Integrand size = 40, antiderivative size = 89 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {B i^3 (c+d x)^4}{16 (b c-a d) g^5 (a+b x)^4}-\frac {i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d) g^5 (a+b x)^4} \] Output:
-1/16*B*i^3*(d*x+c)^4/(-a*d+b*c)/g^5/(b*x+a)^4-1/4*i^3*(d*x+c)^4*(A+B*ln(e *(b*x+a)/(d*x+c)))/(-a*d+b*c)/g^5/(b*x+a)^4
Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(89)=178\).
Time = 0.54 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.80 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {i^3 \left (4 A b^4 c^4+b^4 B c^4-4 a^4 A d^4-a^4 B d^4+16 A b^4 c^3 d x+4 b^4 B c^3 d x-16 a^3 A b d^4 x-4 a^3 b B d^4 x+24 A b^4 c^2 d^2 x^2+6 b^4 B c^2 d^2 x^2-24 a^2 A b^2 d^4 x^2-6 a^2 b^2 B d^4 x^2+16 A b^4 c d^3 x^3+4 b^4 B c d^3 x^3-16 a A b^3 d^4 x^3-4 a b^3 B d^4 x^3+4 B d^4 (a+b x)^4 \log (a+b x)+4 B \left (-a^4 d^4-4 a^3 b d^4 x-6 a^2 b^2 d^4 x^2-4 a b^3 d^4 x^3+b^4 c \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-4 a^4 B d^4 \log (c+d x)-16 a^3 b B d^4 x \log (c+d x)-24 a^2 b^2 B d^4 x^2 \log (c+d x)-16 a b^3 B d^4 x^3 \log (c+d x)-4 b^4 B d^4 x^4 \log (c+d x)\right )}{16 b^4 (b c-a d) g^5 (a+b x)^4} \] Input:
Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b* g*x)^5,x]
Output:
-1/16*(i^3*(4*A*b^4*c^4 + b^4*B*c^4 - 4*a^4*A*d^4 - a^4*B*d^4 + 16*A*b^4*c ^3*d*x + 4*b^4*B*c^3*d*x - 16*a^3*A*b*d^4*x - 4*a^3*b*B*d^4*x + 24*A*b^4*c ^2*d^2*x^2 + 6*b^4*B*c^2*d^2*x^2 - 24*a^2*A*b^2*d^4*x^2 - 6*a^2*b^2*B*d^4* x^2 + 16*A*b^4*c*d^3*x^3 + 4*b^4*B*c*d^3*x^3 - 16*a*A*b^3*d^4*x^3 - 4*a*b^ 3*B*d^4*x^3 + 4*B*d^4*(a + b*x)^4*Log[a + b*x] + 4*B*(-(a^4*d^4) - 4*a^3*b *d^4*x - 6*a^2*b^2*d^4*x^2 - 4*a*b^3*d^4*x^3 + b^4*c*(c^3 + 4*c^2*d*x + 6* c*d^2*x^2 + 4*d^3*x^3))*Log[(e*(a + b*x))/(c + d*x)] - 4*a^4*B*d^4*Log[c + d*x] - 16*a^3*b*B*d^4*x*Log[c + d*x] - 24*a^2*b^2*B*d^4*x^2*Log[c + d*x] - 16*a*b^3*B*d^4*x^3*Log[c + d*x] - 4*b^4*B*d^4*x^4*Log[c + d*x]))/(b^4*(b *c - a*d)*g^5*(a + b*x)^4)
Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2962, 2741}
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.
| \(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^3 \int \frac {(c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i^3 \left (-\frac {(c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {B (c+d x)^4}{16 (a+b x)^4}\right )}{g^5 (b c-a d)}\) |
Input:
Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5 ,x]
Output:
(i^3*(-1/16*(B*(c + d*x)^4)/(a + b*x)^4 - ((c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(a + b*x)^4)))/((b*c - a*d)*g^5)
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[(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. \(184\) vs. \(2(85)=170\).
