Integrand size = 42, antiderivative size = 147 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {B^2 i^3 (c+d x)^4}{32 (b c-a d) g^5 (a+b x)^4}-\frac {B i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d) g^5 (a+b x)^4}-\frac {i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d) g^5 (a+b x)^4} \]
-1/32*B^2*i^3*(d*x+c)^4/(-a*d+b*c)/g^5/(b*x+a)^4-1/8*B*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)^4-1/4*i^3*(d*x+c)^4*(A+B*ln( e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)/g^5/(b*x+a)^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.71 (sec) , antiderivative size = 2401, normalized size of antiderivative = 16.33 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Result too large to show} \]
-1/4*((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g^5*( a + b*x)^4) - (d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2) /(b^4*g^5*(a + b*x)^3) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/( c + d*x)])^2)/(2*b^4*g^5*(a + b*x)^2) - (d^3*i^3*(A + B*Log[(e*(a + b*x))/ (c + d*x)])^2)/(b^4*g^5*(a + b*x)) + (B*(b*c - a*d)^4*i^3*(-1/4*A/((b*c - a*d)*(a + b*x)^4) - B/(16*(b*c - a*d)*(a + b*x)^4) + (A*d)/(3*(b*c - a*d)^ 2*(a + b*x)^3) + (7*B*d)/(36*(b*c - a*d)^2*(a + b*x)^3) - (A*d^2)/(2*(b*c - a*d)^3*(a + b*x)^2) - (13*B*d^2)/(24*(b*c - a*d)^3*(a + b*x)^2) + (A*d^3 )/((b*c - a*d)^4*(a + b*x)) + (25*B*d^3)/(12*(b*c - a*d)^4*(a + b*x)) + (A *d^4*Log[a + b*x])/(b*c - a*d)^5 + (25*B*d^4*Log[a + b*x])/(12*(b*c - a*d) ^5) - (B*d^4*Log[a + b*x]^2)/(2*(b*c - a*d)^5) - (B*Log[(e*(a + b*x))/(c + d*x)])/(4*(b*c - a*d)*(a + b*x)^4) + (B*d*Log[(e*(a + b*x))/(c + d*x)])/( 3*(b*c - a*d)^2*(a + b*x)^3) - (B*d^2*Log[(e*(a + b*x))/(c + d*x)])/(2*(b* c - a*d)^3*(a + b*x)^2) + (B*d^3*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d )^4*(a + b*x)) + (B*d^4*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)])/(b*c - a*d)^5 - (A*d^4*Log[c + d*x])/(b*c - a*d)^5 - (25*B*d^4*Log[c + d*x])/(12* (b*c - a*d)^5) + (B*d^4*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b *c - a*d)^5 - (B*d^4*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x])/(b*c - a*d )^5 - (B*d^4*Log[c + d*x]^2)/(2*(b*c - a*d)^5) + (B*d^4*Log[a + b*x]*Log[( b*(c + d*x))/(b*c - a*d)])/(b*c - a*d)^5 + (B*d^4*PolyLog[2, -((d*(a + ...
Time = 0.37 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2742, 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.
| \(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^3 \int \frac {(c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {i^3 \left (\frac {1}{2} B \int \frac {(c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}-\frac {(c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}\right )}{g^5 (b c-a d)}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i^3 \left (\frac {1}{2} B \left (-\frac {(c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {B (c+d x)^4}{16 (a+b x)^4}\right )-\frac {(c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}\right )}{g^5 (b c-a d)}\) |
(i^3*(-1/4*((c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a + b*x)^ 4 + (B*(-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)))/2))/((b*c - a*d)*g^5)
3.1.81.3.1 Defintions of rubi rules used
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_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b , c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
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. \(366\) vs. \(2(141)=282\).
