Integrand size = 43, antiderivative size = 289 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=-\frac {B d (b c-a d) i^2 n x}{2 b^2 g}+\frac {d (b c-a d) i^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {i^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac {B (b c-a d)^2 i^2 n \log \left (\frac {a+b x}{c+d x}\right )}{2 b^3 g}-\frac {3 B (b c-a d)^2 i^2 n \log (c+d x)}{2 b^3 g}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {B (b c-a d)^2 i^2 n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g} \] Output:
-1/2*B*d*(-a*d+b*c)*i^2*n*x/b^2/g+d*(-a*d+b*c)*i^2*(b*x+a)*(A+B*ln(e*((b*x +a)/(d*x+c))^n))/b^3/g+1/2*i^2*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b /g-1/2*B*(-a*d+b*c)^2*i^2*n*ln((b*x+a)/(d*x+c))/b^3/g-3/2*B*(-a*d+b*c)^2*i ^2*n*ln(d*x+c)/b^3/g-(-a*d+b*c)^2*i^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1 -b*(d*x+c)/d/(b*x+a))/b^3/g+B*(-a*d+b*c)^2*i^2*n*polylog(2,b*(d*x+c)/d/(b* x+a))/b^3/g
Time = 0.23 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.91 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {i^2 \left (2 A b d (b c-a d) x-B (b c-a d) n (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 (b c-a d)^2 \log (g (a+b x)) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)+B (b c-a d)^2 n \left (-\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{2 b^3 g} \] Input:
Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x),x]
Output:
(i^2*(2*A*b*d*(b*c - a*d)*x - B*(b*c - a*d)*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) + 2*B*d*(b*c - a*d)*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + b^2* (c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*(b*c - a*d)^2*Log[g *(a + b*x)]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*B*(b*c - a*d)^2*n*L og[c + d*x] + B*(b*c - a*d)^2*n*(-(Log[g*(a + b*x)]*(Log[g*(a + b*x)] - 2* Log[(b*(c + d*x))/(b*c - a*d)])) + 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a* d)])))/(2*b^3*g)
Time = 0.94 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2961, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a g+b g x} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{g}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {c+d x}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B n \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {i^2 (b c-a d)^2 \left (\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}\right )}{g}\) |
Input:
Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x) ,x]
Output:
((b*c - a*d)^2*i^2*((d*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*d*(b - ( d*(a + b*x))/(c + d*x))^2) - (B*n*(1/(b*(b - (d*(a + b*x))/(c + d*x))) + L og[(a + b*x)/(c + d*x)]/b^2 - Log[b - (d*(a + b*x))/(c + d*x)]/b^2))/(2*d) ))/b + ((d*(((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*(c + d*x )*(b - (d*(a + b*x))/(c + d*x))) + (B*n*Log[b - (d*(a + b*x))/(c + d*x)])/ (b*d)))/b + (-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x ))/(d*(a + b*x))])/b) + (B*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b)/b )/b))/g
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
\[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{b g x +a g}d x\]
Input:
int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g),x)
Output:
int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g),x)
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{b g x + a g} \,d x } \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g),x, al gorithm="fricas")
Output:
integral((A*d^2*i^2*x^2 + 2*A*c*d*i^2*x + A*c^2*i^2 + (B*d^2*i^2*x^2 + 2*B *c*d*i^2*x + B*c^2*i^2)*log(e*((b*x + a)/(d*x + c))^n))/(b*g*x + a*g), x)
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {i^{2} \left (\int \frac {A c^{2}}{a + b x}\, dx + \int \frac {A d^{2} x^{2}}{a + b x}\, dx + \int \frac {B c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx + \int \frac {2 A c d x}{a + b x}\, dx + \int \frac {B d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx + \int \frac {2 B c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx\right )}{g} \] Input:
integrate((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g),x)
Output:
i**2*(Integral(A*c**2/(a + b*x), x) + Integral(A*d**2*x**2/(a + b*x), x) + Integral(B*c**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a + b*x), x) + I ntegral(2*A*c*d*x/(a + b*x), x) + Integral(B*d**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a + b*x), x) + Integral(2*B*c*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a + b*x), x))/g
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (280) = 560\).
