4*i^4*m*log(j*x + i) - 3*(j^4*m - 4*j^4*log
(h))*e^4*x^4)*log((e*x + d)^n) + (12*e^4*j^4*x^4*log((e*x + d)^n) + 4*d*e^3*j^4*n*x^3 - 6*d^2*e^2*j^4*n*x^2 +
12*d^3*e*j^4*n*x - 12*d^4*j^4*n*log(e*x + d) - 3*(e^4*j^4*n - 4*e^4*j^4*log(c))*x^4)*log((j*x + i)^m))/(e^4*j^
4) + 48*integrate(-1/48*(6*(2*(j^4*m - 4*j^4*log(h))*e^5*log(c) - (j^4*m*n - 2*j^4*n*log(h))*e^5)*x^5 + (d*e^4
*j^4*m*n + (i*j^3*m*n + 12*i*j^3*n*log(h))*e^5 - 12*(4*e^5*i*j^3*log(h) - (j^4*m - 4*j^4*log(h))*d*e^4)*log(c)
)*x^4 - 2*(e^5*i^2*j^2*m*n + d^2*e^3*j^4*m*n + 24*d*e^4*i*j^3*log(c)*log(h))*x^3 + 6*(e^5*i^3*j*m*n + d^3*e^2*
j^4*m*n)*x^2 + 12*(e^5*i^4*m*n + d^4*e*j^4*m*n)*x + 12*(d*e^4*i^4*m*n - d^5*j^4*m*n + (e^5*i^4*m*n - d^4*e*j^4
*m*n)*x)*log(e*x + d))/(e^5*j^4*x^2 + d*e^4*i*j^3 + (e^5*i*j^3 + d*e^4*j^4)*x), x))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b f x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + a f x^{3} +{\left (b g x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + a g x^{3}\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(x^3*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")

[Out]

integral(b*f*x^3*log((e*x + d)^n*c) + a*f*x^3 + (b*g*x^3*log((e*x + d)^n*c) + a*g*x^3)*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(x**3*(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 )} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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^3, x)