Time = 1.65 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.50
| method | result | size |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{3} d^{2} e^{3} A^{2}}{4 \left (a d -c b \right )^{2} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {2 i^{3} d^{2} e^{3} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{2} g^{5}}+\frac {i^{3} d^{2} e^{3} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{2} g^{5}}\right )}{d^{2}}\) | \(367\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{3} d^{2} e^{3} A^{2}}{4 \left (a d -c b \right )^{2} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {2 i^{3} d^{2} e^{3} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{2} g^{5}}+\frac {i^{3} d^{2} e^{3} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{2} g^{5}}\right )}{d^{2}}\) | \(367\) |
| parts | \(\frac {i^{3} A^{2} \left (-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{4} \left (b x +a \right )^{3}}+\frac {3 d^{2} \left (a d -c b \right )}{2 b^{4} \left (b x +a \right )^{2}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}-\frac {d^{3}}{b^{4} \left (b x +a \right )}\right )}{g^{5}}-\frac {i^{3} B^{2} e^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{g^{5} \left (a d -c b \right )}-\frac {2 i^{3} B A \,e^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{g^{5} \left (a d -c b \right )}\) | \(423\) |
| risch | \(-\frac {i^{3} A^{2} d^{3} a^{2}}{g^{5} b^{4} \left (b x +a \right )^{3}}+\frac {2 i^{3} A^{2} d^{2} a c}{g^{5} b^{3} \left (b x +a \right )^{3}}-\frac {i^{3} A^{2} d \,c^{2}}{g^{5} b^{2} \left (b x +a \right )^{3}}+\frac {3 i^{3} A^{2} d^{3} a}{2 g^{5} b^{4} \left (b x +a \right )^{2}}-\frac {3 i^{3} A^{2} d^{2} c}{2 g^{5} b^{3} \left (b x +a \right )^{2}}+\frac {i^{3} A^{2} a^{3} d^{3}}{4 g^{5} b^{4} \left (b x +a \right )^{4}}-\frac {3 i^{3} A^{2} a^{2} c \,d^{2}}{4 g^{5} b^{3} \left (b x +a \right )^{4}}+\frac {3 i^{3} A^{2} a \,c^{2} d}{4 g^{5} b^{2} \left (b x +a \right )^{4}}-\frac {i^{3} A^{2} c^{3}}{4 g^{5} b \left (b x +a \right )^{4}}-\frac {i^{3} A^{2} d^{3}}{g^{5} b^{4} \left (b x +a \right )}+\frac {i^{3} B^{2} e^{4} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 g^{5} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} B^{2} e^{4} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{8 g^{5} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} B^{2} e^{4}}{32 g^{5} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} B A \,e^{4} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{5} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} B A \,e^{4}}{8 g^{5} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{4}}\) | \(633\) |
| norman | \(\frac {\frac {B^{2} c \,d^{3} i^{3} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{\left (a d -c b \right ) g}+\frac {B^{2} c^{3} d \,i^{3} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a d -c b \right )}+\frac {\left (8 i^{3} A^{2} c^{3}+4 i^{3} c^{3} B A +i^{3} c^{3} B^{2}\right ) x}{8 g a}+\frac {3 \left (8 A^{2} a \,c^{2} d \,i^{3}+8 A^{2} b \,c^{3} i^{3}+4 A B a \,c^{2} d \,i^{3}+4 A B b \,c^{3} i^{3}+B^{2} a \,c^{2} d \,i^{3}+B^{2} b \,c^{3} i^{3}\right ) x^{2}}{16 g \,a^{2}}+\frac {\left (8 A^{2} a^{2} c \,d^{2} i^{3}+8 A^{2} a b \,c^{2} d \,i^{3}+8 A^{2} b^{2} c^{3} i^{3}+4 A B \,a^{2} c \,d^{2} i^{3}+4 A B a b \,c^{2} d \,i^{3}+4 A B \,b^{2} c^{3} i^{3}+B^{2} a^{2} c \,d^{2} i^{3}+B^{2} a b \,c^{2} d \,i^{3}+B^{2} b^{2} c^{3} i^{3}\right ) x^{3}}{8 g \,a^{3}}+\frac {\left (8 A^{2} d^{3} i^{3} a^{3}+8 A^{2} a^{2} b c \,d^{2} i^{3}+8 A^{2} a \,b^{2} c^{2} d \,i^{3}+8 A^{2} b^{3} c^{3} i^{3}+4 A B \,a^{3} d^{3} i^{3}+4 A B \,a^{2} b c \,d^{2} i^{3}+4 A B a \,b^{2} c^{2} d \,i^{3}+4 A B \,b^{3} c^{3} i^{3}+B^{2} a^{3} d^{3} i^{3}+B^{2} a^{2} b c \,d^{2} i^{3}+B^{2} a \,b^{2} c^{2} d \,i^{3}+B^{2} b^{3} c^{3} i^{3}\right ) x^{4}}{32 a^{4} g}+\frac {i^{3} B^{2} c^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{4 g \left (a d -c b \right )}+\frac {B \,c^{4} i^{3} \left (4 A +B \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{8 g \left (a d -c b \right )}+\frac {B^{2} d^{4} i^{3} x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{4 \left (a d -c b \right ) g}+\frac {3 B^{2} c^{2} d^{2} i^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a d -c b \right )}+\frac {d^{4} B \,i^{3} \left (4 A +B \right ) x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{8 g \left (a d -c b \right )}+\frac {c B \,i^{3} d^{3} \left (4 A +B \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a d -c b \right )}+\frac {3 c^{2} B \,i^{3} d^{2} \left (4 A +B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 g \left (a d -c b \right )}+\frac {c^{3} B \,i^{3} d \left (4 A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a d -c b \right )}}{g^{4} \left (b x +a \right )^{4}}\) | \(899\) |
| parallelrisch | \(\frac {B^{2} x^{4} a^{6} c \,d^{4} i^{3}+64 A B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{2} d^{3} i^{3}+96 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{3} d^{2} i^{3}+64 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{4} d \,i^{3}+16 A B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c \,d^{4} i^{3}+4 B^{2} x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c \,d^{4} i^{3}+32 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c^{2} d^{3} i^{3}+4 A B \,x^{4} a^{6} c \,d^{4} i^{3}-4 A B \,x^{4} a^{2} b^{4} c^{5} i^{3}+16 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{2} d^{3} i^{3}-48 A^{2} x^{2} a^{4} b^{2} c^{5} i^{3}+6 B^{2} x^{2} a^{6} c^{3} d^{2} i^{3}-6 B^{2} x^{2} a^{4} b^{2} c^{5} i^{3}+32 A^{2} x \,a^{6} c^{4} d \,i^{3}-32 A^{2} x \,a^{5} b \,c^{5} i^{3}+16 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{5} i^{3}+4 B^{2} x \,a^{6} c^{4} d \,i^{3}-4 B^{2} x \,a^{5} b \,c^{5} i^{3}+32 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c^{4} d \,i^{3}+24 A B \,x^{2} a^{6} c^{3} d^{2} i^{3}-24 A B \,x^{2} a^{4} b^{2} c^{5} i^{3}+16 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{4} d \,i^{3}+16 A B x \,a^{6} c^{4} d \,i^{3}-16 A B x \,a^{5} b \,c^{5} i^{3}+8 B^{2} x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c \,d^{4} i^{3}-B^{2} x^{4} a^{2} b^{4} c^{5} i^{3}+8 A^{2} x^{4} a^{6} c \,d^{4} i^{3}-8 A^{2} x^{4} a^{2} b^{4} c^{5} i^{3}+32 A^{2} x^{3} a^{6} c^{2} d^{3} i^{3}-32 A^{2} x^{3} a^{3} b^{3} c^{5} i^{3}+4 B^{2} x^{3} a^{6} c^{2} d^{3} i^{3}-4 B^{2} x^{3} a^{3} b^{3} c^{5} i^{3}+48 A^{2} x^{2} a^{6} c^{3} d^{2} i^{3}+16 A B \,x^{3} a^{6} c^{2} d^{3} i^{3}-16 A B \,x^{3} a^{3} b^{3} c^{5} i^{3}+24 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{3} d^{2} i^{3}+48 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c^{3} d^{2} i^{3}+8 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c^{5} i^{3}+4 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{5} i^{3}}{32 g^{5} \left (b x +a \right )^{4} a^{6} c \left (a d -c b \right )}\) | \(965\) |
-1/d^2*e*(a*d-b*c)*(-1/4*i^3*d^2*e^3/(a*d-b*c)^2/g^5*A^2/(b*e/d+(a*d-b*c)* e/d/(d*x+c))^4+2*i^3*d^2*e^3/(a*d-b*c)^2/g^5*A*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)+i^3*d^2*e^3/(a*d-b*c)^2/g^5*B^2*(-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))^2-1/8/(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))-1/32/(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. 559 vs. \(2 (141) = 282\).
