3.387 \(\int x^2 (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\)

Optimal. Leaf size=558 \[ \frac{b d^3 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{3 e^3}+\frac{b g i^3 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{3 j^3}+\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{g i^3 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 j^3}+\frac{g i m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac{1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{a g i^2 m x}{3 j^2}-\frac{b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\frac{b d^3 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3}-\frac{b d^2 f n x}{3 e^2}-\frac{b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}-\frac{b d^2 g i m n \log (d+e x)}{6 e^2 j}+\frac{4 b d^2 g m n x}{9 e^2}-\frac{b d^3 g m n \log (d+e x)}{9 e^3}+\frac{b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac{b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac{b d g i m n x}{3 e j}-\frac{5 b d g m n x^2}{36 e}-\frac{1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{4 b g i^2 m n x}{9 j^2}-\frac{b g i^3 m n \log (i+j x)}{9 j^3}-\frac{5 b g i m n x^2}{36 j}+\frac{2}{27} b g m n x^3 \]

[Out]

-(a*g*i^2*m*x)/(3*j^2) - (b*d^2*f*n*x)/(3*e^2) + (4*b*d^2*g*m*n*x)/(9*e^2) + (4*b*g*i^2*m*n*x)/(9*j^2) + (b*d*
g*i*m*n*x)/(3*e*j) - (5*b*d*g*m*n*x^2)/(36*e) - (5*b*g*i*m*n*x^2)/(36*j) + (2*b*g*m*n*x^3)/27 - (b*d^3*g*m*n*L
og[d + e*x])/(9*e^3) - (b*d^2*g*i*m*n*Log[d + e*x])/(6*e^2*j) - (b*g*i^2*m*(d + e*x)*Log[c*(d + e*x)^n])/(3*e*
j^2) + (g*i*m*x^2*(a + b*Log[c*(d + e*x)^n]))/(6*j) - (g*m*x^3*(a + b*Log[c*(d + e*x)^n]))/9 - (b*g*i^3*m*n*Lo
g[i + j*x])/(9*j^3) - (b*d*g*i^2*m*n*Log[i + j*x])/(6*e*j^2) + (g*i^3*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i +
j*x))/(e*i - d*j)])/(3*j^3) - (b*d^2*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(3*e^2*j) + (b*d*n*x^2*(f + g*Log[h*(i
+ j*x)^m]))/(6*e) - (b*n*x^3*(f + g*Log[h*(i + j*x)^m]))/9 + (b*d^3*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f +
g*Log[h*(i + j*x)^m]))/(3*e^3) + (x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/3 + (b*g*i^3*m*n*
PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) + (b*d^3*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(3*e^3
)

________________________________________________________________________________________

Rubi [A]  time = 0.609394, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 29, 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} \[ \frac{b d^3 g m n \text{PolyLog}\left (2,\frac{e (i+j x)}{e i-d j}\right )}{3 e^3}+\frac{b g i^3 m n \text{PolyLog}\left (2,-\frac{j (d+e x)}{e i-d j}\right )}{3 j^3}+\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{g i^3 m \log \left (\frac{e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 j^3}+\frac{g i m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac{1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{a g i^2 m x}{3 j^2}-\frac{b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\frac{b d^3 n \log \left (-\frac{j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3}-\frac{b d^2 f n x}{3 e^2}-\frac{b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}-\frac{b d^2 g i m n \log (d+e x)}{6 e^2 j}+\frac{4 b d^2 g m n x}{9 e^2}-\frac{b d^3 g m n \log (d+e x)}{9 e^3}+\frac{b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac{b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac{b d g i m n x}{3 e j}-\frac{5 b d g m n x^2}{36 e}-\frac{1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac{4 b g i^2 m n x}{9 j^2}-\frac{b g i^3 m n \log (i+j x)}{9 j^3}-\frac{5 b g i m n x^2}{36 j}+\frac{2}{27} b g m n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*g*i^2*m*x)/(3*j^2) - (b*d^2*f*n*x)/(3*e^2) + (4*b*d^2*g*m*n*x)/(9*e^2) + (4*b*g*i^2*m*n*x)/(9*j^2) + (b*d*
g*i*m*n*x)/(3*e*j) - (5*b*d*g*m*n*x^2)/(36*e) - (5*b*g*i*m*n*x^2)/(36*j) + (2*b*g*m*n*x^3)/27 - (b*d^3*g*m*n*L
og[d + e*x])/(9*e^3) - (b*d^2*g*i*m*n*Log[d + e*x])/(6*e^2*j) - (b*g*i^2*m*(d + e*x)*Log[c*(d + e*x)^n])/(3*e*
j^2) + (g*i*m*x^2*(a + b*Log[c*(d + e*x)^n]))/(6*j) - (g*m*x^3*(a + b*Log[c*(d + e*x)^n]))/9 - (b*g*i^3*m*n*Lo
g[i + j*x])/(9*j^3) - (b*d*g*i^2*m*n*Log[i + j*x])/(6*e*j^2) + (g*i^3*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i +
j*x))/(e*i - d*j)])/(3*j^3) - (b*d^2*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(3*e^2*j) + (b*d*n*x^2*(f + g*Log[h*(i
+ j*x)^m]))/(6*e) - (b*n*x^3*(f + g*Log[h*(i + j*x)^m]))/9 + (b*d^3*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f +
g*Log[h*(i + j*x)^m]))/(3*e^3) + (x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/3 + (b*g*i^3*m*n*
PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) + (b*d^3*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(3*e^3
)

