\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^6} \, dx\) [19]

Optimal result

Integrand size = 40, antiderivative size = 281 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {B d^2 i^2 (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \]

[Out]

-1/9*B*d^2*i^2*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/8*b*B*d*i^2*(d*x+c)^4/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/25*b^
2*B*i^2*(d*x+c)^5/(-a*d+b*c)^3/g^6/(b*x+a)^5-1/3*d^2*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^
6/(b*x+a)^3+1/2*b*d*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/5*b^2*i^2*(d*x+c)^5
*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^6/(b*x+a)^5

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 45, 2372, 12, 14} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {b^2 i^2 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac {d^2 i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b d i^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^6 (a+b x)^4 (b c-a d)^3}-\frac {b^2 B i^2 (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac {B d^2 i^2 (c+d x)^3}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b B d i^2 (c+d x)^4}{8 g^6 (a+b x)^4 (b c-a d)^3} \]

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

-1/9*(B*d^2*i^2*(c + d*x)^3)/((b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*(c + d*x)^4)/(8*(b*c - a*d)^3*g^6*(a
+ b*x)^4) - (b^2*B*i^2*(c + d*x)^5)/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[(e*(
a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c +
d*x)]))/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(b*c
- a*d)^3*g^6*(a + b*x)^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(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] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps \begin{align*} \text {integral}& = \frac {i^2 \text {Subst}\left (\int \frac {(b-d x)^2 (A+B \log (e x))}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^6} \\ & = -\frac {d^2 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)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2\right ) \text {Subst}\left (\int \frac {-6 b^2+15 b d x-10 d^2 x^2}{30 x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^6} \\ & = -\frac {d^2 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)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2\right ) \text {Subst}\left (\int \frac {-6 b^2+15 b d x-10 d^2 x^2}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 (b c-a d)^3 g^6} \\ & = -\frac {d^2 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)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2\right ) \text {Subst}\left (\int \left (-\frac {6 b^2}{x^6}+\frac {15 b d}{x^5}-\frac {10 d^2}{x^4}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{30 (b c-a d)^3 g^6} \\ & = -\frac {B d^2 i^2 (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 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)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.22 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\frac {i^2 \left (-\frac {360 A b^2 c^2}{(a+b x)^5}-\frac {72 b^2 B c^2}{(a+b x)^5}+\frac {720 a A b c d}{(a+b x)^5}+\frac {144 a b B c d}{(a+b x)^5}-\frac {360 a^2 A d^2}{(a+b x)^5}-\frac {72 a^2 B d^2}{(a+b x)^5}-\frac {900 A b c d}{(a+b x)^4}-\frac {135 b B c d}{(a+b x)^4}+\frac {900 a A d^2}{(a+b x)^4}+\frac {135 a B d^2}{(a+b x)^4}-\frac {600 A d^2}{(a+b x)^3}-\frac {20 B d^2}{(a+b x)^3}+\frac {30 B d^3}{(b c-a d) (a+b x)^2}-\frac {60 B d^4}{(b c-a d)^2 (a+b x)}-\frac {60 B d^5 \log (a+b x)}{(b c-a d)^3}-\frac {60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5}+\frac {60 B d^5 \log (c+d x)}{(b c-a d)^3}\right )}{1800 b^3 g^6} \]

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

(i^2*((-360*A*b^2*c^2)/(a + b*x)^5 - (72*b^2*B*c^2)/(a + b*x)^5 + (720*a*A*b*c*d)/(a + b*x)^5 + (144*a*b*B*c*d
)/(a + b*x)^5 - (360*a^2*A*d^2)/(a + b*x)^5 - (72*a^2*B*d^2)/(a + b*x)^5 - (900*A*b*c*d)/(a + b*x)^4 - (135*b*
B*c*d)/(a + b*x)^4 + (900*a*A*d^2)/(a + b*x)^4 + (135*a*B*d^2)/(a + b*x)^4 - (600*A*d^2)/(a + b*x)^3 - (20*B*d
^2)/(a + b*x)^3 + (30*B*d^3)/((b*c - a*d)*(a + b*x)^2) - (60*B*d^4)/((b*c - a*d)^2*(a + b*x)) - (60*B*d^5*Log[
a + b*x])/(b*c - a*d)^3 - (60*B*(a^2*d^2 + a*b*d*(3*c + 5*d*x) + b^2*(6*c^2 + 15*c*d*x + 10*d^2*x^2))*Log[(e*(
a + b*x))/(c + d*x)])/(a + b*x)^5 + (60*B*d^5*Log[c + d*x])/(b*c - a*d)^3))/(1800*b^3*g^6)

