Integrand size = 40, antiderivative size = 310 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {B d^2 i^3 (c+d x)}{b^3 g^4 (a+b x)}-\frac {B d i^3 (c+d x)^2}{4 b^2 g^4 (a+b x)^2}-\frac {B i^3 (c+d x)^3}{9 b g^4 (a+b x)^3}-\frac {d^2 i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}-\frac {d i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^4 (a+b x)^2}-\frac {i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b g^4 (a+b x)^3}-\frac {d^3 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^4}+\frac {B d^3 i^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^4} \] Output:
-B*d^2*i^3*(d*x+c)/b^3/g^4/(b*x+a)-1/4*B*d*i^3*(d*x+c)^2/b^2/g^4/(b*x+a)^2 -1/9*B*i^3*(d*x+c)^3/b/g^4/(b*x+a)^3-d^2*i^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d* x+c)))/b^3/g^4/(b*x+a)-1/2*d*i^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2 /g^4/(b*x+a)^2-1/3*i^3*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/g^4/(b*x+a) ^3-d^3*i^3*(A+B*ln(e*(b*x+a)/(d*x+c)))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g^4+B *d^3*i^3*polylog(2,b*(d*x+c)/d/(b*x+a))/b^4/g^4
Time = 0.49 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.99 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\frac {i^3 \left (-\frac {4 B (b c-a d)^3}{(a+b x)^3}-\frac {21 B d (b c-a d)^2}{(a+b x)^2}+\frac {66 B d^2 (-b c+a d)}{a+b x}-66 B d^3 \log (a+b x)-\frac {12 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}-\frac {54 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {108 d^2 (-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+36 d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+66 B d^3 \log (c+d x)-18 B d^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{36 b^4 g^4} \] Input:
Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b* g*x)^4,x]
Output:
(i^3*((-4*B*(b*c - a*d)^3)/(a + b*x)^3 - (21*B*d*(b*c - a*d)^2)/(a + b*x)^ 2 + (66*B*d^2*(-(b*c) + a*d))/(a + b*x) - 66*B*d^3*Log[a + b*x] - (12*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^3 - (54*d*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (108*d^2*(-(b* c) + a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + 36*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 66*B*d^3*Log[c + d*x] - 18*B* d^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*Po lyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/(36*b^4*g^4)
Time = 1.00 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2962, 2780, 2741, 2780, 2741, 2780, 2741, 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.
| \(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^4} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^3 \int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{g^4}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {i^3 \left (\frac {\int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{b}+\frac {d \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i^3 \left (\frac {d \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {i^3 \left (\frac {d \left (\frac {\int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{b}+\frac {d \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i^3 \left (\frac {d \left (\frac {d \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {i^3 \left (\frac {d \left (\frac {d \left (\frac {\int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{b}+\frac {d \int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i^3 \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {B (c+d x)}{a+b x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {i^3 \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {B \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b}\right )}{b}+\frac {-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {B (c+d x)}{a+b x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {i^3 \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {B \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b}\right )}{b}+\frac {-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {B (c+d x)}{a+b x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{b}+\frac {-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}}{b}\right )}{g^4}\) |
Input:
Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^4 ,x]
Output:
(i^3*((-1/9*(B*(c + d*x)^3)/(a + b*x)^3 - ((c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(a + b*x)^3))/b + (d*((-1/4*(B*(c + d*x)^2)/(a + b*x )^2 - ((c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2))/ b + (d*((-((B*(c + d*x))/(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x))/ (c + d*x)]))/(a + b*x))/b + (d*(-(((A + B*Log[(e*(a + b*x))/(c + d*x)])*Lo g[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (B*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b))/b))/b))/b))/g^4
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -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_.)*(x_)^(m_.))/((d_) + (e_.)* (x_)^(r_.)), x_Symbol] :> Simp[1/d Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Simp[e/d Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
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_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs. \(2(302)=604\).