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^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac{1}{3} (g j m) \int \frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{387+j x} \, dx-\frac{1}{3} (b e n) \int \frac{x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac{1}{3} (g j m) \int \left (\frac{149769 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac{387 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}-\frac{57960603 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3 (387+j x)}\right ) \, dx-\frac{1}{3} (b e n) \int \left (\frac{d^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{e^3}-\frac{d x \left (f+g \log \left (h (387+j x)^m\right )\right )}{e^2}+\frac{x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{e}-\frac{d^3 \left (f+g \log \left (h (387+j x)^m\right )\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac{1}{3} (g m) \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-\frac{(49923 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j^2}+\frac{(19320201 g m) \int \frac{a+b \log \left (c (d+e x)^n\right )}{387+j x} \, dx}{j^2}+\frac{(129 g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j}-\frac{1}{3} (b n) \int x^2 \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx-\frac{\left (b d^2 n\right ) \int \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx}{3 e^2}+\frac{\left (b d^3 n\right ) \int \frac{f+g \log \left (h (387+j x)^m\right )}{d+e x} \, dx}{3 e^2}+\frac{(b d n) \int x \left (f+g \log \left (h (387+j x)^m\right )\right ) \, dx}{3 e}\\ &=-\frac{49923 a g m x}{j^2}-\frac{b d^2 f n x}{3 e^2}+\frac{129 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac{1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{19320201 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (387+j x)}{387 e-d j}\right )}{j^3}+\frac{b d n x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{6 e}-\frac{1}{9} b n x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac{b d^3 n \log \left (-\frac{j (d+e x)}{387 e-d j}\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac{(49923 b g m) \int \log \left (c (d+e x)^n\right ) \, dx}{j^2}-\frac{\left (b d^2 g n\right ) \int \log \left (h (387+j x)^m\right ) \, dx}{3 e^2}+\frac{1}{9} (b e g m n) \int \frac{x^3}{d+e x} \, dx-\frac{(19320201 b e g m n) \int \frac{\log \left (\frac{e (387+j x)}{387 e-d j}\right )}{d+e x} \, dx}{j^3}-\frac{(129 b e g m n) \int \frac{x^2}{d+e x} \, dx}{2 j}+\frac{1}{9} (b g j m n) \int \frac{x^3}{387+j x} \, dx-\frac{\left (b d^3 g j m n\right ) \int \frac{\log \left (\frac{j (d+e x)}{-387 e+d j}\right )}{387+j x} \, dx}{3 e^3}-\frac{(b d g j m n) \int \frac{x^2}{387+j x} \, dx}{6 e}\\ &=-\frac{49923 a g m x}{j^2}-\frac{b d^2 f n x}{3 e^2}+\frac{129 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac{1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{19320201 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (387+j x)}{387 e-d j}\right )}{j^3}+\frac{b d n x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{6 e}-\frac{1}{9} b n x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac{b d^3 n \log \left (-\frac{j (d+e x)}{387 e-d j}\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )-\frac{(49923 b g m) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j^2}-\frac{\left (b d^2 g n\right ) \operatorname{Subst}\left (\int \log \left (h x^m\right ) \, dx,x,387+j x\right )}{3 e^2 j}-\frac{\left (b d^3 g m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{-387 e+d j}\right )}{x} \, dx,x,387+j x\right )}{3 e^3}+\frac{1}{9} (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-\frac{(19320201 b g m n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{387 