Integrand size = 40, antiderivative size = 89 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {B i^2 (c+d x)^3}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d) g^4 (a+b x)^3} \] Output:
-1/9*B*i^2*(d*x+c)^3/(-a*d+b*c)/g^4/(b*x+a)^3-1/3*i^2*(d*x+c)^3*(A+B*ln(e* (b*x+a)/(d*x+c)))/(-a*d+b*c)/g^4/(b*x+a)^3
Leaf count is larger than twice the leaf count of optimal. \(315\) vs. \(2(89)=178\).
Time = 0.37 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.54 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {i^2 \left (3 A b^3 c^3+b^3 B c^3-3 a^3 A d^3-a^3 B d^3+9 A b^3 c^2 d x+3 b^3 B c^2 d x-9 a^2 A b d^3 x-3 a^2 b B d^3 x+9 A b^3 c d^2 x^2+3 b^3 B c d^2 x^2-9 a A b^2 d^3 x^2-3 a b^2 B d^3 x^2+3 B d^3 (a+b x)^3 \log (a+b x)+3 B (b c-a d) \left (a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-3 a^3 B d^3 \log (c+d x)-9 a^2 b B d^3 x \log (c+d x)-9 a b^2 B d^3 x^2 \log (c+d x)-3 b^3 B d^3 x^3 \log (c+d x)\right )}{9 b^3 (b c-a d) g^4 (a+b x)^3} \] Input:
Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b* g*x)^4,x]
Output:
-1/9*(i^2*(3*A*b^3*c^3 + b^3*B*c^3 - 3*a^3*A*d^3 - a^3*B*d^3 + 9*A*b^3*c^2 *d*x + 3*b^3*B*c^2*d*x - 9*a^2*A*b*d^3*x - 3*a^2*b*B*d^3*x + 9*A*b^3*c*d^2 *x^2 + 3*b^3*B*c*d^2*x^2 - 9*a*A*b^2*d^3*x^2 - 3*a*b^2*B*d^3*x^2 + 3*B*d^3 *(a + b*x)^3*Log[a + b*x] + 3*B*(b*c - a*d)*(a^2*d^2 + a*b*d*(c + 3*d*x) + b^2*(c^2 + 3*c*d*x + 3*d^2*x^2))*Log[(e*(a + b*x))/(c + d*x)] - 3*a^3*B*d ^3*Log[c + d*x] - 9*a^2*b*B*d^3*x*Log[c + d*x] - 9*a*b^2*B*d^3*x^2*Log[c + d*x] - 3*b^3*B*d^3*x^3*Log[c + d*x]))/(b^3*(b*c - a*d)*g^4*(a + b*x)^3)
Time = 0.31 (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.
| \(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^4} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^2 \int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 (b c-a d)}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i^2 \left (-\frac {(c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}-\frac {B (c+d x)^3}{9 (a+b x)^3}\right )}{g^4 (b c-a d)}\) |
Input:
Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^4 ,x]
Output:
(i^2*(-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*c - a*d)*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[(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 = 1.64 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.08
| method | result | size |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{2} d^{2} e^{2} A}{3 \left (d a -b c \right )^{2} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {i^{2} d^{2} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{2} g^{4}}\right )}{d^{2}}\) | \(185\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{2} d^{2} e^{2} A}{3 \left (d a -b c \right )^{2} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {i^{2} d^{2} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{2} g^{4}}\right )}{d^{2}}\) | \(185\) |
| parts | \(\frac {i^{2} A \left (\frac {d \left (d a -b c \right )}{b^{3} \left (b x +a \right )^{2}}-\frac {d^{2}}{b^{3} \left (b x +a \right )}-\frac {a^{2} d^{2}-2 a c d b +c^{2} b^{2}}{3 b^{3} \left (b x +a \right )^{3}}\right )}{g^{4}}-\frac {i^{2} B \,e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{g^{4} \left (d a -b c \right )}\) | \(192\) |