Time = 0.31 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.80 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {4 \, {\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c d^{3} - {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a b^{3} d^{4}\right )} i^{3} x^{3} + 6 \, {\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} - {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 4 \, {\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{3} d - {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a^{3} b d^{4}\right )} i^{3} x + {\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} - {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a^{4} d^{4}\right )} i^{3} + 8 \, {\left (B^{2} b^{4} d^{4} i^{3} x^{4} + 4 \, B^{2} b^{4} c d^{3} i^{3} x^{3} + 6 \, B^{2} b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B^{2} b^{4} c^{3} d i^{3} x + B^{2} b^{4} c^{4} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 4 \, {\left ({\left (4 \, A B + B^{2}\right )} b^{4} d^{4} i^{3} x^{4} + 4 \, {\left (4 \, A B + B^{2}\right )} b^{4} c d^{3} i^{3} x^{3} + 6 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{3} d i^{3} x + {\left (4 \, A B + B^{2}\right )} b^{4} c^{4} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{32 \, {\left ({\left (b^{9} c - a b^{8} d\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c - a^{2} b^{7} d\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c - a^{3} b^{6} d\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c - a^{4} b^{5} d\right )} g^{5} x + {\left (a^{4} b^{5} c - a^{5} b^{4} d\right )} g^{5}\right )}} \]
-1/32*(4*((8*A^2 + 4*A*B + B^2)*b^4*c*d^3 - (8*A^2 + 4*A*B + B^2)*a*b^3*d^ 4)*i^3*x^3 + 6*((8*A^2 + 4*A*B + B^2)*b^4*c^2*d^2 - (8*A^2 + 4*A*B + B^2)* a^2*b^2*d^4)*i^3*x^2 + 4*((8*A^2 + 4*A*B + B^2)*b^4*c^3*d - (8*A^2 + 4*A*B + B^2)*a^3*b*d^4)*i^3*x + ((8*A^2 + 4*A*B + B^2)*b^4*c^4 - (8*A^2 + 4*A*B + B^2)*a^4*d^4)*i^3 + 8*(B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x + B^2*b^4*c^4*i^3)*log(( b*e*x + a*e)/(d*x + c))^2 + 4*((4*A*B + B^2)*b^4*d^4*i^3*x^4 + 4*(4*A*B + B^2)*b^4*c*d^3*i^3*x^3 + 6*(4*A*B + B^2)*b^4*c^2*d^2*i^3*x^2 + 4*(4*A*B + B^2)*b^4*c^3*d*i^3*x + (4*A*B + B^2)*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. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 11688 vs. \(2 (141) = 282\).
Time = 1.03 (sec) , antiderivative size = 11688, normalized size of antiderivative = 79.51 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-1/4*(4*b*x + a)*B^2*c^2*d*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^6 *g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2 *g^5) - 1/4*(6*b^2*x^2 + 4*a*b*x + a^2)*B^2*c*d^2*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(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) - 1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*B^2*d^3*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(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) + 1/288* (12*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^ 3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 1 3*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3) *g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3) *g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^ 3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2* d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3) *g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a* b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log(b*e *x/(d*x + c) + a*e/(d*x + c)) - (9*b^4*c^4 - 64*a*b^3*c^3*d + 216*a^2*b^2* c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*...