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^3}-\frac{(129 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}+\frac{1}{9} (b g j m n) \int \left (\frac{149769}{j^3}-\frac{387 x}{j^2}+\frac{x^2}{j}-\frac{57960603}{j^3 (387+j x)}\right ) \, dx-\frac{(b d g j m n) \int \left (-\frac{387}{j^2}+\frac{x}{j}+\frac{149769}{j^2 (387+j x)}\right ) \, dx}{6 e}\\ &=-\frac{49923 a g m x}{j^2}-\frac{b d^2 f n x}{3 e^2}+\frac{4 b d^2 g m n x}{9 e^2}+\frac{66564 b g m n x}{j^2}+\frac{129 b d g m n x}{e j}-\frac{5 b d g m n x^2}{36 e}-\frac{215 b g m n x^2}{4 j}+\frac{2}{27} b g m n x^3-\frac{b d^3 g m n \log (d+e x)}{9 e^3}-\frac{129 b d^2 g m n \log (d+e x)}{2 e^2 j}-\frac{49923 b g m (d+e x) \log \left (c (d+e x)^n\right )}{e j^2}+\frac{129 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac{1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{6440067 b g m n \log (387+j x)}{j^3}-\frac{49923 b d g m n \log (387+j x)}{2 e j^2}+\frac{19320201 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (387+j x)}{387 e-d j}\right )}{j^3}-\frac{b d^2 g n (387+j x) \log \left (h (387+j x)^m\right )}{3 e^2 j}+\frac{b d n x^2 \left (f+g \log \left (h (387+j x)^m\right )\right )}{6 e}-\frac{1}{9} b n x^3 \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac{b d^3 n \log \left (-\frac{j (d+e x)}{387 e-d j}\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (387+j x)^m\right )\right )+\frac{19320201 b g m n \text{Li}_2\left (-\frac{j (d+e x)}{387 e-d j}\right )}{j^3}+\frac{b d^3 g m n \text{Li}_2\left (\frac{e (387+j x)}{387 e-d j}\right )}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.921902, size = 492, normalized size = 0.88 \[ \frac{36 b g m n \left (e^3 i^3-d^3 j^3\right ) \text{PolyLog}\left (2,\frac{j (d+e x)}{d j-e i}\right )+e \left (j \left (-6 g j^2 x \left (b n \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 a e^2 x^2\right ) \log \left (h (i+j x)^m\right )+6 a e^2 x \left (6 f j^2 x^2+g m \left (-6 i^2+3 i j x-2 j^2 x^2\right )\right )+b n \left (12 d^2 j^2 x (4 g m-3 f)+3 d e \left (6 f j^2 x^2+g m \left (12 i^2+12 i j x-5 j^2 x^2\right )\right )+e^2 x \left (g m \left (48 i^2-15 i j x+8 j^2 x^2\right )-12 f j^2 x^2\right )\right )\right )+6 g i m \log (i+j x) \left (6 a e^2 i^2-b n \left (6 d^2 j^2+3 d e i j+2 e^2 i^2\right )\right )+6 b e^2 \log \left (c (d+e x)^n\right ) \left (6 f j^3 x^3+6 g j^3 x^3 \log \left (h (i+j x)^m\right )+g j m x \left (-6 i^2+3 i j x-2 j^2 x^2\right )+6 g i^3 m \log (i+j x)\right )\right )+6 b n \log (d+e x) \left (d j \left (2 d^2 j^2 (3 f-g m)+6 d^2 g j^2 \log \left (h (i+j x)^m\right )-3 d e g i j m-6 e^2 g i^2 m\right )+6 g m \left (e^3 i^3-d^3 j^3\right ) \log \left (\frac{e (i+j x)}{e i-d j}\right )-6 e^3 g i^3 m \log (i+j x)\right )}{108 e^3 j^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(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]*(-6*e^3*g*i^3*m*Log[i + j*x] + 6*g*(e^3*i^3 - d^3*j^3)*m*Log[(e*(i + j*x))/(e*i - d*j)] +
d*j*(-6*e^2*g*i^2*m - 3*d*e*g*i*j*m + 2*d^2*j^2*(3*f - g*m) + 6*d^2*g*j^2*Log[h*(i + j*x)^m])) + e*(6*g*i*m*(6
*a*e^2*i^2 - b*(2*e^2*i^2 + 3*d*e*i*j + 6*d^2*j^2)*n)*Log[i + j*x] + 6*b*e^2*Log[c*(d + e*x)^n]*(6*f*j^3*x^3 +
g*j*m*x*(-6*i^2 + 3*i*j*x - 2*j^2*x^2) + 6*g*i^3*m*Log[i + j*x] + 6*g*j^3*x^3*Log[h*(i + j*x)^m]) + j*(6*a*e^
2*x*(6*f*j^2*x^2 + g*m*(-6*i^2 + 3*i*j*x - 2*j^2*x^2)) + b*n*(12*d^2*j^2*(-3*f + 4*g*m)*x + 3*d*e*(6*f*j^2*x^2
+ g*m*(12*i^2 + 12*i*j*x - 5*j^2*x^2)) + e^2*x*(-12*f*j^2*x^2 + g*m*(48*i^2 - 15*i*j*x + 8*j^2*x^2))) - 6*g*j
^2*x*(-6*a*e^2*x^2 + b*n*(6*d^2 - 3*d*e*x + 2*e^2*x^2))*Log[h*(i + j*x)^m])) + 36*b*g*(e^3*i^3 - d^3*j^3)*m*n*
PolyLog[2, (j*(d + e*x))/(-(e*i) + d*j)])/(108*e^3*j^3)