Time = 3.26 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.21
| method | result | size |
| parts | \(\frac {i^{3} A \left (-\frac {3 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{2 b^{4} \left (b x +a \right )^{2}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 d^{2} \left (d a -b c \right )}{b^{4} \left (b x +a \right )}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{3 b^{4} \left (b x +a \right )^{3}}\right )}{g^{4}}-\frac {i^{3} B \left (d a -b c \right )^{4} e^{4} \left (-\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d^{7}}{\left (d a -b c \right )^{4} b^{3} e^{3}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{8}}{2 \left (d a -b c \right )^{4} b^{4} e^{4}}-\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right ) d^{6}}{\left (d a -b c \right )^{4} b^{2} e^{2}}-\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right ) d^{5}}{\left (d a -b c \right )^{4} b e}+\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{9}}{\left (d a -b c \right )^{4} b^{4} e^{4}}\right )}{g^{4} d^{5}}\) | \(685\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{3} d^{2} e^{3} A \left (-\frac {d^{3} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{4} e^{4}}-\frac {1}{3 b e \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{4} e^{4}}-\frac {d^{2}}{b^{3} e^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {d}{2 b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right ) g^{4}}-\frac {i^{3} d^{2} e^{3} B \left (\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right ) d}{b^{2} e^{2}}+\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d^{2}}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}}{b e}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{4}}{b^{4} e^{4}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{3}}{2 b^{4} e^{4}}\right )}{\left (d a -b c \right ) g^{4}}\right )}{d^{2}}\) | \(740\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{3} d^{2} e^{3} A \left (-\frac {d^{3} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{4} e^{4}}-\frac {1}{3 b e \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{4} e^{4}}-\frac {d^{2}}{b^{3} e^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {d}{2 b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right ) g^{4}}-\frac {i^{3} d^{2} e^{3} B \left (\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right ) d}{b^{2} e^{2}}+\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d^{2}}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}}{b e}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{4}}{b^{4} e^{4}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{3}}{2 b^{4} e^{4}}\right )}{\left (d a -b c \right ) g^{4}}\right )}{d^{2}}\) | \(740\) |
| risch | \(\text {Expression too large to display}\) | \(3716\) |
Input:
int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x,method=_RETU RNVERBOSE)
Output:
i^3*A/g^4*(-3/2*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^4/(b*x+a)^2+d^3/b^4*ln(b*x +a)+3/b^4*d^2*(a*d-b*c)/(b*x+a)-1/3*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+ b^3*c^3)/b^4/(b*x+a)^3)-i^3*B/g^4/d^5*(a*d-b*c)^4*e^4*(-(-1/(b*e/d+(a*d-b* c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d* x+c)))/(a*d-b*c)^4*d^7/b^3/e^3-1/2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(a*d- b*c)^4*d^8/b^4/e^4-(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c )*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)/(a*d-b*c)^4*d^6/b^2/e^ 2-(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/ 9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)/(a*d-b*c)^4*d^5/b/e+(dilog(-((b*e/d+(a* d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d+ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-((b* e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d)/(a*d-b*c)^4*d^9/b^4/e^4)
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (b g x + a g\right )}^{4}} \,d x } \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algo rithm="fricas")
Output:
integral((A*d^3*i^3*x^3 + 3*A*c*d^2*i^3*x^2 + 3*A*c^2*d*i^3*x + A*c^3*i^3 + (B*d^3*i^3*x^3 + 3*B*c*d^2*i^3*x^2 + 3*B*c^2*d*i^3*x + B*c^3*i^3)*log((b *e*x + a*e)/(d*x + c)))/(b^4*g^4*x^4 + 4*a*b^3*g^4*x^3 + 6*a^2*b^2*g^4*x^2 + 4*a^3*b*g^4*x + a^4*g^4), x)
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\frac {i^{3} \left (\int \frac {A c^{3}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {A d^{3} x^{3}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {B c^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {3 A c d^{2} x^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {3 A c^{2} d x}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {B d^{3} x^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {3 B c d^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {3 B c^{2} d x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx\right )}{g^{4}} \] Input:
integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4,x)
Output:
i**3*(Integral(A*c**3/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x** 3 + b**4*x**4), x) + Integral(A*d**3*x**3/(a**4 + 4*a**3*b*x + 6*a**2*b**2 *x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(B*c**3*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(3*A*c*d**2*x**2/(a**4 + 4*a**3*b*x + 6*a**2*b* *2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(3*A*c**2*d*x/(a**4 + 4 *a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(B *d**3*x**3*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a**4 + 4*a**3*b*x + 6*a** 2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(3*B*c*d**2*x**2*lo g(a*e/(c + d*x) + b*e*x/(c + d*x))/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(3*B*c**2*d*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x))/g**4
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (b g x + a g\right )}^{4}} \,d x } \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algo rithm="maxima")
Output:
-1/6*B*d^3*i^3*((18*a*b^2*x^2 + 27*a^2*b*x + 11*a^3 + 6*(b^3*x^3 + 3*a*b^2 *x^2 + 3*a^2*b*x + a^3)*log(b*x + a))*log(d*x + c)/(b^7*g^4*x^3 + 3*a*b^6* g^4*x^2 + 3*a^2*b^5*g^4*x + a^3*b^4*g^4) - 6*integrate(1/6*(6*b^4*d*x^4*lo g(e) + 45*a^2*b^2*d*x^2 + 38*a^3*b*d*x + 11*a^4*d + 6*(b^4*c*log(e) + 3*a* b^3*d)*x^3 + 6*(2*b^4*d*x^4 + 6*a^2*b^2*d*x^2 + 4*a^3*b*d*x + a^4*d + (b^4 *c + 4*a*b^3*d)*x^3)*log(b*x + a))/(b^8*d*g^4*x^5 + a^4*b^4*c*g^4 + (b^8*c *g^4 + 4*a*b^7*d*g^4)*x^4 + 2*(2*a*b^7*c*g^4 + 3*a^2*b^6*d*g^4)*x^3 + 2*(3 *a^2*b^6*c*g^4 + 2*a^3*b^5*d*g^4)*x^2 + (4*a^3*b^5*c*g^4 + a^4*b^4*d*g^4)* x), x)) - 1/6*B*c*d^2*i^3*(6*(3*b^2*x^2 + 3*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3 *b^3*g^4) + (11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a *b^3*c*d + a^2*b^2*d^2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2 )*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7*c^2 - 2*a^2 *b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4 *d^2)*g^4*x + (a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2* c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 - 3*a*b^5*c^2*d + 3* a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3 )*log(d*x + c)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)* g^4)) - 1/12*B*c^2*d*i^3*(6*(3*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c ))/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) + (5...
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (b g x + a g\right )}^{4}} \,d x } \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algo rithm="giac")
Output:
integrate((d*i*x + c*i)^3*(B*log((b*x + a)*e/(d*x + c)) + A)/(b*g*x + a*g) ^4, x)
Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^4} \,d x \] Input:
int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^4 ,x)
Output:
int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^4 , x)
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\text {too large to display} \] Input:
int((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x)
Output:
(i*( - 36*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a **2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x)*a**8*b**5*d**6 + 108*int((lo g((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4 *a*b**3*x**3 + b**4*x**4),x)*a**7*b**6*c*d**5 - 108*int((log((a*e + b*e*x) /(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x)*a**7*b**6*d**6*x - 108*int((log((a*e + b*e*x)/(c + d*x))*x** 3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x)*a **6*b**7*c**2*d**4 + 324*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4 *a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x)*a**6*b**7*c*d **5*x - 108*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6 *a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x)*a**6*b**7*d**6*x**2 + 36*i nt((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x* *2 + 4*a*b**3*x**3 + b**4*x**4),x)*a**5*b**8*c**3*d**3 - 324*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b** 3*x**3 + b**4*x**4),x)*a**5*b**8*c**2*d**4*x + 324*int((log((a*e + b*e*x)/ (c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b **4*x**4),x)*a**5*b**8*c*d**5*x**2 - 36*int((log((a*e + b*e*x)/(c + d*x))* x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x )*a**5*b**8*d**6*x**3 + 108*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4),x)*a**4*b*...