\(\int x^2 (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [387]

Optimal result

Integrand size = 32, antiderivative size = 558 \[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-\frac {a g i^2 m x}{3 j^2}-\frac {b d^2 f n x}{3 e^2}+\frac {4 b d^2 g m n x}{9 e^2}+\frac {4 b g i^2 m n x}{9 j^2}+\frac {b d g i m n x}{3 e j}-\frac {5 b d g m n x^2}{36 e}-\frac {5 b g i m n x^2}{36 j}+\frac {2}{27} b g m n x^3-\frac {b d^3 g m n \log (d+e x)}{9 e^3}-\frac {b d^2 g i m n \log (d+e x)}{6 e^2 j}-\frac {b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\frac {g i m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b g i^3 m n \log (i+j x)}{9 j^3}-\frac {b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac {g i^3 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{3 j^3}-\frac {b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i^3 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{3 j^3}+\frac {b d^3 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{3 e^3} \]

[Out]

-1/3*a*g*i^2*m*x/j^2-1/3*b*d^2*f*n*x/e^2+4/9*b*d^2*g*m*n*x/e^2+4/9*b*g*i^2*m*n*x/j^2+1/3*b*d*g*i*m*n*x/e/j-5/3
6*b*d*g*m*n*x^2/e-5/36*b*g*i*m*n*x^2/j+2/27*b*g*m*n*x^3-1/9*b*d^3*g*m*n*ln(e*x+d)/e^3-1/6*b*d^2*g*i*m*n*ln(e*x
+d)/e^2/j-1/3*b*g*i^2*m*(e*x+d)*ln(c*(e*x+d)^n)/e/j^2+1/6*g*i*m*x^2*(a+b*ln(c*(e*x+d)^n))/j-1/9*g*m*x^3*(a+b*l
n(c*(e*x+d)^n))-1/9*b*g*i^3*m*n*ln(j*x+i)/j^3-1/6*b*d*g*i^2*m*n*ln(j*x+i)/e/j^2+1/3*g*i^3*m*(a+b*ln(c*(e*x+d)^
n))*ln(e*(j*x+i)/(-d*j+e*i))/j^3-1/3*b*d^2*g*n*(j*x+i)*ln(h*(j*x+i)^m)/e^2/j+1/6*b*d*n*x^2*(f+g*ln(h*(j*x+i)^m
))/e-1/9*b*n*x^3*(f+g*ln(h*(j*x+i)^m))+1/3*b*d^3*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^3+1/3*x^3
*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))+1/3*b*g*i^3*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^3+1/3*b*d^3*g*
m*n*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^3

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2489, 45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {g i^3 m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 j^3}+\frac {g i m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{9} g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {a g i^2 m x}{3 j^2}-\frac {b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\frac {b d^3 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3}+\frac {b d^3 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{3 e^3}-\frac {b d^3 g m n \log (d+e x)}{9 e^3}-\frac {b d^2 f n x}{3 e^2}-\frac {b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}-\frac {b d^2 g i m n \log (d+e x)}{6 e^2 j}+\frac {4 b d^2 g m n x}{9 e^2}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}+\frac {b g i^3 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{3 j^3}-\frac {b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac {b d g i m n x}{3 e j}-\frac {5 b d g m n x^2}{36 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^3 m n \log (i+j x)}{9 j^3}+\frac {4 b g i^2 m n x}{9 j^2}-\frac {5 b g i m n x^2}{36 j}+\frac {2}{27} b g m n x^3 \]

[In]

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

[Out]