________________________________________________________________________________________

Maple [C]  time = 1.842, size = 3680, normalized size = 6.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*f*x^3-1/6*I*Pi*x^3*a*g*csgn(I*h*(j*x+i)^m)^3-1/9*n*b*f*x^3+1/6*I*Pi*b*f*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n
)^2+1/6*I*Pi*b*f*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/9*ln(h)*x^3*b*g*n+1/3*ln(c)*ln(h)*x^3*b*g-1/9*l
n(c)*x^3*b*g*m+1/3*a*g*x^3*ln((j*x+i)^m)+1/3*ln(c)*b*f*x^3-1/12*I/j*Pi*x^2*b*g*i*m*csgn(I*c)*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)+1/3*ln(h)*x^3*a*g-1/3/e^3*b*d^3*g*m*n*ln(e*x+d)*ln(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))+1/6*I*
ln(c)*Pi*x^3*b*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/6*I*Pi*ln(h)*x^3*b*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^
2-1/6*I*Pi*x^3*a*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/18*I*Pi*x^3*b*g*n*csgn(I*(j*x+i)^m)*csgn(
I*h*(j*x+i)^m)^2-1/6*I*Pi*b*f*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3/j^3*b*g*i^3*m*n*dilog(((
j*x+i)*e+d*j-e*i)/(d*j-e*i))+1/3*a*g/j^3*m*i^3*ln(j*x+i)-1/9*x^3*a*g*m+1/6/j*x^2*a*g*i*m-1/3/e^2*n*b*g*ln((j*x
+i)^m)*x*d^2+1/6*I*Pi*ln(h)*x^3*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/3*b*d*g*i*m*n*x/e/j+1/18*I*Pi*x^
3*b*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/6*I*Pi*ln(h)*x^3*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csg
n(I*c*(e*x+d)^n)-1/6*I*ln(c)*Pi*x^3*b*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-1/6*I*b*Pi*csgn(I*c)*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g*x^3*ln((j*x+i)^m)+1/6/e*b*d*f*n*x^2+1/3/e^3*ln(e*x+d)*b*d^3*f*n-1/6*I*P
i*b*f*x^3*csgn(I*c*(e*x+d)^n)^3+2/9/e^2/j*g*i*m*b*d^2*n+1/6*I*ln(c)*Pi*x^3*b*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2
-1/18*I*Pi*x^3*b*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+2/27*b*g*m*n*x^3-1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*
csgn(I*c*(e*x+d)^n)*x^3*g*csgn(I*h*(j*x+i)^m)^3-1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h)*cs
gn(I*h*(j*x+i)^m)^2-1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-
1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/12*b*Pi^2*csgn(I*(
e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/3/j^3*b*g*i^3*m*n*ln(j*x+i)*ln
(((j*x+i)*e+d*j-e*i)/(d*j-e*i))+(1/3*g*b*x^3*ln((j*x+i)^m)+1/18*b*(-3*I*Pi*g*j^3*x^3*csgn(I*h)*csgn(I*(j*x+i)^
m)*csgn(I*h*(j*x+i)^m)+3*I*Pi*g*j^3*x^3*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+3*I*Pi*g*j^3*x^3*csgn(I*(j*x+i)^m)*csg
n(I*h*(j*x+i)^m)^2-3*I*Pi*g*j^3*x^3*csgn(I*h*(j*x+i)^m)^3+6*ln(h)*g*j^3*x^3-2*g*j^3*m*x^3+6*f*j^3*x^3+3*g*i*j^
2*m*x^2+6*g*i^3*m*ln(j*x+i)-6*g*i^2*j*m*x)/j^3)*ln((e*x+d)^n)-1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c
*(e*x+d)^n)*x^3*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/6*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d
)^n)^2*g/j^3*m*i^3*ln(j*x+i)+49/108/e^3*b*d^3*g*m*n-1/18*I*Pi*x^3*b*g*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/18*I
*Pi*x^3*b*g*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/3/e/j^2*ln(e*x+d)*b*d*g*i^2*m*n-1/6/e/j^2*g*i^2*m*ln((
e*x+d)*j-d*j+e*i)*b*d*n-1/3/e^2/j*g*i*m*ln((e*x+d)*j-d*j+e*i)*b*d^2*n+1/12*I/j*Pi*x^2*b*g*i*m*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)^2-1/6*I/j^2*Pi*x*b*g*i^2*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/6*I/j^2*Pi*x*b*g*i^