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.53

method	result	size
parts	\(\frac {i^{2} A \left (-\frac {d^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {d \left (a d -c b \right )}{2 b^{3} \left (b x +a \right )^{4}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{5 b^{3} \left (b x +a \right )^{5}}\right )}{g^{6}}-\frac {i^{2} B \left (a d -c b \right )^{3} e^{3} \left (\frac {d^{6} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{6}}-\frac {2 d^{5} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{6}}+\frac {d^{4} e^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (a d -c b \right )^{6}}\right )}{g^{6} d^{4}}\)	\(430\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{2} d^{2} e^{4} A \,b^{2}}{5 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}+\frac {i^{2} d^{3} e^{3} A b}{2 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{4} e^{2} A}{3 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {i^{2} d^{2} e^{4} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (a d -c b \right )^{4} g^{6}}-\frac {2 i^{2} d^{3} e^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{4} g^{6}}+\frac {i^{2} d^{4} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4} g^{6}}\right )}{d^{2}}\)	\(532\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{2} d^{2} e^{4} A \,b^{2}}{5 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}+\frac {i^{2} d^{3} e^{3} A b}{2 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{4} e^{2} A}{3 \left (a d -c b \right )^{4} g^{6} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {i^{2} d^{2} e^{4} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (a d -c b \right )^{4} g^{6}}-\frac {2 i^{2} d^{3} e^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{4} g^{6}}+\frac {i^{2} d^{4} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4} g^{6}}\right )}{d^{2}}\)	\(532\)
risch	\(-\frac {i^{2} B \left (10 d^{2} x^{2} b^{2}+5 a b \,d^{2} x +15 b^{2} c d x +a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{30 \left (b x +a \right )^{5} g^{6} b^{3}}-\frac {\left (60 B \ln \left (d x +c \right ) a^{5} d^{5}-72 B \,b^{5} c^{5}+60 B a \,b^{4} d^{5} x^{4}-60 B \,b^{5} c \,d^{4} x^{4}+270 B \,a^{2} b^{3} d^{5} x^{3}+30 B \,b^{5} c^{2} d^{3} x^{3}+600 A \,a^{3} b^{2} d^{5} x^{2}+470 B \,a^{3} b^{2} d^{5} x^{2}-20 B \,b^{5} c^{3} d^{2} x^{2}+300 A \,a^{4} b \,d^{5} x -1800 A \,a^{2} b^{3} c \,d^{4} x^{2}-300 B a \,b^{4} c \,d^{4} x^{3}-600 B \,a^{2} b^{3} c \,d^{4} x^{2}+150 B a \,b^{4} c^{2} d^{3} x^{2}-600 B \,a^{2} b^{3} c^{2} d^{3} x +500 B a \,b^{4} c^{3} d^{2} x -60 B \ln \left (-b x -a \right ) a^{5} d^{5}+235 B \,a^{4} b \,d^{5} x -135 B \,b^{5} c^{4} d x +1800 A a \,b^{4} c^{2} d^{3} x^{2}-1800 A \,a^{2} b^{3} c^{2} d^{3} x +2400 A a \,b^{4} c^{3} d^{2} x -300 B \ln \left (-b x -a \right ) a \,b^{4} d^{5} x^{4}+300 B \ln \left (d x +c \right ) a \,b^{4} d^{5} x^{4}-600 B \ln \left (-b x -a \right ) a^{2} b^{3} d^{5} x^{3}+600 B \ln \left (d x +c \right ) a^{2} b^{3} d^{5} x^{3}-600 B \ln \left (-b x -a \right ) a^{3} b^{2} d^{5} x^{2}+60 A \,a^{5} d^{5}+600 B \ln \left (d x +c \right ) a^{3} b^{2} d^{5} x^{2}-300 B \ln \left (-b x -a \right ) a^{4} b \,d^{5} x +300 B \ln \left (d x +c \right ) a^{4} b \,d^{5} x -360 A \,b^{5} c^{5}+47 B \,a^{5} d^{5}-600 A \,a^{2} b^{3} c^{3} d^{2}+900 A a \,b^{4} c^{4} d -200 B \,a^{2} b^{3} c^{3} d^{2}+225 B a \,b^{4} c^{4} d -600 A \,b^{5} c^{3} d^{2} x^{2}-900 A \,b^{5} c^{4} d x -60 B \ln \left (-b x -a \right ) b^{5} d^{5} x^{5}+60 B \ln \left (d x +c \right ) b^{5} d^{5} x^{5}\right ) i^{2}}{1800 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{5} g^{6} b^{3}}\)	\(768\)
parallelrisch	\(-\frac {-1800 A \,x^{2} a^{2} b^{7} c \,d^{5} i^{2}+1800 A \,x^{2} a \,b^{8} c^{2} d^{4} i^{2}-600 B \,x^{2} a^{2} b^{7} c \,d^{5} i^{2}-1800 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} c^{2} d^{4} i^{2}-300 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} d^{6} i^{2}-600 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} d^{6} i^{2}-300 B \,x^{3} a \,b^{8} c \,d^{5} i^{2}-600 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} c^{3} d^{3} i^{2}+150 B \,x^{2} a \,b^{8} c^{2} d^{4} i^{2}-900 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} c^{4} d^{2} i^{2}-1800 A x \,a^{2} b^{7} c^{2} d^{4} i^{2}+2400 A x a \,b^{8} c^{3} d^{3} i^{2}-600 B x \,a^{2} b^{7} c^{2} d^{4} i^{2}+500 B x a \,b^{8} c^{3} d^{3} i^{2}-600 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} c^{3} d^{3} i^{2}+900 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} c^{4} d^{2} i^{2}-200 B \,a^{2} b^{7} c^{3} d^{3} i^{2}+225 B a \,b^{8} c^{4} d^{2} i^{2}-60 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} d^{6} i^{2}+60 B \,x^{4} a \,b^{8} d^{6} i^{2}-60 B \,x^{4} b^{9} c \,d^{5} i^{2}+270 B \,x^{3} a^{2} b^{7} d^{6} i^{2}+30 B \,x^{3} b^{9} c^{2} d^{4} i^{2}+600 A \,x^{2} a^{3} b^{6} d^{6} i^{2}-600 A \,x^{2} b^{9} c^{3} d^{3} i^{2}+470 B \,x^{2} a^{3} b^{6} d^{6} i^{2}-20 B \,x^{2} b^{9} c^{3} d^{3} i^{2}+300 A x \,a^{4} b^{5} d^{6} i^{2}-900 A x \,b^{9} c^{4} d^{2} i^{2}+235 B x \,a^{4} b^{5} d^{6} i^{2}-135 B x \,b^{9} c^{4} d^{2} i^{2}-360 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{9} c^{5} d \,i^{2}+2400 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} c^{3} d^{3} i^{2}-600 A \,a^{2} b^{7} c^{3} d^{3} i^{2}+900 A a \,b^{8} c^{4} d^{2} i^{2}-1800 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{7} c \,d^{5} i^{2}+1800 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{8} c^{2} d^{4} i^{2}+60 A \,a^{5} b^{4} d^{6} i^{2}-360 A \,b^{9} c^{5} d \,i^{2}+47 B \,a^{5} b^{4} d^{6} i^{2}-72 B \,b^{9} c^{5} d \,i^{2}}{1800 g^{6} \left (b x +a \right )^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{7} d}\)	\(935\)
norman	\(\text {Expression too large to display}\)	\(1043\)