Time = 2.14 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.08
| method | result | size |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{3} d^{2} e^{3} A}{4 \left (d a -b c \right )^{2} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} d^{2} e^{3} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{2} g^{5}}\right )}{d^{2}}\) | \(185\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{3} d^{2} e^{3} A}{4 \left (d a -b c \right )^{2} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} d^{2} e^{3} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{2} g^{5}}\right )}{d^{2}}\) | \(185\) |
| parts | \(\frac {i^{3} A \left (-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}+\frac {3 d^{2} \left (d a -b c \right )}{2 b^{4} \left (b x +a \right )^{2}}-\frac {d^{3}}{b^{4} \left (b x +a \right )}-\frac {d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{b^{4} \left (b x +a \right )^{3}}\right )}{g^{5}}-\frac {i^{3} B \,e^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{g^{5} \left (d a -b c \right )}\) | \(244\) |
| parallelrisch | \(\frac {4 B \,x^{3} a^{6} c^{2} d^{3} i^{3}-4 B \,x^{3} a^{3} b^{3} c^{5} i^{3}+24 A \,x^{2} a^{6} c^{3} d^{2} i^{3}-24 A \,x^{2} a^{4} b^{2} c^{5} i^{3}+6 B \,x^{2} a^{6} c^{3} d^{2} i^{3}-6 B \,x^{2} a^{4} b^{2} c^{5} i^{3}+16 A x \,a^{6} c^{4} d \,i^{3}-16 A x \,a^{5} b \,c^{5} i^{3}+4 B x \,a^{6} c^{4} d \,i^{3}-4 B x \,a^{5} b \,c^{5} i^{3}+4 A \,x^{4} a^{6} c \,d^{4} i^{3}-4 A \,x^{4} a^{2} b^{4} c^{5} i^{3}+B \,x^{4} a^{6} c \,d^{4} i^{3}-B \,x^{4} a^{2} b^{4} c^{5} i^{3}+16 A \,x^{3} a^{6} c^{2} d^{3} i^{3}-16 A \,x^{3} a^{3} b^{3} c^{5} i^{3}+4 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{5} i^{3}+4 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c \,d^{4} i^{3}+16 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{2} d^{3} i^{3}+24 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{3} d^{2} i^{3}+16 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{4} d \,i^{3}}{16 g^{5} \left (b x +a \right )^{4} a^{6} c \left (d a -b c \right )}\) | \(450\) |
| norman | \(\frac {\frac {B c \,d^{3} i^{3} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (d a -b c \right ) g}+\frac {B \,c^{3} d \,i^{3} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (d a -b c \right )}+\frac {\left (4 A \,c^{3} i^{3}+i^{3} c^{3} B \right ) x}{4 g a}+\frac {3 \left (4 A a \,c^{2} d \,i^{3}+4 A b \,c^{3} i^{3}+B a \,c^{2} d \,i^{3}+B b \,c^{3} i^{3}\right ) x^{2}}{8 g \,a^{2}}+\frac {\left (4 A \,a^{2} c \,d^{2} i^{3}+4 A a b \,c^{2} d \,i^{3}+4 A \,b^{2} c^{3} i^{3}+B \,a^{2} c \,d^{2} i^{3}+B a b \,c^{2} d \,i^{3}+B \,b^{2} c^{3} i^{3}\right ) x^{3}}{4 g \,a^{3}}+\frac {\left (4 d^{3} a^{3} A \,i^{3}+4 A \,a^{2} b c \,d^{2} i^{3}+4 A a \,b^{2} c^{2} d \,i^{3}+4 A \,b^{3} c^{3} i^{3}+d^{3} a^{3} B \,i^{3}+B \,a^{2} b c \,d^{2} i^{3}+B a \,b^{2} c^{2} d \,i^{3}+B \,b^{3} c^{3} i^{3}\right ) x^{4}}{16 a^{4} g}+\frac {i^{3} B \,c^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 g \left (d a -b c \right )}+\frac {B \,d^{4} i^{3} x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 \left (d a -b c \right ) g}+\frac {3 B \,c^{2} d^{2} i^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (d a -b c \right )}}{g^{4} \left (b x +a \right )^{4}}\) | \(485\) |