2*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/6*I/j^2*Pi*x*b*g*i^2*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1
/6*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g/j^3*m*i^3*ln(j*x+i)-1/6*b*d^2*g*i*m*n*ln(e*x+d)/e^
2/j-1/3/j^2*ln(c)*x*b*g*i^2*m-1/9/j^3*g*i^3*m*ln((e*x+d)*j-d*j+e*i)*b*n-1/3/e^3*b*d^3*g*m*n*dilog(((e*x+d)*j-d
*j+e*i)/(-d*j+e*i))-1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*csgn(I*h*(j*x+i)^m)^3+1/3/e^3*ln(e*x+d)*ln(h)*b*d^
3*g*n+1/6/e*ln(h)*x^2*b*d*g*n-1/3/e^2*ln(h)*x*b*d^2*g*n+1/6/j*ln(c)*x^2*b*g*i*m+1/3*b*ln(c)*g/j^3*m*i^3*ln(j*x
+i)-1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g/j^3*m*i^3*ln(j*x+i)-1/9*n*b*g*ln((j*x+i)^m)*x^3+1/3*b*ln(c)*g*x^3*ln((j
*x+i)^m)+1/6*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g*x^3*ln((j*x+i)^m)+1/6*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*
(e*x+d)^n)^2*g*x^3*ln((j*x+i)^m)+1/6*I*Pi*x^3*a*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/6*I*Pi*x^3*a*g*csgn(I*(j*x
+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x
+i)^m)+1/6*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/6*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(
I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/12*I/e*Pi*x^2*b*d*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/12*I/j*Pi*x^2*b*g
*i*m*csgn(I*c*(e*x+d)^n)^3+1/6*I/j^2*Pi*x*b*g*i^2*m*csgn(I*c*(e*x+d)^n)^3+1/18*I*Pi*x^3*b*g*m*csgn(I*c)*csgn(I
*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/9/e/j^2*b*d*g*i^2*m*n+1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^
3*g*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)*x^3*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*x^3*g*csgn
(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h)*csgn(I*(j*x+i)
^m)*csgn(I*h*(j*x+i)^m)-1/3*a*g*i^2*m*x/j^2-1/3*b*d^2*f*n*x/e^2-1/12*I/e*Pi*x^2*b*d*g*n*csgn(I*h*(j*x+i)^m)^3+
1/6*I/e^2*Pi*x*b*d^2*g*n*csgn(I*h*(j*x+i)^m)^3-1/6*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*h*(j*x+i)^m)^3-1/6*I*ln
(c)*Pi*x^3*b*g*csgn(I*h*(j*x+i)^m)^3-1/6*I*Pi*ln(h)*x^3*b*g*csgn(I*c*(e*x+d)^n)^3+1/18*I*Pi*x^3*b*g*m*csgn(I*c
*(e*x+d)^n)^3+1/18*I*Pi*x^3*b*g*n*csgn(I*h*(j*x+i)^m)^3-1/6*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*h)*csgn(I*(j*x
+i)^m)*csgn(I*h*(j*x+i)^m)-1/12*I/e*Pi*x^2*b*d*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/6*I/e^2*P
i*x*b*d^2*g*n*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+1/3/e^3*n*b*g*ln((j*x+i)^m)*d^3*ln(e*x+d)+1/6/e*
n*b*g*ln((j*x+i)^m)*x^2*d-1/6*I/e^2*Pi*x*b*d^2*g*n*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/6*I*b*Pi*csgn(I*c)*csgn(I
*c*(e*x+d)^n)^2*g/j^3*m*i^3*ln(j*x+i)+1/12*I/e*Pi*x^2*b*d*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/6*I/e^
2*Pi*x*b*d^2*g*n*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/12*I/j*Pi*x^2*b*g*i*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)
^2-1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g*x^3*ln((j*x+i)^m)+1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*csgn(I*(j*x+i)
^m)*csgn(I*h*(j*x+i)^m)^2+1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h*(j*x+i)^m)^3+1/12*b*Pi^2*
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*x^3*g*csgn(I*h*(j*x+i)^m)^3+1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3*x^3*g*cs
gn(I*h)*csgn(I*h*(j*x+i)^m)^2+4/9*b*d^2*g*m*n*x/e^2+4/9*b*g*i^2*m*n*x/j^2-5/36*b*d*g*m*n*x^2/e-5/36*b*g*i*m*n*
x^2/j-1/9*b*d^3*g*m*n*ln(e*x+d)/e^3