[In]

int((d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x,method=_RETURNVERBOSE)

[Out]

i^2*A/g^6*(-1/3*d^2/b^3/(b*x+a)^3+1/2*d*(a*d-b*c)/b^3/(b*x+a)^4-1/5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3/(b*x+a)^5)
-i^2*B/g^6/d^4*(a*d-b*c)^3*e^3*(d^6/(a*d-b*c)^6*(-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)-2*d^5/(a*d-b*c)^6*b*e*(-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)+d^4/(a*d-b*c)^6*e^2*b^2*(-1/5/(b*e/d+(a*d-b*c
)*e/d/(d*x+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (269) = 538\).

Time = 0.37 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.87 \[ \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)^6} \, dx=-\frac {60 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} i^{2} x^{4} - 30 \, {\left (B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} + 9 \, B a^{2} b^{3} d^{5}\right )} i^{2} x^{3} + 10 \, {\left (2 \, {\left (30 \, A + B\right )} b^{5} c^{3} d^{2} - 15 \, {\left (12 \, A + B\right )} a b^{4} c^{2} d^{3} + 60 \, {\left (3 \, A + B\right )} a^{2} b^{3} c d^{4} - {\left (60 \, A + 47 \, B\right )} a^{3} b^{2} d^{5}\right )} i^{2} x^{2} + 5 \, {\left (9 \, {\left (20 \, A + 3 \, B\right )} b^{5} c^{4} d - 20 \, {\left (24 \, A + 5 \, B\right )} a b^{4} c^{3} d^{2} + 120 \, {\left (3 \, A + B\right )} a^{2} b^{3} c^{2} d^{3} - {\left (60 \, A + 47 \, B\right )} a^{4} b d^{5}\right )} i^{2} x + {\left (72 \, {\left (5 \, A + B\right )} b^{5} c^{5} - 225 \, {\left (4 \, A + B\right )} a b^{4} c^{4} d + 200 \, {\left (3 \, A + B\right )} a^{2} b^{3} c^{3} d^{2} - {\left (60 \, A + 47 \, B\right )} a^{5} d^{5}\right )} i^{2} + 60 \, {\left (B b^{5} d^{5} i^{2} x^{5} + 5 \, B a b^{4} d^{5} i^{2} x^{4} + 10 \, B a^{2} b^{3} d^{5} i^{2} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3} + 3 \, B a^{2} b^{3} c d^{4}\right )} i^{2} x^{2} + 5 \, {\left (3 \, B b^{5} c^{4} d - 8 \, B a b^{4} c^{3} d^{2} + 6 \, B a^{2} b^{3} c^{2} d^{3}\right )} i^{2} x + {\left (6 \, B b^{5} c^{5} - 15 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{1800 \, {\left ({\left (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}\right )} g^{6} x^{5} + 5 \, {\left (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}\right )} g^{6} x^{4} + 10 \, {\left (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}\right )} g^{6} x^{3} + 10 \, {\left (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}\right )} g^{6} x^{2} + 5 \, {\left (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}\right )} g^{6} x + {\left (a^{5} b^{6} c^{3} - 3 \, a^{6} b^{5} c^{2} d + 3 \, a^{7} b^{4} c d^{2} - a^{8} b^{3} d^{3}\right )} g^{6}\right )}} \]

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="fricas")

[Out]

-1/1800*(60*(B*b^5*c*d^4 - B*a*b^4*d^5)*i^2*x^4 - 30*(B*b^5*c^2*d^3 - 10*B*a*b^4*c*d^4 + 9*B*a^2*b^3*d^5)*i^2*
x^3 + 10*(2*(30*A + B)*b^5*c^3*d^2 - 15*(12*A + B)*a*b^4*c^2*d^3 + 60*(3*A + B)*a^2*b^3*c*d^4 - (60*A + 47*B)*
a^3*b^2*d^5)*i^2*x^2 + 5*(9*(20*A + 3*B)*b^5*c^4*d - 20*(24*A + 5*B)*a*b^4*c^3*d^2 + 120*(3*A + B)*a^2*b^3*c^2
*d^3 - (60*A + 47*B)*a^4*b*d^5)*i^2*x + (72*(5*A + B)*b^5*c^5 - 225*(4*A + B)*a*b^4*c^4*d + 200*(3*A + B)*a^2*
b^3*c^3*d^2 - (60*A + 47*B)*a^5*d^5)*i^2 + 60*(B*b^5*d^5*i^2*x^5 + 5*B*a*b^4*d^5*i^2*x^4 + 10*B*a^2*b^3*d^5*i^
2*x^3 + 10*(B*b^5*c^3*d^2 - 3*B*a*b^4*c^2*d^3 + 3*B*a^2*b^3*c*d^4)*i^2*x^2 + 5*(3*B*b^5*c^4*d - 8*B*a*b^4*c^3*
d^2 + 6*B*a^2*b^3*c^2*d^3)*i^2*x + (6*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2)*i^2)*log((b*e*x + a
*e)/(d*x + c)))/((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^6*x^5 + 5*(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^6*x^4 + 10*(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^6*x^3 + 10*(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^6*x^2 + 5*(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^6*x + (a^5*b^6*c^3 - 3*a^6*b^5*c^2*d + 3*a^7*b^4*c*d^2 -
a^8*b^3*d^3)*g^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1300 vs. \(2 (258) = 516\).

