Integrand size = 30, antiderivative size = 397 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {a g i m x}{2 j}+\frac {b d f n x}{2 e}-\frac {3 b d g m n x}{4 e}-\frac {3 b g i m n x}{4 j}+\frac {1}{4} b g m n x^2+\frac {b d^2 g m n \log (d+e x)}{4 e^2}+\frac {b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac {1}{4} g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g i^2 m n \log (i+j x)}{4 j^2}-\frac {g i^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}+\frac {b d g n (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac {1}{4} b n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}-\frac {b d^2 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 e^2} \]
1/2*a*g*i*m*x/j+1/2*b*d*f*n*x/e-3/4*b*d*g*m*n*x/e-3/4*b*g*i*m*n*x/j+1/4*b* g*m*n*x^2+1/4*b*d^2*g*m*n*ln(e*x+d)/e^2+1/2*b*g*i*m*(e*x+d)*ln(c*(e*x+d)^n )/e/j-1/4*g*m*x^2*(a+b*ln(c*(e*x+d)^n))+1/4*b*g*i^2*m*n*ln(j*x+i)/j^2-1/2* g*i^2*m*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j^2+1/2*b*d*g*n*(j* x+i)*ln(h*(j*x+i)^m)/e/j-1/4*b*n*x^2*(f+g*ln(h*(j*x+i)^m))-1/2*b*d^2*n*ln( -j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^2+1/2*x^2*(a+b*ln(c*(e*x+d) ^n))*(f+g*ln(h*(j*x+i)^m))-1/2*b*g*i^2*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i) )/j^2-1/2*b*d^2*g*m*n*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^2
Time = 0.33 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.86 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {b n \log (d+e x) \left (2 e^2 g i^2 m \log (i+j x)+2 g \left (-e^2 i^2+d^2 j^2\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (-2 d f j+2 e g i m+d g j m-2 d g j \log \left (h (i+j x)^m\right )\right )\right )+e \left (g i m (-2 a e i+b (e i+2 d j) n) \log (i+j x)+j \left (a e x (2 f j x+g m (2 i-j x))-b n (e x (3 g i m+f j x-g j m x)+d (2 g i m-2 f j x+3 g j m x))+g j x (2 a e x+b n (2 d-e x)) \log \left (h (i+j x)^m\right )\right )+b e \log \left (c (d+e x)^n\right ) \left (-2 g i^2 m \log (i+j x)+j x \left (2 g i m+2 f j x-g j m x+2 g j x \log \left (h (i+j x)^m\right )\right )\right )\right )+2 b g \left (-e^2 i^2+d^2 j^2\right ) m n \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{4 e^2 j^2} \]
(b*n*Log[d + e*x]*(2*e^2*g*i^2*m*Log[i + j*x] + 2*g*(-(e^2*i^2) + d^2*j^2) *m*Log[(e*(i + j*x))/(e*i - d*j)] + d*j*(-2*d*f*j + 2*e*g*i*m + d*g*j*m - 2*d*g*j*Log[h*(i + j*x)^m])) + e*(g*i*m*(-2*a*e*i + b*(e*i + 2*d*j)*n)*Log [i + j*x] + j*(a*e*x*(2*f*j*x + g*m*(2*i - j*x)) - b*n*(e*x*(3*g*i*m + f*j *x - g*j*m*x) + d*(2*g*i*m - 2*f*j*x + 3*g*j*m*x)) + g*j*x*(2*a*e*x + b*n* (2*d - e*x))*Log[h*(i + j*x)^m]) + b*e*Log[c*(d + e*x)^n]*(-2*g*i^2*m*Log[ i + j*x] + j*x*(2*g*i*m + 2*f*j*x - g*j*m*x + 2*g*j*x*Log[h*(i + j*x)^m])) ) + 2*b*g*(-(e^2*i^2) + d^2*j^2)*m*n*PolyLog[2, (j*(d + e*x))/(-(e*i) + d* j)])/(4*e^2*j^2)
Time = 0.74 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2889 |
| \(\displaystyle -\frac {1}{2} g j m \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x}dx-\frac {1}{2} b e n \int \frac {x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x}dx+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle -\frac {1}{2} g j m \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) i^2}{j^2 (i+j x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) i}{j^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}\right )dx-\frac {1}{2} b e n \int \left (\frac {\left (f+g \log \left (h (i+j x)^m\right )\right ) d^2}{e^2 (d+e x)}-\frac {\left (f+g \log \left (h (i+j x)^m\right )\right ) d}{e^2}+\frac {x \left (f+g \log \left (h (i+j x)^m\right )\right )}{e}\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} g j m \left (\frac {i^2 \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j}-\frac {a i x}{j^2}-\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e j^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 j}+\frac {b i^2 n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j^3}+\frac {b d n x}{2 e j}+\frac {b i n x}{j^2}-\frac {b n x^2}{4 j}\right )+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {1}{2} b e n \left (\frac {d^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{e^3}+\frac {d^2 g m \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e^3}-\frac {d f x}{e^2}-\frac {d g (i+j x) \log \left (h (i+j x)^m\right )}{e^2 j}+\frac {d g m x}{e^2}+\frac {x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e}-\frac {g i^2 m \log (i+j x)}{2 e j^2}+\frac {g i m x}{2 e j}-\frac {g m x^2}{4 e}\right )\) |