-1/3*(a*g*i^2*m*x)/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 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}{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 {1}{3} (g j m) \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x} \, dx-\frac {1}{3} (b e n) \int \frac {x^3 \left (f+g \log \left (h (i+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 (i+j x)^m\right )\right )-\frac {1}{3} (g j m) \int \left (\frac {i^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac {i 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 {i^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3 (i+j x)}\right ) \, dx-\frac {1}{3} (b e n) \int \left (\frac {d^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^3}-\frac {d x \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^2}+\frac {x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}-\frac {d^3 \left (f+g \log \left (h (i+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 (i+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 {\left (g i^2 m\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{3 j^2}+\frac {\left (g i^3 m\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{i+j x} \, dx}{3 j^2}+\frac {(g i m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{3 j}-\frac {1}{3} (b n) \int x^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx-\frac {\left (b d^2 n\right ) \int \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx}{3 e^2}+\frac {\left (b d^3 n\right ) \int \frac {f+g \log \left (h (i+j x)^m\right )}{d+e x} \, dx}{3 e^2}+\frac {(b d n) \int x \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx}{3 e} \\ & = -\frac {a g i^2 m x}{3 j^2}-\frac {b d^2 f n x}{3 e^2}+\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 {g i^3 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 )}{3 j^3}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\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 {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 {\left (b g i^2 m\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{3 j^2}-\frac {\left (b d^2 g n\right ) \int \log \left (h (i+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 {\left (b e g i^3 m n\right ) \int \frac {\log \left (\frac {e (i+j x)}{e i-d j}\right )}{d+e x} \, dx}{3 j^3}-\frac {(b e g i m n) \int \frac {x^2}{d+e x} \, dx}{6 j}+\frac {1}{9} (b g j m n) \int \frac {x^3}{i+j x} \, dx-\frac {\left (b d^3 g j m n\right ) \int \frac {\log \left (\frac {j (d+e x)}{-e i+d j}\right )}{i+j x} \, dx}{3 e^3}-\frac {(b d g j m n) \int \frac {x^2}{i+j x} \, dx}{6 e} \\ & = -\frac {a g i^2 m x}{3 j^2}-\frac {b d^2 f n x}{3 e^2}+\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 {g i^3 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 )}{3 j^3}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\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 {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 {\left (b g i^2 m\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{3 e j^2}-\frac {\left (b d^2 g n\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,i+j x\right )}{3 e^2 j}-\frac {\left (b d^3 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 )}{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 {\left (b g i^3 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 )}{3 j^3}-\frac {(b e g i m n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{6 j}+\frac {1}{9} (b g j m n) \int \left (\frac {i^2}{j^3}-\frac {i x}{j^2}+\frac {x^2}{j}-\frac {i^3}{j^3 (i+j x)}\right ) \, dx-\frac {(b d g j m n) \int \left (-\frac {i}{j^2}+\frac {x}{j}+\frac {i^2}{j^2 (i+j x)}\right ) \, dx}{6 e} \\ & = -\frac {a g i^2 m x}{3 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 {4 b g i^2 m n x}{9 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 {5 b g i m n x^2}{36 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 {b d^2 g i m n \log (d+e x)}{6 e^2 j}-\frac {b g i^2 m (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}+\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 {b g i^3 m n \log (i+j x)}{9 j^3}-\frac {b d g i^2 m n \log (i+j x)}{6 e j^2}+\frac {g i^3 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 )}{3 j^3}-\frac {b d^2 g n (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}+\frac {b d n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e}-\frac {1}{9} b n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\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 {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 {b g i^3 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{3 j^3}+\frac {b d^3 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{3 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 492, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {6 b n \log (d+e x) \left (-6 e^3 g i^3 m \log (i+j x)+6 g \left (e^3 i^3-d^3 j^3\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (-6 e^2 g i^2 m-3 d e g i j m+2 d^2 j^2 (3 f-g m)+6 d^2 g j^2 \log \left (h (i+j x)^m\right )\right )\right )+e \left (6 g i m \left (6 a e^2 i^2-b \left (2 e^2 i^2+3 d e i j+6 d^2 j^2\right ) n\right ) \log (i+j x)+6 b e^2 \log \left (c (d+e x)^n\right ) \left (6 f j^3 x^3+g j m x \left (-6 i^2+3 i j x-2 j^2 x^2\right )+6 g i^3 m \log (i+j x)+6 g j^3 x^3 \log \left (h (i+j x)^m\right )\right )+j \left (6 a e^2 x \left (6 f j^2 x^2+g m \left (-6 i^2+3 i j x-2 j^2 x^2\right )\right )+b n \left (12 d^2 j^2 (-3 f+4 g m) x+3 d e \left (6 f j^2 x^2+g m \left (12 i^2+12 i j x-5 j^2 x^2\right )\right )+e^2 x \left (-12 f j^2 x^2+g m \left (48 i^2-15 i j x+8 j^2 x^2\right )\right )\right )-6 g j^2 x \left (-6 a e^2 x^2+b n \left (6 d^2-3 d e x+2 e^2 x^2\right )\right ) \log \left (h (i+j x)^m\right )\right )\right )+36 b g \left (e^3 i^3-d^3 j^3\right ) m n \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{108 e^3 j^3} \]

[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] (warning: unable to verify)

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

Time = 234.84 (sec) , antiderivative size = 1724, normalized size of antiderivative = 3.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(1724\)

[In]

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

[Out]

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

Fricas [F]

\[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{2} \,d x } \]

[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)]

Timed out. \[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{2} \,d x } \]

[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))

Giac [F]

\[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \]

[In]

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

[Out]

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