Integrand size = 32, antiderivative size = 742 \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {a g i^3 m x}{4 j^3}+\frac {b d^3 f n x}{4 e^3}-\frac {5 b d^3 g m n x}{16 e^3}-\frac {5 b g i^3 m n x}{16 j^3}-\frac {5 b d g i^2 m n x}{24 e j^2}-\frac {5 b d^2 g i m n x}{24 e^2 j}+\frac {3 b d^2 g m n x^2}{32 e^2}+\frac {3 b g i^2 m n x^2}{32 j^2}+\frac {b d g i m n x^2}{12 e j}-\frac {7 b d g m n x^3}{144 e}-\frac {7 b g i m n x^3}{144 j}+\frac {1}{32} b g m n x^4+\frac {b d^4 g m n \log (d+e x)}{16 e^4}+\frac {b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac {b d^3 g i m n \log (d+e x)}{12 e^3 j}+\frac {b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\frac {g i^2 m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 j^2}+\frac {g i m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{12 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g i^4 m n \log (i+j x)}{16 j^4}+\frac {b d g i^3 m n \log (i+j x)}{12 e j^3}+\frac {b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac {g i^4 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{4 j^4}+\frac {b d^3 g n (i+j x) \log \left (h (i+j x)^m\right )}{4 e^3 j}-\frac {b d^2 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^4 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{4 j^4}-\frac {b d^4 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{4 e^4} \] Output:
1/4*x^4*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))+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+1/8*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-5/24*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+1/32*b*g*m*n*x^4-1/ 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*g*i^4*m*n*ln(j*x+i)/j^4+1/16*b*d^4*g*m *n*ln(e*x+d)/e^4-5/16*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+3/32*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+1/4*b*d^3*g*n*(j*x+i)*ln(h*(j*x+i)^m)/e^3/j+1/4*b*g*i^3*m*(e*x+d) *ln(c*(e*x+d)^n)/e/j^3+1/4*a*g*i^3*m*x/j^3+1/4*b*d^3*f*n*x/e^3-1/8*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^4-1/8*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^4-1/16*b*n*x^4*(f+g* ln(h*(j*x+i)^m))-1/16*g*m*x^4*(a+b*ln(c*(e*x+d)^n))
Time = 1.12 (sec) , antiderivative size = 605, normalized size of antiderivative = 0.82 \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {6 b n \log (d+e x) \left (12 e^4 g i^4 m \log (i+j x)-12 g \left (e^4 i^4-d^4 j^4\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (12 e^3 g i^3 m+6 d e^2 g i^2 j m+4 d^2 e g i j^2 m+3 d^3 j^3 (-4 f+g m)-12 d^3 g j^3 \log \left (h (i+j x)^m\right )\right )\right )+e \left (6 g i m \left (-12 a e^3 i^3+b \left (3 e^3 i^3+4 d e^2 i^2 j+6 d^2 e i j^2+12 d^3 j^3\right ) n\right ) \log (i+j x)-6 b e^3 \log \left (c (d+e x)^n\right ) \left (-12 f j^4 x^4+g j m x \left (-12 i^3+6 i^2 j x-4 i j^2 x^2+3 j^3 x^3\right )+12 g i^4 m \log (i+j x)-12 g j^4 x^4 \log \left (h (i+j x)^m\right )\right )+j \left (6 a e^3 x \left (12 f j^3 x^3+g m \left (12 i^3-6 i^2 j x+4 i j^2 x^2-3 j^3 x^3\right )\right )-b n \left (18 d^3 j^3 (-4 f+5 g m) x+3 d^2 e j^2 x (12 f j x+g m (20 i-9 j x))+e^3 x \left (18 f j^3 x^3+g m \left (90 i^3-27 i^2 j x+14 i j^2 x^2-9 j^3 x^3\right )\right )+2 d e^2 \left (-12 f j^3 x^3+g m \left (36 i^3+30 i^2 j x-12 i j^2 x^2+7 j^3 x^3\right )\right )\right )-6 g j^3 x \left (-12 a e^3 x^3+b n \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right ) \log \left (h (i+j x)^m\right )\right )\right )-72 b g \left (e^4 i^4-d^4 j^4\right ) m n \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{288 e^4 j^4} \] Input:
Integrate[x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]),x]
Output:
(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)
Time = 2.20 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2889, 2863, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2889 |
| \(\displaystyle -\frac {1}{4} g j m \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x}dx-\frac {1}{4} b e n \int \frac {x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x}dx+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle -\frac {1}{4} g j m \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) i^4}{j^4 (i+j x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) i^3}{j^4}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) i^2}{j^3}-\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) i}{j^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}\right )dx-\frac {1}{4} b e n \int \left (\frac {\left (f+g \log \left (h (i+j x)^m\right )\right ) d^4}{e^4 (d+e x)}-\frac {\left (f+g \log \left (h (i+j x)^m\right )\right ) d^3}{e^4}+\frac {x \left (f+g \log \left (h (i+j x)^m\right )\right ) d^2}{e^3}-\frac {x^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) d}{e^2}+\frac {x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}\right )dx+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) x^4-\frac {1}{4} g j m \left (-\frac {b n \log (d+e x) d^4}{4 e^4 j}+\frac {b n x d^3}{4 e^3 j}-\frac {b i n \log (d+e x) d^3}{3 e^3 j^2}-\frac {b n x^2 d^2}{8 e^2 j}+\frac {b i n x d^2}{3 e^2 j^2}-\frac {b i^2 n \log (d+e x) d^2}{2 e^2 j^3}+\frac {b n x^3 d}{12 e j}-\frac {b i n x^2 d}{6 e j^2}+\frac {b i^2 n x d}{2 e j^3}-\frac {b n x^4}{16 j}+\frac {b i n x^3}{9 j^2}-\frac {b i^2 n x^2}{4 j^3}+\frac {b i^3 n x}{j^4}-\frac {a i^3 x}{j^4}-\frac {b i^3 (d+e x) \log \left (c (d+e x)^n\right )}{e j^4}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 j}-\frac {i x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 j^2}+\frac {i^2 x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^3}+\frac {i^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j^5}+\frac {b i^4 n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j^5}\right )-\frac {1}{4} b e n \left (\frac {\log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) d^4}{e^5}+\frac {g m \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right ) d^4}{e^5}-\frac {f x d^3}{e^4}+\frac {g m x d^3}{e^4}-\frac {g (i+j x) \log \left (h (i+j x)^m\right ) d^3}{e^4 j}-\frac {g m x^2 d^2}{4 e^3}+\frac {g i m x d^2}{2 e^3 j}-\frac {g i^2 m \log (i+j x) d^2}{2 e^3 j^2}+\frac {x^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) d^2}{2 e^3}+\frac {g m x^3 d}{9 e^2}-\frac {g i m x^2 d}{6 e^2 j}+\frac {g i^2 m x d}{3 e^2 j^2}-\frac {g i^3 m \log (i+j x) d}{3 e^2 j^3}-\frac {x^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) d}{3 e^2}-\frac {g m x^4}{16 e}+\frac {g i m x^3}{12 e j}-\frac {g i^2 m x^2}{8 e j^2}+\frac {g i^3 m x}{4 e j^3}-\frac {g i^4 m \log (i+j x)}{4 e j^4}+\frac {x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e}\right )\) |
Input:
Int[x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]),x]
Output:
(x^4*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/4 - (g*j*m*(-( (a*i^3*x)/j^4) + (b*i^3*n*x)/j^4 + (b*d*i^2*n*x)/(2*e*j^3) + (b*d^2*i*n*x) /(3*e^2*j^2) + (b*d^3*n*x)/(4*e^3*j) - (b*i^2*n*x^2)/(4*j^3) - (b*d*i*n*x^ 2)/(6*e*j^2) - (b*d^2*n*x^2)/(8*e^2*j) + (b*i*n*x^3)/(9*j^2) + (b*d*n*x^3) /(12*e*j) - (b*n*x^4)/(16*j) - (b*d^2*i^2*n*Log[d + e*x])/(2*e^2*j^3) - (b *d^3*i*n*Log[d + e*x])/(3*e^3*j^2) - (b*d^4*n*Log[d + e*x])/(4*e^4*j) - (b *i^3*(d + e*x)*Log[c*(d + e*x)^n])/(e*j^4) + (i^2*x^2*(a + b*Log[c*(d + e* x)^n]))/(2*j^3) - (i*x^3*(a + b*Log[c*(d + e*x)^n]))/(3*j^2) + (x^4*(a + b *Log[c*(d + e*x)^n]))/(4*j) + (i^4*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/j^5 + (b*i^4*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j)) ])/j^5))/4 - (b*e*n*(-((d^3*f*x)/e^4) + (d^3*g*m*x)/e^4 + (g*i^3*m*x)/(4*e *j^3) + (d*g*i^2*m*x)/(3*e^2*j^2) + (d^2*g*i*m*x)/(2*e^3*j) - (d^2*g*m*x^2 )/(4*e^3) - (g*i^2*m*x^2)/(8*e*j^2) - (d*g*i*m*x^2)/(6*e^2*j) + (d*g*m*x^3 )/(9*e^2) + (g*i*m*x^3)/(12*e*j) - (g*m*x^4)/(16*e) - (g*i^4*m*Log[i + j*x ])/(4*e*j^4) - (d*g*i^3*m*Log[i + j*x])/(3*e^2*j^3) - (d^2*g*i^2*m*Log[i + j*x])/(2*e^3*j^2) - (d^3*g*(i + j*x)*Log[h*(i + j*x)^m])/(e^4*j) + (d^2*x ^2*(f + g*Log[h*(i + j*x)^m]))/(2*e^3) - (d*x^3*(f + g*Log[h*(i + j*x)^m]) )/(3*e^2) + (x^4*(f + g*Log[h*(i + j*x)^m]))/(4*e) + (d^4*Log[-((j*(d + e* x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/e^5 + (d^4*g*m*PolyLog[2, (e *(i + j*x))/(e*i - d*j)])/e^5))/4