(x^2*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/2 - (g*j*m*(-( (a*i*x)/j^2) + (b*i*n*x)/j^2 + (b*d*n*x)/(2*e*j) - (b*n*x^2)/(4*j) - (b*d^ 2*n*Log[d + e*x])/(2*e^2*j) - (b*i*(d + e*x)*Log[c*(d + e*x)^n])/(e*j^2) + (x^2*(a + b*Log[c*(d + e*x)^n]))/(2*j) + (i^2*(a + b*Log[c*(d + e*x)^n])* Log[(e*(i + j*x))/(e*i - d*j)])/j^3 + (b*i^2*n*PolyLog[2, -((j*(d + e*x))/ (e*i - d*j))])/j^3))/2 - (b*e*n*(-((d*f*x)/e^2) + (d*g*m*x)/e^2 + (g*i*m*x )/(2*e*j) - (g*m*x^2)/(4*e) - (g*i^2*m*Log[i + j*x])/(2*e*j^2) - (d*g*(i + j*x)*Log[h*(i + j*x)^m])/(e^2*j) + (x^2*(f + g*Log[h*(i + j*x)^m]))/(2*e) + (d^2*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/e^3 + (d^2*g*m*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e^3))/2
3.4.88.3.1 Defintions of rubi rules used
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 = 111.26 (sec) , antiderivative size = 1372, normalized size of antiderivative = 3.46
1/2/e*n*b*g*ln((j*x+i)^m)*x*d-1/2*n*b*d^2*f/e^2*ln(e*x+d)-1/4*b*f*n*x^2-1/ 4*ln(h)*x^2*b*g*n+(-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)*(1/2*(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*c sgn(I*h*(j*x+i)^m)^2*csgn(I*h)+2*g*ln(h)+2*f)*x^2+g*ln((j*x+i)^m)*x^2-1/2* g*m*x^2+g*m/j*x*i-g*m/j^2*i^2*ln(j*x+i))+1/2/e*n*b/j*d*ln(e*x+d)*g*i*m+1/2 /e*n*b*g*m/j*i*ln((e*x+d)*j-d*j+e*i)*d-1/4*I/e*n*b*x*Pi*d*g*csgn(I*h*(j*x+ i)^m)^3+1/4*I/e^2*n*b*d^2*ln(e*x+d)*Pi*g*csgn(I*h*(j*x+i)^m)^3+1/8*I*n*b*P i*x^2*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+(1/2*x^2*b*g*ln((j *x+i)^m)-1/4*b*(-I*Pi*g*j^2*x^2*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2+I* Pi*g*j^2*x^2*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+I*Pi*g*j^2*x^ 2*csgn(I*h*(j*x+i)^m)^3-I*Pi*g*j^2*x^2*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-2*j ^2*x^2*ln(h)*g+g*j^2*m*x^2+2*g*i^2*m*ln(j*x+i)-2*f*j^2*x^2-2*g*i*j*m*x)/j^ 2)*ln((e*x+d)^n)-1/4*b*d*g*i*m*n/e/j+1/2/e^2*n*b*g*m*d^2*ln(e*x+d)*ln(((e* x+d)*j-d*j+e*i)/(-d*j+e*i))+1/2*n*b*g*i^2*m/j^2*ln(j*x+i)*ln(((j*x+i)*e+d* j-e*i)/(d*j-e*i))-1/8*I*n*b*Pi*x^2*g*csgn(I*h*(j*x+i)^m)^2*csgn(I*h)-1/8*I *n*b*Pi*x^2*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-5/8*b*d^2*g*m*n/e^2- 1/4*I/e*n*b*x*Pi*d*g*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)*csgn(I*h)+1/...
\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]
integral(b*f*x*log((e*x + d)^n*c) + a*f*x + (b*g*x*log((e*x + d)^n*c) + a* g*x)*log((j*x + i)^m*h), x)
Timed out. \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \]
\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]
-1/4*b*e*f*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) - 1/4*a*g*j*m* (2*i^2*log(j*x + i)/j^3 + (j*x^2 - 2*i*x)/j^2) + 1/2*b*f*x^2*log((e*x + d) ^n*c) + 1/2*a*g*x^2*log((j*x + i)^m*h) + 1/2*a*f*x^2 + 1/4*b*g*((2*e^2*i^2 *m*n*log(e*x + d)*log(j*x + i) + (2*e^2*i*j*m*x - 2*e^2*i^2*m*log(j*x + i) - (j^2*m - 2*j^2*log(h))*e^2*x^2)*log((e*x + d)^n) + (2*e^2*j^2*x^2*log(( e*x + d)^n) + 2*d*e*j^2*n*x - 2*d^2*j^2*n*log(e*x + d) - (e^2*j^2*n - 2*e^ 2*j^2*log(c))*x^2)*log((j*x + i)^m))/(e^2*j^2) + 4*integrate(-1/4*(2*((j^2 *m - 2*j^2*log(h))*e^3*log(c) - (j^2*m*n - j^2*n*log(h))*e^3)*x^3 + (d*e^2 *j^2*m*n + (i*j*m*n + 2*i*j*n*log(h))*e^3 - 2*(2*e^3*i*j*log(h) - (j^2*m - 2*j^2*log(h))*d*e^2)*log(c))*x^2 + 2*(e^3*i^2*m*n + d^2*e*j^2*m*n - 2*d*e ^2*i*j*log(c)*log(h))*x + 2*(d*e^2*i^2*m*n - d^3*j^2*m*n + (e^3*i^2*m*n - d^2*e*j^2*m*n)*x)*log(e*x + d))/(e^3*j^2*x^2 + d*e^2*i*j + (e^3*i*j + d*e^ 2*j^2)*x), x))
\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int x\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \]