\(\int x^3 (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [386]

Optimal result

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

[Out]

-1/4*b*g*i^4*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^4-1/4*b*d^4*g*m*n*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^4+1/16
*b*d^4*g*m*n*ln(e*x+d)/e^4+1/16*b*g*i^4*m*n*ln(j*x+i)/j^4+3/32*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-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+1/32*b*g*m*n*x^4+1/4*a
*g*i^3*m*x/j^3+1/4*b*d^3*f*n*x/e^3-5/24*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+1/8*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+1/12*b*d*g*i^3*m*n*ln(j*x+i)/e/j^
3+1/8*b*d^2*g*i^2*m*n*ln(j*x+i)/e^2/j^2+1/4*b*g*i^3*m*(e*x+d)*ln(c*(e*x+d)^n)/e/j^3+1/4*b*d^3*g*n*(j*x+i)*ln(h
*(j*x+i)^m)/e^3/j-1/16*g*m*x^4*(a+b*ln(c*(e*x+d)^n))-1/16*b*n*x^4*(f+g*ln(h*(j*x+i)^m))+1/4*x^4*(a+b*ln(c*(e*x
+d)^n))*(f+g*ln(h*(j*x+i)^m))-1/8*g*i^2*m*x^2*(a+b*ln(c*(e*x+d)^n))/j^2+1/12*g*i*m*x^3*(a+b*ln(c*(e*x+d)^n))/j
-1/4*g*i^4*m*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j^4-1/8*b*d^2*n*x^2*(f+g*ln(h*(j*x+i)^m))/e^2+1/12
*b*d*n*x^3*(f+g*ln(h*(j*x+i)^m))/e-1/4*b*d^4*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^4

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.00, number of steps used = 35, 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} \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {g i^4 m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 j^4}-\frac {g i^2 m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 j^2}+\frac {g i m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{12 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {a g i^3 m x}{4 j^3}+\frac {b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e^4}-\frac {b d^4 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{4 e^4}+\frac {b d^4 g m n \log (d+e x)}{16 e^4}+\frac {b d^3 f n x}{4 e^3}+\frac {b d^3 g n (i+j x) \log \left (h (i+j x)^m\right )}{4 e^3 j}+\frac {b d^3 g i m n \log (d+e x)}{12 e^3 j}-\frac {5 b d^3 g m n x}{16 e^3}-\frac {b d^2 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}+\frac {b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac {b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac {5 b d^2 g i m n x}{24 e^2 j}+\frac {3 b d^2 g m n x^2}{32 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac {b g i^4 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{4 j^4}+\frac {b d g i^3 m n \log (i+j x)}{12 e j^3}-\frac {5 b d g i^2 m n x}{24 e j^2}+\frac {b d g i m n x^2}{12 e j}-\frac {7 b d g m n x^3}{144 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i^4 m n \log (i+j x)}{16 j^4}-\frac {5 b g i^3 m n x}{16 j^3}+\frac {3 b g i^2 m n x^2}{32 j^2}-\frac {7 b g i m n x^3}{144 j}+\frac {1}{32} b g m n x^4 \]