Time = 0.38 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.01 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=2 \, A c d i^{2} {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac {1}{2} \, A d^{2} i^{2} {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3} g} + \frac {b x^{2} - 2 \, a x}{b^{2} g}\right )} + \frac {A c^{2} i^{2} \log \left (b g x + a g\right )}{b g} - \frac {{\left (3 \, b c^{2} i^{2} n - 2 \, a c d i^{2} n\right )} B \log \left (d x + c\right )}{2 \, b^{2} g} + \frac {{\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{3} g} + \frac {B b^{2} d^{2} i^{2} x^{2} \log \left (e\right ) - {\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} B \log \left (b x + a\right )^{2} - {\left ({\left (i^{2} n - 4 \, i^{2} \log \left (e\right )\right )} b^{2} c d - {\left (i^{2} n - 2 \, i^{2} \log \left (e\right )\right )} a b d^{2}\right )} B x + {\left (2 \, b^{2} c^{2} i^{2} \log \left (e\right ) + 4 \, {\left (i^{2} n - i^{2} \log \left (e\right )\right )} a b c d - {\left (3 \, i^{2} n - 2 \, i^{2} \log \left (e\right )\right )} a^{2} d^{2}\right )} B \log \left (b x + a\right ) + {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{3} g} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g),x, al gorithm="maxima")
Output:
2*A*c*d*i^2*(x/(b*g) - a*log(b*x + a)/(b^2*g)) + 1/2*A*d^2*i^2*(2*a^2*log( b*x + a)/(b^3*g) + (b*x^2 - 2*a*x)/(b^2*g)) + A*c^2*i^2*log(b*g*x + a*g)/( b*g) - 1/2*(3*b*c^2*i^2*n - 2*a*c*d*i^2*n)*B*log(d*x + c)/(b^2*g) + (b^2*c ^2*i^2*n - 2*a*b*c*d*i^2*n + a^2*d^2*i^2*n)*(log(b*x + a)*log((b*d*x + a*d )/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/(b^3*g) + 1/2*(B *b^2*d^2*i^2*x^2*log(e) - (b^2*c^2*i^2*n - 2*a*b*c*d*i^2*n + a^2*d^2*i^2*n )*B*log(b*x + a)^2 - ((i^2*n - 4*i^2*log(e))*b^2*c*d - (i^2*n - 2*i^2*log( e))*a*b*d^2)*B*x + (2*b^2*c^2*i^2*log(e) + 4*(i^2*n - i^2*log(e))*a*b*c*d - (3*i^2*n - 2*i^2*log(e))*a^2*d^2)*B*log(b*x + a) + (B*b^2*d^2*i^2*x^2 + 2*(2*b^2*c*d*i^2 - a*b*d^2*i^2)*B*x + 2*(b^2*c^2*i^2 - 2*a*b*c*d*i^2 + a^2 *d^2*i^2)*B*log(b*x + a))*log((b*x + a)^n) - (B*b^2*d^2*i^2*x^2 + 2*(2*b^2 *c*d*i^2 - a*b*d^2*i^2)*B*x + 2*(b^2*c^2*i^2 - 2*a*b*c*d*i^2 + a^2*d^2*i^2 )*B*log(b*x + a))*log((d*x + c)^n))/(b^3*g)
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{b g x + a g} \,d x } \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g),x, al gorithm="giac")
Output:
integrate((d*i*x + c*i)^2*(B*log(e*((b*x + a)/(d*x + c))^n) + A)/(b*g*x + a*g), x)
Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{a\,g+b\,g\,x} \,d x \] Input:
int(((c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)))/(a*g + b*g*x) ,x)
Output:
int(((c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)))/(a*g + b*g*x) , x)
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b x +a}d x \right ) b^{4} c^{2}-2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x^{2}}{b x +a}d x \right ) b^{4} d^{2}-4 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x}{b x +a}d x \right ) b^{4} c d -2 \,\mathrm {log}\left (b x +a \right ) a^{3} d^{2}+4 \,\mathrm {log}\left (b x +a \right ) a^{2} b c d -2 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c^{2}+2 a^{2} b \,d^{2} x -4 a \,b^{2} c d x -a \,b^{2} d^{2} x^{2}}{2 b^{3} g} \] Input:
int((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g),x)
Output:
( - 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a + b*x),x)*b**4*c**2 - 2*in t((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(a + b*x),x)*b**4*d**2 - 4*int ((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a + b*x),x)*b**4*c*d - 2*log(a + b*x)*a**3*d**2 + 4*log(a + b*x)*a**2*b*c*d - 2*log(a + b*x)*a*b**2*c**2 + 2*a**2*b*d**2*x - 4*a*b**2*c*d*x - a*b**2*d**2*x**2)/(2*b**3*g)