Time = 76.59 (sec) , antiderivative size = 1300, normalized size of antiderivative = 4.63 \[ \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)^6} \, dx=- \frac {B d^{5} i^{2} \log {\left (x + \frac {- \frac {B a^{4} d^{9} i^{2}}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{8} i^{2}}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{7} i^{2}}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{6} i^{2}}{\left (a d - b c\right )^{3}} + B a d^{6} i^{2} - \frac {B b^{4} c^{4} d^{5} i^{2}}{\left (a d - b c\right )^{3}} + B b c d^{5} i^{2}}{2 B b d^{6} i^{2}} \right )}}{30 b^{3} g^{6} \left (a d - b c\right )^{3}} + \frac {B d^{5} i^{2} \log {\left (x + \frac {\frac {B a^{4} d^{9} i^{2}}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{8} i^{2}}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{7} i^{2}}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{6} i^{2}}{\left (a d - b c\right )^{3}} + B a d^{6} i^{2} + \frac {B b^{4} c^{4} d^{5} i^{2}}{\left (a d - b c\right )^{3}} + B b c d^{5} i^{2}}{2 B b d^{6} i^{2}} \right )}}{30 b^{3} g^{6} \left (a d - b c\right )^{3}} + \frac {- 60 A a^{4} d^{4} i^{2} - 60 A a^{3} b c d^{3} i^{2} - 60 A a^{2} b^{2} c^{2} d^{2} i^{2} + 540 A a b^{3} c^{3} d i^{2} - 360 A b^{4} c^{4} i^{2} - 47 B a^{4} d^{4} i^{2} - 47 B a^{3} b c d^{3} i^{2} - 47 B a^{2} b^{2} c^{2} d^{2} i^{2} + 153 B a b^{3} c^{3} d i^{2} - 72 B b^{4} c^{4} i^{2} - 60 B b^{4} d^{4} i^{2} x^{4} + x^{3} \left (- 270 B a b^{3} d^{4} i^{2} + 30 B b^{4} c d^{3} i^{2}\right ) + x^{2} \left (- 600 A a^{2} b^{2} d^{4} i^{2} + 1200 A a b^{3} c d^{3} i^{2} - 600 A b^{4} c^{2} d^{2} i^{2} - 470 B a^{2} b^{2} d^{4} i^{2} + 130 B a b^{3} c d^{3} i^{2} - 20 B b^{4} c^{2} d^{2} i^{2}\right ) + x \left (- 300 A a^{3} b d^{4} i^{2} - 300 A a^{2} b^{2} c d^{3} i^{2} + 1500 A a b^{3} c^{2} d^{2} i^{2} - 900 A b^{4} c^{3} d i^{2} - 235 B a^{3} b d^{4} i^{2} - 235 B a^{2} b^{2} c d^{3} i^{2} + 365 B a b^{3} c^{2} d^{2} i^{2} - 135 B b^{4} c^{3} d i^{2}\right )}{1800 a^{7} b^{3} d^{2} g^{6} - 3600 a^{6} b^{4} c d g^{6} + 1800 a^{5} b^{5} c^{2} g^{6} + x^{5} \cdot \left (1800 a^{2} b^{8} d^{2} g^{6} - 3600 a b^{9} c d g^{6} + 1800 b^{10} c^{2} g^{6}\right ) + x^{4} \cdot \left (9000 a^{3} b^{7} d^{2} g^{6} - 18000 a^{2} b^{8} c d g^{6} + 9000 a b^{9} c^{2} g^{6}\right ) + x^{3} \cdot \left (18000 a^{4} b^{6} d^{2} g^{6} - 36000 a^{3} b^{7} c d g^{6} + 18000 a^{2} b^{8} c^{2} g^{6}\right ) + x^{2} \cdot \left (18000 a^{5} b^{5} d^{2} g^{6} - 36000 a^{4} b^{6} c d g^{6} + 18000 a^{3} b^{7} c^{2} g^{6}\right ) + x \left (9000 a^{6} b^{4} d^{2} g^{6} - 18000 a^{5} b^{5} c d g^{6} + 9000 a^{4} b^{6} c^{2} g^{6}\right )} + \frac {\left (- B a^{2} d^{2} i^{2} - 3 B a b c d i^{2} - 5 B a b d^{2} i^{2} x - 6 B b^{2} c^{2} i^{2} - 15 B b^{2} c d i^{2} x - 10 B b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{30 a^{5} b^{3} g^{6} + 150 a^{4} b^{4} g^{6} x + 300 a^{3} b^{5} g^{6} x^{2} + 300 a^{2} b^{6} g^{6} x^{3} + 150 a b^{7} g^{6} x^{4} + 30 b^{8} g^{6} x^{5}} \]