| norman | \(\frac {\frac {B c \,d^{2} i^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (d a -b c \right )}+\frac {B \,c^{2} d \,i^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (d a -b c \right )}-\frac {3 A a c d \,i^{2}+3 A b \,c^{2} i^{2}+B a c d \,i^{2}+B b \,c^{2} i^{2}}{9 g \,b^{2}}+\frac {\left (3 i^{2} A \,d^{2}+B \,d^{2} i^{2}\right ) x^{3}}{9 a g}-\frac {\left (3 A c d \,i^{2}+B c d \,i^{2}\right ) x}{3 g b}+\frac {B \,i^{2} c^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (d a -b c \right )}+\frac {B \,d^{3} i^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (d a -b c \right ) g}}{g^{3} \left (b x +a \right )^{3}}\) | \(267\) |
| parallelrisch | \(-\frac {-3 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{4} i^{2}+9 A \,x^{2} a \,b^{4} d^{4} i^{2}-9 A \,x^{2} b^{5} c \,d^{3} i^{2}+3 B \,x^{2} a \,b^{4} d^{4} i^{2}-3 B \,x^{2} b^{5} c \,d^{3} i^{2}+9 A x \,a^{2} b^{3} d^{4} i^{2}-9 A x \,b^{5} c^{2} d^{2} i^{2}+3 B x \,a^{2} b^{3} d^{4} i^{2}-3 B x \,b^{5} c^{2} d^{2} i^{2}-3 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{3} d \,i^{2}+B \,a^{3} b^{2} d^{4} i^{2}-B \,b^{5} c^{3} d \,i^{2}+3 A \,a^{3} b^{2} d^{4} i^{2}-3 A \,b^{5} c^{3} d \,i^{2}-9 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d^{2} i^{2}-9 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c \,d^{3} i^{2}}{9 g^{4} \left (b x +a \right )^{3} b^{5} d \left (d a -b c \right )}\) | \(333\) |
| risch | \(-\frac {B \,i^{2} \left (3 d^{2} b^{2} x^{2}+3 a b \,d^{2} x +3 b^{2} c d x +a^{2} d^{2}+a c d b +c^{2} b^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (b x +a \right )^{3} g^{4} b^{3}}-\frac {i^{2} \left (3 B \ln \left (d x +c \right ) b^{3} d^{3} x^{3}-3 B \ln \left (-b x -a \right ) b^{3} d^{3} x^{3}+9 B \ln \left (d x +c \right ) a \,b^{2} d^{3} x^{2}-9 B \ln \left (-b x -a \right ) a \,b^{2} d^{3} x^{2}+9 A a \,b^{2} d^{3} x^{2}-9 A \,b^{3} c \,d^{2} x^{2}+9 B \ln \left (d x +c \right ) a^{2} b \,d^{3} x -9 B \ln \left (-b x -a \right ) a^{2} b \,d^{3} x +3 B a \,b^{2} d^{3} x^{2}-3 B \,b^{3} c \,d^{2} x^{2}+9 A \,a^{2} b \,d^{3} x -9 A \,b^{3} c^{2} d x +3 B \ln \left (d x +c \right ) a^{3} d^{3}-3 B \ln \left (-b x -a \right ) a^{3} d^{3}+3 B \,a^{2} b \,d^{3} x -3 B \,b^{3} c^{2} d x +3 A \,a^{3} d^{3}-3 A \,b^{3} c^{3}+B \,a^{3} d^{3}-B \,c^{3} b^{3}\right )}{9 \left (b x +a \right )^{3} g^{4} b^{3} \left (d a -b c \right )}\) | \(393\) |
| orering | \(-\frac {\left (b x +a \right ) \left (2 a \,b^{2} d^{3} x^{4}+2 b^{3} c \,d^{2} x^{4}-5 a^{2} b \,d^{3} x^{3}+8 a \,b^{2} c \,d^{2} x^{3}+4 b^{3} c^{2} d \,x^{3}-15 a^{2} b c \,d^{2} x^{2}+12 a \,b^{2} c^{2} d \,x^{2}-11 a^{2} b \,c^{2} d x +2 a^{3} c^{2} d -7 a^{2} b \,c^{3}\right ) \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{9 a^{2} b \left (d a -b c \right ) \left (d x +c \right )^{2} \left (b g x +a g \right )^{4}}+\frac {\left (-a b \,d^{2} x^{3}-b^{2} c d \,x^{3}-3 a b c d \,x^{2}+a^{2} c^{2}\right ) \left (b x +a \right )^{2} \left (\frac {2 \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (b g x +a g \right )^{4}}+\frac {\left (d i x +c i \right )^{2} B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{4}}-\frac {4 \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{5}}\right )}{9 a^{2} b \left (d a -b c \right ) \left (d x +c \right )}\) | \(406\) |
Input:
int((d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x,method=_RETU RNVERBOSE)
Output:
-1/d^2*e*(a*d-b*c)*(-1/3*i^2*d^2*e^2/(a*d-b*c)^2/g^4*A/(b*e/d+(a*d-b*c)*e/ d/(d*x+c))^3+i^2*d^2*e^2/(a*d-b*c)^2/g^4*B*(-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 ))
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (85) = 170\).