| risch | \(-\frac {i^{3} B \left (4 b^{3} d^{3} x^{3}+6 a \,b^{2} d^{3} x^{2}+6 b^{3} c \,d^{2} x^{2}+4 a^{2} b \,d^{3} x +4 a \,b^{2} c \,d^{2} x +4 b^{3} c^{2} d x +a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 \left (b x +a \right )^{4} g^{5} b^{4}}-\frac {i^{3} \left (-6 B \,b^{4} c^{2} d^{2} x^{2}+16 A a \,b^{3} d^{4} x^{3}+4 B a \,b^{3} d^{4} x^{3}-4 B \,b^{4} c \,d^{3} x^{3}+24 A \,a^{2} b^{2} d^{4} x^{2}+4 B \,a^{3} b \,d^{4} x -4 B \,b^{4} c^{3} d x +4 B \ln \left (d x +c \right ) a^{4} d^{4}-4 B \ln \left (-b x -a \right ) a^{4} d^{4}+4 B \ln \left (d x +c \right ) b^{4} d^{4} x^{4}-4 B \ln \left (-b x -a \right ) b^{4} d^{4} x^{4}+6 B \,a^{2} b^{2} d^{4} x^{2}+16 A \,a^{3} b \,d^{4} x -24 A \,b^{4} c^{2} d^{2} x^{2}-16 A \,b^{4} c \,d^{3} x^{3}+16 B \ln \left (d x +c \right ) a \,b^{3} d^{4} x^{3}-16 B \ln \left (-b x -a \right ) a \,b^{3} d^{4} x^{3}+24 B \ln \left (d x +c \right ) a^{2} b^{2} d^{4} x^{2}-24 B \ln \left (-b x -a \right ) a^{2} b^{2} d^{4} x^{2}+16 B \ln \left (d x +c \right ) a^{3} b \,d^{4} x -16 B \ln \left (-b x -a \right ) a^{3} b \,d^{4} x +4 A \,a^{4} d^{4}-16 A \,b^{4} c^{3} d x +B \,a^{4} d^{4}-B \,b^{4} c^{4}-4 A \,b^{4} c^{4}\right )}{16 \left (b x +a \right )^{4} g^{5} b^{4} \left (d a -b c \right )}\) | \(550\) |
| orering | \(\frac {\left (b x +a \right ) \left (-2 a^{3} b \,d^{4} x^{5}-2 a^{2} b^{2} c \,d^{3} x^{5}-2 a \,b^{3} c^{2} d^{2} x^{5}-2 b^{4} c^{3} d \,x^{5}+7 a^{4} d^{4} x^{4}-10 a^{3} b c \,d^{3} x^{4}-10 a^{2} b^{2} c^{2} d^{2} x^{4}-10 a \,b^{3} c^{3} d \,x^{4}-5 b^{4} c^{4} x^{4}+28 a^{4} c \,d^{3} x^{3}-20 a^{3} b \,c^{2} d^{2} x^{3}-20 a^{2} b^{2} c^{3} d \,x^{3}-20 a \,b^{3} c^{4} x^{3}+42 a^{4} c^{2} d^{2} x^{2}-20 a^{3} b \,c^{3} d \,x^{2}-30 a^{2} b^{2} c^{4} x^{2}+28 a^{4} c^{3} d x -20 a^{3} b \,c^{4} x +4 a^{4} c^{4}\right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{16 \left (d a -b c \right ) a^{4} \left (d x +c \right )^{3} \left (b g x +a g \right )^{5}}-\frac {x \left (a^{3} d^{3} x^{3}+a^{2} b c \,d^{2} x^{3}+a \,b^{2} c^{2} d \,x^{3}+b^{3} c^{3} x^{3}+4 a^{3} c \,d^{2} x^{2}+4 a^{2} b \,c^{2} d \,x^{2}+4 a \,b^{2} c^{3} x^{2}+6 a^{3} c^{2} d x +6 a^{2} b \,c^{3} x +4 a^{3} c^{3}\right ) \left (b x +a \right )^{2} \left (\frac {3 \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (b g x +a g \right )^{5}}+\frac {\left (d i x +c i \right )^{3} B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{5}}-\frac {5 \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{6}}\right )}{16 \left (d a -b c \right ) a^{4} \left (d x +c \right )^{2}}\) | \(602\) |
Input:
int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x,method=_RETU RNVERBOSE)
Output:
-1/d^2*e*(a*d-b*c)*(-1/4*i^3*d^2*e^3/(a*d-b*c)^2/g^5*A/(b*e/d+(a*d-b*c)*e/ d/(d*x+c))^4+i^3*d^2*e^3/(a*d-b*c)^2/g^5*B*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x +c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^ 4))
Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (85) = 170\).
Time = 0.09 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.99 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {4 \, {\left ({\left (4 \, A + B\right )} b^{4} c d^{3} - {\left (4 \, A + B\right )} a b^{3} d^{4}\right )} i^{3} x^{3} + 6 \, {\left ({\left (4 \, A + B\right )} b^{4} c^{2} d^{2} - {\left (4 \, A + B\right )} a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 4 \, {\left ({\left (4 \, A + B\right )} b^{4} c^{3} d - {\left (4 \, A + B\right )} a^{3} b d^{4}\right )} i^{3} x + {\left ({\left (4 \, A + B\right )} b^{4} c^{4} - {\left (4 \, A + B\right )} a^{4} d^{4}\right )} i^{3} + 4 \, {\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x + B b^{4} c^{4} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{16 \, {\left ({\left (b^{9} c - a b^{8} d\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c - a^{2} b^{7} d\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c - a^{3} b^{6} d\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c - a^{4} b^{5} d\right )} g^{5} x + {\left (a^{4} b^{5} c - a^{5} b^{4} d\right )} g^{5}\right )}} \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algo rithm="fricas")
Output:
-1/16*(4*((4*A + B)*b^4*c*d^3 - (4*A + B)*a*b^3*d^4)*i^3*x^3 + 6*((4*A + B )*b^4*c^2*d^2 - (4*A + B)*a^2*b^2*d^4)*i^3*x^2 + 4*((4*A + B)*b^4*c^3*d - (4*A + B)*a^3*b*d^4)*i^3*x + ((4*A + B)*b^4*c^4 - (4*A + B)*a^4*d^4)*i^3 + 4*(B*b^4*d^4*i^3*x^4 + 4*B*b^4*c*d^3*i^3*x^3 + 6*B*b^4*c^2*d^2*i^3*x^2 + 4*B*b^4*c^3*d*i^3*x + B*b^4*c^4*i^3)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c - a*b^8*d)*g^5*x^4 + 4*(a*b^8*c - a^2*b^7*d)*g^5*x^3 + 6*(a^2*b^7*c - a^3 *b^6*d)*g^5*x^2 + 4*(a^3*b^6*c - a^4*b^5*d)*g^5*x + (a^4*b^5*c - a^5*b^4*d )*g^5)
Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\text {Timed out} \] Input:
integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 3107 vs. \(2 (85) = 170\).
Time = 0.18 (sec) , antiderivative size = 3107, normalized size of antiderivative = 34.91 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algo rithm="maxima")
Output:
-1/48*B*d^3*i^3*(12*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*log(b*e*x/ (d*x + c) + a*e/(d*x + c))/(b^8*g^5*x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5* x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) + (25*a^3*b^3*c^3 - 23*a^4*b^2*c^2*d + 13*a^5*b*c*d^2 - 3*a^6*d^3 + 12*(4*b^6*c^3 - 6*a*b^5*c^2*d + 4*a^2*b^4*c *d^2 - a^3*b^3*d^3)*x^3 + 6*(18*a*b^5*c^3 - 22*a^2*b^4*c^2*d + 13*a^3*b^3* c*d^2 - 3*a^4*b^2*d^3)*x^2 + 4*(22*a^2*b^4*c^3 - 23*a^3*b^3*c^2*d + 13*a^4 *b^2*c*d^2 - 3*a^5*b*d^3)*x)/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^5*x^4 + 4*(a*b^10*c^3 - 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^ 2 - a^4*b^7*d^3)*g^5*x^3 + 6*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c* d^2 - a^5*b^6*d^3)*g^5*x^2 + 4*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6* c*d^2 - a^6*b^5*d^3)*g^5*x + (a^4*b^7*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*c* d^2 - a^7*b^4*d^3)*g^5) + 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^ 3 - a^3*d^4)*log(b*x + a)/((b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*g^5) - 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*log(d*x + c)/((b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^ 6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*g^5)) - 1/48*B*c*d^2*i^3*(12*(6 *b^2*x^2 + 4*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^5*x^ 4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) + (13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b ^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c...
Time = 0.39 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.57 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {1}{16} \, {\left (\frac {4 \, {\left (d x + c\right )}^{4} B e^{5} i^{3} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{4} g^{5}} + \frac {{\left (4 \, A e^{5} i^{3} + B e^{5} i^{3}\right )} {\left (d x + c\right )}^{4}}{{\left (b e x + a e\right )}^{4} g^{5}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algo rithm="giac")
Output:
-1/16*(4*(d*x + c)^4*B*e^5*i^3*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e) ^4*g^5) + (4*A*e^5*i^3 + B*e^5*i^3)*(d*x + c)^4/((b*e*x + a*e)^4*g^5))*(b* c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 29.16 (sec) , antiderivative size = 780, normalized size of antiderivative = 8.76 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {x^3\,\left (4\,A\,b^3\,d^3\,i^3+B\,b^3\,d^3\,i^3\right )+x^2\,\left (6\,A\,a\,b^2\,d^3\,i^3+\frac {3\,B\,a\,b^2\,d^3\,i^3}{2}+6\,A\,b^3\,c\,d^2\,i^3+\frac {3\,B\,b^3\,c\,d^2\,i^3}{2}\right )+x\,\left (4\,A\,a^2\,b\,d^3\,i^3+B\,a^2\,b\,d^3\,i^3+4\,A\,b^3\,c^2\,d\,i^3+B\,b^3\,c^2\,d\,i^3+4\,A\,a\,b^2\,c\,d^2\,i^3+B\,a\,b^2\,c\,d^2\,i^3\right )+A\,a^3\,d^3\,i^3+A\,b^3\,c^3\,i^3+\frac {B\,a^3\,d^3\,i^3}{4}+\frac {B\,b^3\,c^3\,i^3}{4}+A\,a\,b^2\,c^2\,d\,i^3+A\,a^2\,b\,c\,d^2\,i^3+\frac {B\,a\,b^2\,c^2\,d\,i^3}{4}+\frac {B\,a^2\,b\,c\,d^2\,i^3}{4}}{4\,a^4\,b^4\,g^5+16\,a^3\,b^5\,g^5\,x+24\,a^2\,b^6\,g^5\,x^2+16\,a\,b^7\,g^5\,x^3+4\,b^8\,g^5\,x^4}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x^2\,\left (b\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,a\,d^3\,i^3}{2\,b^4\,g^5}+\frac {B\,c\,d^2\,i^3}{2\,b^3\,g^5}\right )+\frac {3\,B\,a\,d^3\,i^3}{4\,b^3\,g^5}+\frac {3\,B\,c\,d^2\,i^3}{4\,b^2\,g^5}\right )+x\,\left (b\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,c^2\,d\,i^3}{4\,b^3\,g^5}\right )+a\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,a\,d^3\,i^3}{2\,b^4\,g^5}+\frac {B\,c\,d^2\,i^3}{2\,b^3\,g^5}\right )+\frac {3\,B\,c^2\,d\,i^3}{4\,b^2\,g^5}\right )+a\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,c^2\,d\,i^3}{4\,b^3\,g^5}\right )+\frac {B\,c^3\,i^3}{4\,b^2\,g^5}+\frac {B\,d^3\,i^3\,x^3}{b^2\,g^5}\right )}{4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3}-\frac {B\,d^4\,i^3\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^4\,g^5\,\left (a\,d-b\,c\right )} \] Input:
int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^5 ,x)
Output:
- (x^3*(4*A*b^3*d^3*i^3 + B*b^3*d^3*i^3) + x^2*(6*A*a*b^2*d^3*i^3 + (3*B*a *b^2*d^3*i^3)/2 + 6*A*b^3*c*d^2*i^3 + (3*B*b^3*c*d^2*i^3)/2) + x*(4*A*a^2* b*d^3*i^3 + B*a^2*b*d^3*i^3 + 4*A*b^3*c^2*d*i^3 + B*b^3*c^2*d*i^3 + 4*A*a* b^2*c*d^2*i^3 + B*a*b^2*c*d^2*i^3) + A*a^3*d^3*i^3 + A*b^3*c^3*i^3 + (B*a^ 3*d^3*i^3)/4 + (B*b^3*c^3*i^3)/4 + A*a*b^2*c^2*d*i^3 + A*a^2*b*c*d^2*i^3 + (B*a*b^2*c^2*d*i^3)/4 + (B*a^2*b*c*d^2*i^3)/4)/(4*a^4*b^4*g^5 + 4*b^8*g^5 *x^4 + 16*a^3*b^5*g^5*x + 16*a*b^7*g^5*x^3 + 24*a^2*b^6*g^5*x^2) - (log((e *(a + b*x))/(c + d*x))*(x^2*(b*(b*((B*a*d^3*i^3)/(4*b^5*g^5) + (B*c*d^2*i^ 3)/(4*b^4*g^5)) + (B*a*d^3*i^3)/(2*b^4*g^5) + (B*c*d^2*i^3)/(2*b^3*g^5)) + (3*B*a*d^3*i^3)/(4*b^3*g^5) + (3*B*c*d^2*i^3)/(4*b^2*g^5)) + x*(b*(a*((B* a*d^3*i^3)/(4*b^5*g^5) + (B*c*d^2*i^3)/(4*b^4*g^5)) + (B*c^2*d*i^3)/(4*b^3 *g^5)) + a*(b*((B*a*d^3*i^3)/(4*b^5*g^5) + (B*c*d^2*i^3)/(4*b^4*g^5)) + (B *a*d^3*i^3)/(2*b^4*g^5) + (B*c*d^2*i^3)/(2*b^3*g^5)) + (3*B*c^2*d*i^3)/(4* b^2*g^5)) + a*(a*((B*a*d^3*i^3)/(4*b^5*g^5) + (B*c*d^2*i^3)/(4*b^4*g^5)) + (B*c^2*d*i^3)/(4*b^3*g^5)) + (B*c^3*i^3)/(4*b^2*g^5) + (B*d^3*i^3*x^3)/(b ^2*g^5)))/(4*a^3*x + a^4/b + b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3) - (B*d^4 *i^3*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*1i)/(2*b^4*g^5*(a*d - b*c) )
Time = 0.19 (sec) , antiderivative size = 727, normalized size of antiderivative = 8.17 \[ \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)^5} \, dx=\frac {i \left (4 a^{5} c \,d^{3}-4 a^{2} b^{3} c^{4}-4 a^{2} b^{3} d^{4} x^{4}-4 \,\mathrm {log}\left (b x +a \right ) a^{4} b c \,d^{3}-4 \,\mathrm {log}\left (b x +a \right ) b^{5} c \,d^{3} x^{4}+4 \,\mathrm {log}\left (d x +c \right ) b^{5} c \,d^{3} x^{4}+4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{4} b c \,d^{3}+4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} c \,d^{3} x^{4}+16 a^{4} b c \,d^{3} x +24 a^{3} b^{2} c \,d^{3} x^{2}-16 a^{2} b^{3} c^{3} d x -24 a^{2} b^{3} c^{2} d^{2} x^{2}+4 a \,b^{4} c \,d^{3} x^{4}+4 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{3}-4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} d^{4} x^{4}+4 a^{3} b^{2} c \,d^{3} x +6 a^{2} b^{3} c \,d^{3} x^{2}-4 a \,b^{4} c^{3} d x -6 a \,b^{4} c^{2} d^{2} x^{2}+a^{4} b c \,d^{3}-a \,b^{4} c^{4}-a \,b^{4} d^{4} x^{4}+b^{5} c \,d^{3} x^{4}-4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{4}-24 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c \,d^{3} x^{2}-16 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c \,d^{3} x^{3}+16 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c \,d^{3} x +24 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c \,d^{3} x^{2}+16 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c \,d^{3} x^{3}+16 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b^{2} c \,d^{3} x +24 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{3} c \,d^{3} x^{2}-16 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{3} d x -24 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{2} d^{2} x^{2}-16 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c \,d^{3} x \right )}{16 a \,b^{3} g^{5} \left (a \,b^{4} d \,x^{4}-b^{5} c \,x^{4}+4 a^{2} b^{3} d \,x^{3}-4 a \,b^{4} c \,x^{3}+6 a^{3} b^{2} d \,x^{2}-6 a^{2} b^{3} c \,x^{2}+4 a^{4} b d x -4 a^{3} b^{2} c x +a^{5} d -a^{4} b c \right )} \] Input:
int((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x)
Output:
(i*( - 4*log(a + b*x)*a**4*b*c*d**3 - 16*log(a + b*x)*a**3*b**2*c*d**3*x - 24*log(a + b*x)*a**2*b**3*c*d**3*x**2 - 16*log(a + b*x)*a*b**4*c*d**3*x** 3 - 4*log(a + b*x)*b**5*c*d**3*x**4 + 4*log(c + d*x)*a**4*b*c*d**3 + 16*lo g(c + d*x)*a**3*b**2*c*d**3*x + 24*log(c + d*x)*a**2*b**3*c*d**3*x**2 + 16 *log(c + d*x)*a*b**4*c*d**3*x**3 + 4*log(c + d*x)*b**5*c*d**3*x**4 + 4*log ((a*e + b*e*x)/(c + d*x))*a**4*b*c*d**3 + 16*log((a*e + b*e*x)/(c + d*x))* a**3*b**2*c*d**3*x + 24*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c*d**3*x**2 - 4*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**4 - 16*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**3*d*x - 24*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**2*d**2*x **2 - 4*log((a*e + b*e*x)/(c + d*x))*a*b**4*d**4*x**4 + 4*log((a*e + b*e*x )/(c + d*x))*b**5*c*d**3*x**4 + 4*a**5*c*d**3 + 16*a**4*b*c*d**3*x + a**4* b*c*d**3 + 24*a**3*b**2*c*d**3*x**2 + 4*a**3*b**2*c*d**3*x - 4*a**2*b**3*c **4 - 16*a**2*b**3*c**3*d*x - 24*a**2*b**3*c**2*d**2*x**2 + 6*a**2*b**3*c* d**3*x**2 - 4*a**2*b**3*d**4*x**4 - a*b**4*c**4 - 4*a*b**4*c**3*d*x - 6*a* b**4*c**2*d**2*x**2 + 4*a*b**4*c*d**3*x**4 - a*b**4*d**4*x**4 + b**5*c*d** 3*x**4))/(16*a*b**3*g**5*(a**5*d - a**4*b*c + 4*a**4*b*d*x - 4*a**3*b**2*c *x + 6*a**3*b**2*d*x**2 - 6*a**2*b**3*c*x**2 + 4*a**2*b**3*d*x**3 - 4*a*b* *4*c*x**3 + a*b**4*d*x**4 - b**5*c*x**4))