[In]

integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**6,x)

[Out]

-B*d**5*i**2*log(x + (-B*a**4*d**9*i**2/(a*d - b*c)**3 + 4*B*a**3*b*c*d**8*i**2/(a*d - b*c)**3 - 6*B*a**2*b**2
*c**2*d**7*i**2/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**6*i**2/(a*d - b*c)**3 + B*a*d**6*i**2 - B*b**4*c**4*d**5*i
**2/(a*d - b*c)**3 + B*b*c*d**5*i**2)/(2*B*b*d**6*i**2))/(30*b**3*g**6*(a*d - b*c)**3) + B*d**5*i**2*log(x + (
B*a**4*d**9*i**2/(a*d - b*c)**3 - 4*B*a**3*b*c*d**8*i**2/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**7*i**2/(a*d -
b*c)**3 - 4*B*a*b**3*c**3*d**6*i**2/(a*d - b*c)**3 + B*a*d**6*i**2 + B*b**4*c**4*d**5*i**2/(a*d - b*c)**3 + B*
b*c*d**5*i**2)/(2*B*b*d**6*i**2))/(30*b**3*g**6*(a*d - b*c)**3) + (-60*A*a**4*d**4*i**2 - 60*A*a**3*b*c*d**3*i
**2 - 60*A*a**2*b**2*c**2*d**2*i**2 + 540*A*a*b**3*c**3*d*i**2 - 360*A*b**4*c**4*i**2 - 47*B*a**4*d**4*i**2 -
47*B*a**3*b*c*d**3*i**2 - 47*B*a**2*b**2*c**2*d**2*i**2 + 153*B*a*b**3*c**3*d*i**2 - 72*B*b**4*c**4*i**2 - 60*
B*b**4*d**4*i**2*x**4 + x**3*(-270*B*a*b**3*d**4*i**2 + 30*B*b**4*c*d**3*i**2) + x**2*(-600*A*a**2*b**2*d**4*i
**2 + 1200*A*a*b**3*c*d**3*i**2 - 600*A*b**4*c**2*d**2*i**2 - 470*B*a**2*b**2*d**4*i**2 + 130*B*a*b**3*c*d**3*
i**2 - 20*B*b**4*c**2*d**2*i**2) + x*(-300*A*a**3*b*d**4*i**2 - 300*A*a**2*b**2*c*d**3*i**2 + 1500*A*a*b**3*c*
*2*d**2*i**2 - 900*A*b**4*c**3*d*i**2 - 235*B*a**3*b*d**4*i**2 - 235*B*a**2*b**2*c*d**3*i**2 + 365*B*a*b**3*c*
*2*d**2*i**2 - 135*B*b**4*c**3*d*i**2))/(1800*a**7*b**3*d**2*g**6 - 3600*a**6*b**4*c*d*g**6 + 1800*a**5*b**5*c
**2*g**6 + x**5*(1800*a**2*b**8*d**2*g**6 - 3600*a*b**9*c*d*g**6 + 1800*b**10*c**2*g**6) + x**4*(9000*a**3*b**
7*d**2*g**6 - 18000*a**2*b**8*c*d*g**6 + 9000*a*b**9*c**2*g**6) + x**3*(18000*a**4*b**6*d**2*g**6 - 36000*a**3
*b**7*c*d*g**6 + 18000*a**2*b**8*c**2*g**6) + x**2*(18000*a**5*b**5*d**2*g**6 - 36000*a**4*b**6*c*d*g**6 + 180
00*a**3*b**7*c**2*g**6) + x*(9000*a**6*b**4*d**2*g**6 - 18000*a**5*b**5*c*d*g**6 + 9000*a**4*b**6*c**2*g**6))
+ (-B*a**2*d**2*i**2 - 3*B*a*b*c*d*i**2 - 5*B*a*b*d**2*i**2*x - 6*B*b**2*c**2*i**2 - 15*B*b**2*c*d*i**2*x - 10
*B*b**2*d**2*i**2*x**2)*log(e*(a + b*x)/(c + d*x))/(30*a**5*b**3*g**6 + 150*a**4*b**4*g**6*x + 300*a**3*b**5*g
**6*x**2 + 300*a**2*b**6*g**6*x**3 + 150*a*b**7*g**6*x**4 + 30*b**8*g**6*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3029 vs. \(2 (269) = 538\).