Time = 0.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.04 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {3 \, {\left ({\left (3 \, A + B\right )} b^{3} c d^{2} - {\left (3 \, A + B\right )} a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left ({\left (3 \, A + B\right )} b^{3} c^{2} d - {\left (3 \, A + B\right )} a^{2} b d^{3}\right )} i^{2} x + {\left ({\left (3 \, A + B\right )} b^{3} c^{3} - {\left (3 \, A + B\right )} a^{3} d^{3}\right )} i^{2} + 3 \, {\left (B b^{3} d^{3} i^{2} x^{3} + 3 \, B b^{3} c d^{2} i^{2} x^{2} + 3 \, B b^{3} c^{2} d i^{2} x + B b^{3} c^{3} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{9 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algo rithm="fricas")
Output:
-1/9*(3*((3*A + B)*b^3*c*d^2 - (3*A + B)*a*b^2*d^3)*i^2*x^2 + 3*((3*A + B) *b^3*c^2*d - (3*A + B)*a^2*b*d^3)*i^2*x + ((3*A + B)*b^3*c^3 - (3*A + B)*a ^3*d^3)*i^2 + 3*(B*b^3*d^3*i^2*x^3 + 3*B*b^3*c*d^2*i^2*x^2 + 3*B*b^3*c^2*d *i^2*x + B*b^3*c^3*i^2)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c - a*b^6*d)*g ^4*x^3 + 3*(a*b^6*c - a^2*b^5*d)*g^4*x^2 + 3*(a^2*b^5*c - a^3*b^4*d)*g^4*x + (a^3*b^4*c - a^4*b^3*d)*g^4)
Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (76) = 152\).
Time = 9.79 (sec) , antiderivative size = 614, normalized size of antiderivative = 6.90 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=- \frac {B d^{3} i^{2} \log {\left (x + \frac {- \frac {B a^{2} d^{5} i^{2}}{a d - b c} + \frac {2 B a b c d^{4} i^{2}}{a d - b c} + B a d^{4} i^{2} - \frac {B b^{2} c^{2} d^{3} i^{2}}{a d - b c} + B b c d^{3} i^{2}}{2 B b d^{4} i^{2}} \right )}}{3 b^{3} g^{4} \left (a d - b c\right )} + \frac {B d^{3} i^{2} \log {\left (x + \frac {\frac {B a^{2} d^{5} i^{2}}{a d - b c} - \frac {2 B a b c d^{4} i^{2}}{a d - b c} + B a d^{4} i^{2} + \frac {B b^{2} c^{2} d^{3} i^{2}}{a d - b c} + B b c d^{3} i^{2}}{2 B b d^{4} i^{2}} \right )}}{3 b^{3} g^{4} \left (a d - b c\right )} + \frac {- 3 A a^{2} d^{2} i^{2} - 3 A a b c d i^{2} - 3 A b^{2} c^{2} i^{2} - B a^{2} d^{2} i^{2} - B a b c d i^{2} - B b^{2} c^{2} i^{2} + x^{2} \left (- 9 A b^{2} d^{2} i^{2} - 3 B b^{2} d^{2} i^{2}\right ) + x \left (- 9 A a b d^{2} i^{2} - 9 A b^{2} c d i^{2} - 3 B a b d^{2} i^{2} - 3 B b^{2} c d i^{2}\right )}{9 a^{3} b^{3} g^{4} + 27 a^{2} b^{4} g^{4} x + 27 a b^{5} g^{4} x^{2} + 9 b^{6} g^{4} x^{3}} + \frac {\left (- B a^{2} d^{2} i^{2} - B a b c d i^{2} - 3 B a b d^{2} i^{2} x - B b^{2} c^{2} i^{2} - 3 B b^{2} c d i^{2} x - 3 B b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{3 a^{3} b^{3} g^{4} + 9 a^{2} b^{4} g^{4} x + 9 a b^{5} g^{4} x^{2} + 3 b^{6} g^{4} x^{3}} \] Input:
integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4,x)
Output:
-B*d**3*i**2*log(x + (-B*a**2*d**5*i**2/(a*d - b*c) + 2*B*a*b*c*d**4*i**2/ (a*d - b*c) + B*a*d**4*i**2 - B*b**2*c**2*d**3*i**2/(a*d - b*c) + B*b*c*d* *3*i**2)/(2*B*b*d**4*i**2))/(3*b**3*g**4*(a*d - b*c)) + B*d**3*i**2*log(x + (B*a**2*d**5*i**2/(a*d - b*c) - 2*B*a*b*c*d**4*i**2/(a*d - b*c) + B*a*d* *4*i**2 + B*b**2*c**2*d**3*i**2/(a*d - b*c) + B*b*c*d**3*i**2)/(2*B*b*d**4 *i**2))/(3*b**3*g**4*(a*d - b*c)) + (-3*A*a**2*d**2*i**2 - 3*A*a*b*c*d*i** 2 - 3*A*b**2*c**2*i**2 - B*a**2*d**2*i**2 - B*a*b*c*d*i**2 - B*b**2*c**2*i **2 + x**2*(-9*A*b**2*d**2*i**2 - 3*B*b**2*d**2*i**2) + x*(-9*A*a*b*d**2*i **2 - 9*A*b**2*c*d*i**2 - 3*B*a*b*d**2*i**2 - 3*B*b**2*c*d*i**2))/(9*a**3* b**3*g**4 + 27*a**2*b**4*g**4*x + 27*a*b**5*g**4*x**2 + 9*b**6*g**4*x**3) + (-B*a**2*d**2*i**2 - B*a*b*c*d*i**2 - 3*B*a*b*d**2*i**2*x - B*b**2*c**2* i**2 - 3*B*b**2*c*d*i**2*x - 3*B*b**2*d**2*i**2*x**2)*log(e*(a + b*x)/(c + d*x))/(3*a**3*b**3*g**4 + 9*a**2*b**4*g**4*x + 9*a*b**5*g**4*x**2 + 3*b** 6*g**4*x**3)
Leaf count of result is larger than twice the leaf count of optimal. 1515 vs. \(2 (85) = 170\).
Time = 0.10 (sec) , antiderivative size = 1515, normalized size of antiderivative = 17.02 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algo rithm="maxima")
Output:
-1/18*B*d^2*i^2*(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/ 18*B*c*d*i^2*(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*a*b^2*c^2 - 22 *a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16 *a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^ 3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b ^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2* d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/18 *B*c^2*i^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^...
Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.57 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {1}{9} \, {\left (\frac {3 \, {\left (d x + c\right )}^{3} B e^{4} i^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{3} g^{4}} + \frac {{\left (3 \, A e^{4} i^{2} + B e^{4} i^{2}\right )} {\left (d x + c\right )}^{3}}{{\left (b e x + a e\right )}^{3} g^{4}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algo rithm="giac")
Output:
-1/9*(3*(d*x + c)^3*B*e^4*i^2*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^ 3*g^4) + (3*A*e^4*i^2 + B*e^4*i^2)*(d*x + c)^3/((b*e*x + a*e)^3*g^4))*(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 = 28.26 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.75 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {x^2\,\left (3\,A\,b^2\,d^2\,i^2+B\,b^2\,d^2\,i^2\right )+x\,\left (3\,A\,a\,b\,d^2\,i^2+B\,a\,b\,d^2\,i^2+3\,A\,b^2\,c\,d\,i^2+B\,b^2\,c\,d\,i^2\right )+A\,a^2\,d^2\,i^2+A\,b^2\,c^2\,i^2+\frac {B\,a^2\,d^2\,i^2}{3}+\frac {B\,b^2\,c^2\,i^2}{3}+A\,a\,b\,c\,d\,i^2+\frac {B\,a\,b\,c\,d\,i^2}{3}}{3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^4\,g^4}+\frac {B\,c\,d\,i^2}{3\,b^3\,g^4}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^4\,g^4}+\frac {B\,c\,d\,i^2}{3\,b^3\,g^4}\right )+\frac {2\,B\,a\,d^2\,i^2}{3\,b^3\,g^4}+\frac {2\,B\,c\,d\,i^2}{3\,b^2\,g^4}\right )+\frac {B\,c^2\,i^2}{3\,b^2\,g^4}+\frac {B\,d^2\,i^2\,x^2}{b^2\,g^4}\right )}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B\,d^3\,i^2\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )} \] Input:
int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^4 ,x)
Output:
- (x^2*(3*A*b^2*d^2*i^2 + B*b^2*d^2*i^2) + x*(3*A*a*b*d^2*i^2 + B*a*b*d^2* i^2 + 3*A*b^2*c*d*i^2 + B*b^2*c*d*i^2) + A*a^2*d^2*i^2 + A*b^2*c^2*i^2 + ( B*a^2*d^2*i^2)/3 + (B*b^2*c^2*i^2)/3 + A*a*b*c*d*i^2 + (B*a*b*c*d*i^2)/3)/ (3*a^3*b^3*g^4 + 3*b^6*g^4*x^3 + 9*a^2*b^4*g^4*x + 9*a*b^5*g^4*x^2) - (log ((e*(a + b*x))/(c + d*x))*(a*((B*a*d^2*i^2)/(3*b^4*g^4) + (B*c*d*i^2)/(3*b ^3*g^4)) + x*(b*((B*a*d^2*i^2)/(3*b^4*g^4) + (B*c*d*i^2)/(3*b^3*g^4)) + (2 *B*a*d^2*i^2)/(3*b^3*g^4) + (2*B*c*d*i^2)/(3*b^2*g^4)) + (B*c^2*i^2)/(3*b^ 2*g^4) + (B*d^2*i^2*x^2)/(b^2*g^4)))/(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^ 2) - (B*d^3*i^2*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*2i)/(3*b^3*g^4* (a*d - b*c))
Time = 0.19 (sec) , antiderivative size = 534, normalized size of antiderivative = 6.00 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\frac {3 a^{4} c \,d^{2}-3 a^{2} b^{2} c^{3}+a^{3} b c \,d^{2}-a \,b^{3} c^{3}-a \,b^{3} d^{3} x^{3}+b^{4} c \,d^{2} x^{3}-3 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c^{3}-3 a^{2} b^{2} d^{3} x^{3}-9 a^{2} b^{2} c^{2} d x +3 a \,b^{3} c \,d^{2} x^{3}+3 \,\mathrm {log}\left (d x +c \right ) a^{3} b c \,d^{2}-3 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} d^{3} x^{3}+3 a^{2} b^{2} c \,d^{2} x -3 a \,b^{3} c^{2} d x -3 \,\mathrm {log}\left (b x +a \right ) a^{3} b c \,d^{2}-3 \,\mathrm {log}\left (b x +a \right ) b^{4} c \,d^{2} x^{3}+3 \,\mathrm {log}\left (d x +c \right ) b^{4} c \,d^{2} x^{3}+3 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b c \,d^{2}+3 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{4} c \,d^{2} x^{3}+9 a^{3} b c \,d^{2} x -9 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} c \,d^{2} x -9 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} c \,d^{2} x^{2}+9 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c \,d^{2} x +9 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c \,d^{2} x^{2}+9 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} c \,d^{2} x -9 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c^{2} d x}{9 a \,b^{2} g^{4} \left (a \,b^{3} d \,x^{3}-b^{4} c \,x^{3}+3 a^{2} b^{2} d \,x^{2}-3 a \,b^{3} c \,x^{2}+3 a^{3} b d x -3 a^{2} b^{2} c x +a^{4} d -a^{3} b c \right )} \] Input:
int((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x)
Output:
( - 3*log(a + b*x)*a**3*b*c*d**2 - 9*log(a + b*x)*a**2*b**2*c*d**2*x - 9*l og(a + b*x)*a*b**3*c*d**2*x**2 - 3*log(a + b*x)*b**4*c*d**2*x**3 + 3*log(c + d*x)*a**3*b*c*d**2 + 9*log(c + d*x)*a**2*b**2*c*d**2*x + 9*log(c + d*x) *a*b**3*c*d**2*x**2 + 3*log(c + d*x)*b**4*c*d**2*x**3 + 3*log((a*e + b*e*x )/(c + d*x))*a**3*b*c*d**2 + 9*log((a*e + b*e*x)/(c + d*x))*a**2*b**2*c*d* *2*x - 3*log((a*e + b*e*x)/(c + d*x))*a*b**3*c**3 - 9*log((a*e + b*e*x)/(c + d*x))*a*b**3*c**2*d*x - 3*log((a*e + b*e*x)/(c + d*x))*a*b**3*d**3*x**3 + 3*log((a*e + b*e*x)/(c + d*x))*b**4*c*d**2*x**3 + 3*a**4*c*d**2 + 9*a** 3*b*c*d**2*x + a**3*b*c*d**2 - 3*a**2*b**2*c**3 - 9*a**2*b**2*c**2*d*x + 3 *a**2*b**2*c*d**2*x - 3*a**2*b**2*d**3*x**3 - a*b**3*c**3 - 3*a*b**3*c**2* d*x + 3*a*b**3*c*d**2*x**3 - a*b**3*d**3*x**3 + b**4*c*d**2*x**3)/(9*a*b** 2*g**4*(a**4*d - a**3*b*c + 3*a**3*b*d*x - 3*a**2*b**2*c*x + 3*a**2*b**2*d *x**2 - 3*a*b**3*c*x**2 + a*b**3*d*x**3 - b**4*c*x**3))