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]
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 ] + (-Simp[g*j*(m/(r + 1)) Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Simp[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] && (E qQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 2094, normalized size of antiderivative = 2.82
\[\text {Expression too large to display}\]
Input:
int(x^3*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m)),x)
Output:
-1/16*x^4*b*f*n+(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*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-1/4*I*b*Pi*csgn(I*c*( e*x+d)^n)^3+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+1/2*b*ln(c)+1/2*a)* (1/4*(-I*g*Pi*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+I*g*Pi*csgn( I*h)*csgn(I*h*(j*x+i)^m)^2+I*g*Pi*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2- I*g*Pi*csgn(I*h*(j*x+i)^m)^3+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-x*i^3)+i^4/j^5*ln (j*x+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-205/576/e^4*b*d^4*g*m*n+(1/4*b*g*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*c sgn(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/e/j^3*b*d*g*i^3*m*n-11/96/e^ 2/j^2*b*d^2*g*i^2*m*n-3/16/e^3/j*b*d^3*g*i*m*n+1/4/j^4*b*g*i^4*m*n*dilog(( (j*x+i)*e+d*j-e*i)/(d*j-e*i))+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))-1/32*I*n*b*Pi*x^4 *g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2-1/32*I*n*b*Pi*x^4*g*csgn(I*(j*x+i)^m)*c sgn(I*h*(j*x+i)^m)^2+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)^2+1/8*I/e^3*n*b*x*Pi*d^3*g*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/...
\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm=" fricas")
Output:
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)
Timed out. \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*ln(c*(e*x+d)**n))*(f+g*ln(h*(j*x+i)**m)),x)
Output:
Timed out
\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm=" maxima")
Output:
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) + 4 8*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)*l og(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))
\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x, algorithm=" giac")
Output:
integrate((b*log((e*x + d)^n*c) + a)*(g*log((j*x + i)^m*h) + f)*x^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int x^3\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \] Input:
int(x^3*(a + b*log(c*(d + e*x)^n))*(f + g*log(h*(i + j*x)^m)),x)
Output:
int(x^3*(a + b*log(c*(d + e*x)^n))*(f + g*log(h*(i + j*x)^m)), x)
\[ \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {too large to display} \] Input:
int(x^3*(a+b*log(c*(e*x+d)^n))*(f+g*log(h*(j*x+i)^m)),x)
Output:
(72*atan(j*x)*b*d**5*e*g*j**5*m*n - 108*atan(j*x)*b*d**4*e**2*g*i*j**4*m*n - 24*atan(j*x)*b*d**3*e**3*g*j**3*m*n - 6*atan(j*x)*b*d**2*e**4*g*i*j**2* m*n - 12*atan(j*x)*b*d*e**5*g*j*m*n + 18*atan(j*x)*b*e**6*g*i*m*n - 72*int (log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2* i + 2*d*e**2*j*x - e**3*i*x),x)*b*d**8*e*g*i*j**8*n - 288*int(log((i + j*x )**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2* j*x - e**3*i*x),x)*b*d**7*e**2*g*j**7*n + 432*int(log((i + j*x)**m*h)/(d** 3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i *x),x)*b*d**6*e**3*g*i*j**6*n + 288*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d **5*e**4*g*j**5*n + 288*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j* *2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d**3*e**6*g*j **3*n - 432*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d** 2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d**2*e**7*g*i*j**2*n - 28 8*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d* e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d*e**8*g*j*n + 72*int(log((i + j*x) **m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j *x - e**3*i*x),x)*b*e**9*g*i*n + 36*log(j**2*x**2 + 1)*b*d**5*e*g*i*j**5*m *n + 54*log(j**2*x**2 + 1)*b*d**4*e**2*g*j**4*m*n - 12*log(j**2*x**2 + 1)* b*d**3*e**3*g*i*j**3*m*n + 3*log(j**2*x**2 + 1)*b*d**2*e**4*g*j**2*m*n ...