Time = 0.35 (sec) , antiderivative size = 3029, normalized size of antiderivative = 10.78 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\text {Too large to display} \]

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

-1/1800*B*d^2*i^2*(60*(10*b^2*x^2 + 5*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^8*g^6*x^5 + 5*a*b^7
*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) + (47*a^2*b^4*c^4 - 278*a^
3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^
4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 14
0*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*
c^3*d + 702*a^3*b^3*c^2*d^2 - 278*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*x)/((b^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^
2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^4 - 4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*
b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b^9*c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 +
a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*
g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b
^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 + a^9*b^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*
b*c*d^4 + a^2*d^5)*log(b*x + a)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^
4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4
*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6)) - 1/600*B*c*d*i^2*(60*(5*b
*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*
g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) + (27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b
*c*d^3 + 77*a^5*d^4 - 60*(5*b^5*c*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x
^3 - 10*(10*b^5*c^3*d - 67*a*b^4*c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*
c^3*d + 232*a^2*b^3*c^2*d^2 - 428*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2
*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^
7*c*d^3 + a^5*b^6*d^4)*g^6*x^4 + 10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6
*b^5*d^4)*g^6*x^3 + 10*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6
*x^2 + 5*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*
c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b
*x + a)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g
^6) + 60*(5*b*c*d^4 - a*d^5)*log(d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3
+ 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6)) - 1/300*B*c^2*i^2*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*
a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*
b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^
10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8
*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^
4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2
- 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c
*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*
g^6) + 60*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^6*x^5 + 5*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^
3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2
- 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*x + c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*
a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6)) - 1/10*(5*b*x + a)*A*c*d*i^2/(b^7*g^
6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(10*
b^2*x^2 + 5*a*b*x + a^2)*A*d^2*i^2/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 +
5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/5*A*c^2*i^2/(b^6*g^6*x^5 + 5*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^
3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6)

Giac [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.58 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {1}{1800} \, {\left (\frac {60 \, {\left (6 \, B b^{2} e^{6} i^{2} - \frac {15 \, {\left (b e x + a e\right )} B b d e^{5} i^{2}}{d x + c} + \frac {10 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{4} i^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b e x + a e\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b e x + a e\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {360 \, A b^{2} e^{6} i^{2} + 72 \, B b^{2} e^{6} i^{2} - \frac {900 \, {\left (b e x + a e\right )} A b d e^{5} i^{2}}{d x + c} - \frac {225 \, {\left (b e x + a e\right )} B b d e^{5} i^{2}}{d x + c} + \frac {600 \, {\left (b e x + a e\right )}^{2} A d^{2} e^{4} i^{2}}{{\left (d x + c\right )}^{2}} + \frac {200 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{4} i^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b e x + a e\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b e x + a e\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

-1/1800*(60*(6*B*b^2*e^6*i^2 - 15*(b*e*x + a*e)*B*b*d*e^5*i^2/(d*x + c) + 10*(b*e*x + a*e)^2*B*d^2*e^4*i^2/(d*
x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*e*x + a*e)^5*a*b*c*d*g^
6/(d*x + c)^5 + (b*e*x + a*e)^5*a^2*d^2*g^6/(d*x + c)^5) + (360*A*b^2*e^6*i^2 + 72*B*b^2*e^6*i^2 - 900*(b*e*x
+ a*e)*A*b*d*e^5*i^2/(d*x + c) - 225*(b*e*x + a*e)*B*b*d*e^5*i^2/(d*x + c) + 600*(b*e*x + a*e)^2*A*d^2*e^4*i^2
/(d*x + c)^2 + 200*(b*e*x + a*e)^2*B*d^2*e^4*i^2/(d*x + c)^2)/((b*e*x + a*e)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*
e*x + a*e)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*e*x + a*e)^5*a^2*d^2*g^6/(d*x + c)^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)))

Mupad [B] (verification not implemented)