Time = 0.64 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.48 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {1}{32} \, {\left (\frac {8 \, {\left (d x + c\right )}^{4} B^{2} e^{5} i^{3} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )}^{4} g^{5}} + \frac {4 \, {\left (4 \, A B e^{5} i^{3} + B^{2} e^{5} i^{3}\right )} {\left (d x + c\right )}^{4} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{4} g^{5}} + \frac {{\left (8 \, A^{2} e^{5} i^{3} + 4 \, A B e^{5} i^{3} + B^{2} e^{5} i^{3}\right )} {\left (d x + c\right )}^{4}}{{\left (b e x + a e\right )}^{4} g^{5}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
-1/32*(8*(d*x + c)^4*B^2*e^5*i^3*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^4*g^5) + 4*(4*A*B*e^5*i^3 + B^2*e^5*i^3)*(d*x + c)^4*log((b*e*x + a*e )/(d*x + c))/((b*e*x + a*e)^4*g^5) + (8*A^2*e^5*i^3 + 4*A*B*e^5*i^3 + B^2* 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 = 6.02 (sec) , antiderivative size = 1565, normalized size of antiderivative = 10.65 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-(24*A^2*a^4*d^4*i^3 - 24*A^2*b^4*c^4*i^3 + 3*B^2*a^4*d^4*i^3 - 3*B^2*b^4* c^4*i^3 + 12*A*B*a^4*d^4*i^3 - 12*A*B*b^4*c^4*i^3 - 24*B^2*b^4*c^4*i^3*log ((e*(a + b*x))/(c + d*x))^2 + B^2*a^4*d^4*i^3*atan((a*d*1i + b*c*1i + b*d* x*2i)/(a*d - b*c))*24i + 12*B^2*a^4*d^4*i^3*log((e*(a + b*x))/(c + d*x)) - 12*B^2*b^4*c^4*i^3*log((e*(a + b*x))/(c + d*x)) - 24*B^2*b^4*d^4*i^3*x^4* log((e*(a + b*x))/(c + d*x))^2 + 96*A^2*a^3*b*d^4*i^3*x + 12*B^2*a^3*b*d^4 *i^3*x - 96*A^2*b^4*c^3*d*i^3*x - 12*B^2*b^4*c^3*d*i^3*x + B^2*b^4*d^4*i^3 *x^4*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*24i + A*B*a^4*d^4*i^3* atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*96i + 96*A^2*a*b^3*d^4*i^3* x^3 + 12*B^2*a*b^3*d^4*i^3*x^3 - 96*A^2*b^4*c*d^3*i^3*x^3 - 12*B^2*b^4*c*d ^3*i^3*x^3 + 48*A*B*a^4*d^4*i^3*log((e*(a + b*x))/(c + d*x)) - 48*A*B*b^4* c^4*i^3*log((e*(a + b*x))/(c + d*x)) + 144*A^2*a^2*b^2*d^4*i^3*x^2 + 18*B^ 2*a^2*b^2*d^4*i^3*x^2 - 144*A^2*b^4*c^2*d^2*i^3*x^2 - 18*B^2*b^4*c^2*d^2*i ^3*x^2 + 48*A*B*a^3*b*d^4*i^3*x - 48*A*B*b^4*c^3*d*i^3*x + 48*B^2*a*b^3*d^ 4*i^3*x^3*log((e*(a + b*x))/(c + d*x)) - 96*B^2*b^4*c^3*d*i^3*x*log((e*(a + b*x))/(c + d*x))^2 - 48*B^2*b^4*c*d^3*i^3*x^3*log((e*(a + b*x))/(c + d*x )) + A*B*b^4*d^4*i^3*x^4*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*96 i + B^2*a^3*b*d^4*i^3*x*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*96i + 48*A*B*a*b^3*d^4*i^3*x^3 - 48*A*B*b^4*c*d^3*i^3*x^3 + 72*B^2*a^2*b^2*d^ 4*i^3*x^2*log((e*(a + b*x))/(c + d*x)) - 72*B^2*b^4*c^2*d^2*i^3*x^2*log...