[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 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}{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 {1}{4} (g j m) \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x} \, dx-\frac {1}{4} (b e n) \int \frac {x^4 \left (f+g \log \left (h (i+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 (i+j x)^m\right )\right )-\frac {1}{4} (g j m) \int \left (-\frac {i^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^4}+\frac {i^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac {i 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 {i^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^4 (i+j x)}\right ) \, dx-\frac {1}{4} (b e n) \int \left (-\frac {d^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^4}+\frac {d^2 x \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^3}-\frac {d x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^2}+\frac {x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}+\frac {d^4 \left (f+g \log \left (h (i+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 (i+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 {\left (g i^3 m\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{4 j^3}-\frac {\left (g i^4 m\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{i+j x} \, dx}{4 j^3}-\frac {\left (g i^2 m\right ) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{4 j^2}+\frac {(g i m) \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{4 j}-\frac {1}{4} (b n) \int x^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx+\frac {\left (b d^3 n\right ) \int \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx}{4 e^3}-\frac {\left (b d^4 n\right ) \int \frac {f+g \log \left (h (i+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 (i+j x)^m\right )\right ) \, dx}{4 e^2}+\frac {(b d n) \int x^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx}{4 e} \\ & = \frac {a g i^3 m x}{4 j^3}+\frac {b d^3 f n x}{4 e^3}-\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 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 {g i^4 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 )}{4 j^4}-\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 n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )-\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 {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 {\left (b g i^3 m\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{4 j^3}+\frac {\left (b d^3 g n\right ) \int \log \left (h (i+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 {\left (b e g i^4 m n\right ) \int \frac {\log \left (\frac {e (i+j x)}{e i-d j}\right )}{d+e x} \, dx}{4 j^4}+\frac {\left (b e g i^2 m n\right ) \int \frac {x^2}{d+e x} \, dx}{8 j^2}-\frac {(b e g i m n) \int \frac {x^3}{d+e x} \, dx}{12 j}+\frac {1}{16} (b g j m n) \int \frac {x^4}{i+j x} \, dx+\frac {\left (b d^4 g j m n\right ) \int \frac {\log \left (\frac {j (d+e x)}{-e i+d j}\right )}{i+j x} \, dx}{4 e^4}+\frac {\left (b d^2 g j m n\right ) \int \frac {x^2}{i+j x} \, dx}{8 e^2}-\frac {(b d g j m n) \int \frac {x^3}{i+j x} \, dx}{12 e} \\ & = \frac {a g i^3 m x}{4 j^3}+\frac {b d^3 f n x}{4 e^3}-\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 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 {g i^4 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 )}{4 j^4}-\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 n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )-\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 {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 {\left (b g i^3 m\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{4 e j^3}+\frac {\left (b d^3 g n\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,i+j x\right )}{4 e^3 j}+\frac {\left (b d^4 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 )}{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 {\left (b g i^4 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 )}{4 j^4}+\frac {\left (b e g i^2 m n\right ) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{8 j^2}-\frac {(b e g i 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}{12 j}+\frac {1}{16} (b g j m n) \int \left (-\frac {i^3}{j^4}+\frac {i^2 x}{j^3}-\frac {i x^2}{j^2}+\frac {x^3}{j}+\frac {i^4}{j^4 (i+j x)}\right ) \, dx+\frac {\left (b d^2 g j m n\right ) \int \left (-\frac {i}{j^2}+\frac {x}{j}+\frac {i^2}{j^2 (i+j x)}\right ) \, dx}{8 e^2}-\frac {(b d 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}{12 e} \\ & = \frac {a g i^3 m x}{4 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 {5 b g i^3 m n x}{16 j^3}-\frac {5 b d g i^2 m n x}{24 e j^2}-\frac {5 b d^2 g i m n x}{24 e^2 j}+\frac {3 b d^2 g m n x^2}{32 e^2}+\frac {3 b g i^2 m n x^2}{32 j^2}+\frac {b d g i m n x^2}{12 e j}-\frac {7 b d g m n x^3}{144 e}-\frac {7 b g i m n x^3}{144 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 {b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac {b d^3 g i m n \log (d+e x)}{12 e^3 j}+\frac {b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\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 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 {b g i^4 m n \log (i+j x)}{16 j^4}+\frac {b d g i^3 m n \log (i+j x)}{12 e j^3}+\frac {b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac {g i^4 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 )}{4 j^4}+\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 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )-\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 {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 {b g i^4 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{4 j^4}-\frac {b d^4 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{4 e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 605, normalized size of antiderivative = 0.82 \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {6 b n \log (d+e x) \left (12 e^4 g i^4 m \log (i+j x)-12 g \left (e^4 i^4-d^4 j^4\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (12 e^3 g i^3 m+6 d e^2 g i^2 j m+4 d^2 e g i j^2 m+3 d^3 j^3 (-4 f+g m)-12 d^3 g j^3 \log \left (h (i+j x)^m\right )\right )\right )+e \left (6 g i m \left (-12 a e^3 i^3+b \left (3 e^3 i^3+4 d e^2 i^2 j+6 d^2 e i j^2+12 d^3 j^3\right ) n\right ) \log (i+j x)-6 b e^3 \log \left (c (d+e x)^n\right ) \left (-12 f j^4 x^4+g j m x \left (-12 i^3+6 i^2 j x-4 i j^2 x^2+3 j^3 x^3\right )+12 g i^4 m \log (i+j x)-12 g j^4 x^4 \log \left (h (i+j x)^m\right )\right )+j \left (6 a e^3 x \left (12 f j^3 x^3+g m \left (12 i^3-6 i^2 j x+4 i j^2 x^2-3 j^3 x^3\right )\right )-b n \left (18 d^3 j^3 (-4 f+5 g m) x+3 d^2 e j^2 x (12 f j x+g m (20 i-9 j x))+e^3 x \left (18 f j^3 x^3+g m \left (90 i^3-27 i^2 j x+14 i j^2 x^2-9 j^3 x^3\right )\right )+2 d e^2 \left (-12 f j^3 x^3+g m \left (36 i^3+30 i^2 j x-12 i j^2 x^2+7 j^3 x^3\right )\right )\right )-6 g j^3 x \left (-12 a e^3 x^3+b n \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right ) \log \left (h (i+j x)^m\right )\right )\right )-72 b g \left (e^4 i^4-d^4 j^4\right ) m n \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{288 e^4 j^4} \]