________________________________________________________________________________________

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^2*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

1/3*b*f*x^3*log((e*x + d)^n*c) + 1/3*a*g*x^3*log((j*x + i)^m*h) + 1/3*a*f*x^3 + 1/18*b*e*f*n*(6*d^3*log(e*x +
d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + 1/18*a*g*j*m*(6*i^3*log(j*x + i)/j^4 - (2*j^2*x^3 - 3*i*j*x^
2 + 6*i^2*x)/j^3) - 1/18*b*g*((6*e^3*i^3*m*n*log(e*x + d)*log(j*x + i) - (3*e^3*i*j^2*m*x^2 - 6*e^3*i^2*j*m*x
+ 6*e^3*i^3*m*log(j*x + i) - 2*(j^3*m - 3*j^3*log(h))*e^3*x^3)*log((e*x + d)^n) - (6*e^3*j^3*x^3*log((e*x + d)
^n) + 3*d*e^2*j^3*n*x^2 - 6*d^2*e*j^3*n*x + 6*d^3*j^3*n*log(e*x + d) - 2*(e^3*j^3*n - 3*e^3*j^3*log(c))*x^3)*l
og((j*x + i)^m))/(e^3*j^3) + 18*integrate(1/18*(2*(3*(j^3*m - 3*j^3*log(h))*e^4*log(c) - (2*j^3*m*n - 3*j^3*n*
log(h))*e^4)*x^4 + (d*e^3*j^3*m*n + (i*j^2*m*n + 6*i*j^2*n*log(h))*e^4 - 6*(3*e^4*i*j^2*log(h) - (j^3*m - 3*j^
3*log(h))*d*e^3)*log(c))*x^3 - 3*(e^4*i^2*j*m*n + d^2*e^2*j^3*m*n + 6*d*e^3*i*j^2*log(c)*log(h))*x^2 - 6*(e^4*
i^3*m*n + d^3*e*j^3*m*n)*x - 6*(d*e^3*i^3*m*n - d^4*j^3*m*n + (e^4*i^3*m*n - d^3*e*j^3*m*n)*x)*log(e*x + d))/(
e^4*j^3*x^2 + d*e^3*i*j^2 + (e^4*i*j^2 + d*e^3*j^3)*x), x))

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b f x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a f x^{2} +{\left (b g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a g x^{2}\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^2*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(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^2, x)