[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] (warning: unable to verify)

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

Time = 0.06 (sec) , antiderivative size = 2094, normalized size of antiderivative = 2.82

\[\text {Expression too large to display}\]

[In]

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

[Out]

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))-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+1/4/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*l
n((e*x+d)*j-d*j+e*i)*b*d^2*n+1/12/e/j^3*g*i^3*m*ln((e*x+d)*j-d*j+e*i)*b*d*n+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))-1/16/e/j^3*b*d*g*i^3*m*n-205/576/e^4*b*d^4*
g*m*n+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))-1/16*ln(h)*x^4*b*g*n-1/16*n*b*g*ln((j*x
+i)^m)*x^4+(1/4*x^4*b*g*ln((j*x+i)^m)-1/48*b*(-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*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+6*I*Pi*g*j^4*x^4*csgn(I*h*(j*x+i)^m)^3-6*I*Pi*g*j^4
*x^4*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-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)+1/32*b*g*m*n*x^4+1/16*I/e^2*n*b*Pi*x^2*d^2*g*cs
gn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)-1/24*I/e*n*b*Pi*x^3*d*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*cs
gn(I*h)+1/4/j^4*b*g*i^4*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))+3/32*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+1/16*b*d^4*g*m*n*ln(e*x+d)/e^4+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/
4/e^3*ln(h)*b*d^3*g*n*x+1/12/e*x^3*b*d*f*n-1/8/e^2*x^2*b*d^2*f*n-1/4/e^4*ln(e*x+d)*b*d^4*f*n+1/4*b*d^3*f*n*x/e
^3-1/16*x^4*b*f*n-1/8*I/e^4*n*b*d^4*ln(e*x+d)*Pi*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/8*I/e^4*n*b*d^4*l
n(e*x+d)*Pi*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-1/16*I/e^2*n*b*Pi*x^2*d^2*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m
)^2-1/16*I/e^2*n*b*Pi*x^2*d^2*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)+1/8*I/e^3*n*b*x*Pi*d^3*g*csgn(I*(j*x+i)^m)*csg
n(I*h*(j*x+i)^m)^2+1/8*I/e^3*n*b*x*Pi*d^3*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)+1/24*I/e*n*b*Pi*x^3*d*g*csgn(I*(j*
x+i)^m)*csgn(I*h*(j*x+i)^m)^2+1/24*I/e*n*b*Pi*x^3*d*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)+1/8*I/e^4*n*b*d^4*ln(e*x
+d)*Pi*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)-1/8*I/e^3*n*b*x*Pi*d^3*g*csgn(I*(j*x+i)^m)*csgn(I*h*(
j*x+i)^m)*csgn(I*h)+1/12/e*n*b*g*ln((j*x+i)^m)*x^3*d-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^3-1/4/e^4*n*b*g*ln((j*x+i)^m)*d^4*ln(e*x+d)-1/4/e^4*n*b*d^4*ln(e*x+d)*ln(h)*g+1/32*I*n*b*Pi*x^4*g*c
sgn(I*h*(j*x+i)^m)^3-5/24*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+1/8*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-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^2-1/32*I*n*b*Pi*x^4*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-1/32*I*n*b*Pi*x^4*g*csgn(I
*h*(j*x+i)^m)^2*csgn(I*h)+1/32*I*n*b*Pi*x^4*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+1/8*I/e^4*n*b*d^
4*ln(e*x+d)*Pi*g*csgn(I*h*(j*x+i)^m)^3-1/24*I/e*n*b*Pi*x^3*d*g*csgn(I*h*(j*x+i)^m)^3+1/16*I/e^2*n*b*Pi*x^2*d^2
*g*csgn(I*h*(j*x+i)^m)^3-1/8*I/e^3*n*b*x*Pi*d^3*g*csgn(I*h*(j*x+i)^m)^3+(-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)*(1/4*(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^4+2*g*(1/4*ln((j*x+i)^m)*x^4-1/4*m*j*(1/j^4*(1/4*x^4*j^3-1/3*x^3*i*j^2+1/2*j*i
^2*x^2-i^3*x)+i^4/j^5*ln(j*x+i))))

Fricas [F]

\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{3} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{3} \,d x } \]

[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))

Giac [F]

\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{3} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int x^3\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \]

[In]

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

[Out]

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