Time = 4.61 (sec) , antiderivative size = 941, normalized size of antiderivative = 3.35 \[ \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)^6} \, dx=\frac {B\,d^5\,i^2\,\mathrm {atanh}\left (\frac {30\,a^3\,b^3\,d^3\,g^6-30\,a^2\,b^4\,c\,d^2\,g^6-30\,a\,b^5\,c^2\,d\,g^6+30\,b^6\,c^3\,g^6}{30\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{15\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^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}{30\,b^4\,g^6}+\frac {B\,c\,d\,i^2}{10\,b^3\,g^6}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^4\,g^6}+\frac {B\,c\,d\,i^2}{10\,b^3\,g^6}\right )+\frac {2\,B\,a\,d^2\,i^2}{15\,b^3\,g^6}+\frac {2\,B\,c\,d\,i^2}{5\,b^2\,g^6}\right )+\frac {B\,c^2\,i^2}{5\,b^2\,g^6}+\frac {B\,d^2\,i^2\,x^2}{3\,b^2\,g^6}\right )}{5\,a^4\,x+\frac {a^5}{b}+b^4\,x^5+10\,a^3\,b\,x^2+5\,a\,b^3\,x^4+10\,a^2\,b^2\,x^3}-\frac {\frac {60\,A\,a^4\,d^4\,i^2+360\,A\,b^4\,c^4\,i^2+47\,B\,a^4\,d^4\,i^2+72\,B\,b^4\,c^4\,i^2+60\,A\,a^2\,b^2\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,c^2\,d^2\,i^2-540\,A\,a\,b^3\,c^3\,d\,i^2+60\,A\,a^3\,b\,c\,d^3\,i^2-153\,B\,a\,b^3\,c^3\,d\,i^2+47\,B\,a^3\,b\,c\,d^3\,i^2}{60\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (60\,A\,a^2\,b^2\,d^4\,i^2+47\,B\,a^2\,b^2\,d^4\,i^2+60\,A\,b^4\,c^2\,d^2\,i^2+2\,B\,b^4\,c^2\,d^2\,i^2-120\,A\,a\,b^3\,c\,d^3\,i^2-13\,B\,a\,b^3\,c\,d^3\,i^2\right )}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (60\,A\,a^3\,b\,d^4\,i^2+47\,B\,a^3\,b\,d^4\,i^2+180\,A\,b^4\,c^3\,d\,i^2+27\,B\,b^4\,c^3\,d\,i^2-300\,A\,a\,b^3\,c^2\,d^2\,i^2+60\,A\,a^2\,b^2\,c\,d^3\,i^2-73\,B\,a\,b^3\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,c\,d^3\,i^2\right )}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^3\,\left (9\,B\,a\,b^3\,d^3\,i^2-B\,b^4\,c\,d^2\,i^2\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^4\,d^4\,i^2\,x^4}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{30\,a^5\,b^3\,g^6+150\,a^4\,b^4\,g^6\,x+300\,a^3\,b^5\,g^6\,x^2+300\,a^2\,b^6\,g^6\,x^3+150\,a\,b^7\,g^6\,x^4+30\,b^8\,g^6\,x^5} \]

[In]

int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^6,x)

[Out]

(B*d^5*i^2*atanh((30*b^6*c^3*g^6 + 30*a^3*b^3*d^3*g^6 - 30*a*b^5*c^2*d*g^6 - 30*a^2*b^4*c*d^2*g^6)/(30*b^3*g^6
*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(15*b^3*g^6*(a*d - b*c)^3) - (log(
(e*(a + b*x))/(c + d*x))*(a*((B*a*d^2*i^2)/(30*b^4*g^6) + (B*c*d*i^2)/(10*b^3*g^6)) + x*(b*((B*a*d^2*i^2)/(30*
b^4*g^6) + (B*c*d*i^2)/(10*b^3*g^6)) + (2*B*a*d^2*i^2)/(15*b^3*g^6) + (2*B*c*d*i^2)/(5*b^2*g^6)) + (B*c^2*i^2)
/(5*b^2*g^6) + (B*d^2*i^2*x^2)/(3*b^2*g^6)))/(5*a^4*x + a^5/b + b^4*x^5 + 10*a^3*b*x^2 + 5*a*b^3*x^4 + 10*a^2*
b^2*x^3) - ((60*A*a^4*d^4*i^2 + 360*A*b^4*c^4*i^2 + 47*B*a^4*d^4*i^2 + 72*B*b^4*c^4*i^2 + 60*A*a^2*b^2*c^2*d^2
*i^2 + 47*B*a^2*b^2*c^2*d^2*i^2 - 540*A*a*b^3*c^3*d*i^2 + 60*A*a^3*b*c*d^3*i^2 - 153*B*a*b^3*c^3*d*i^2 + 47*B*
a^3*b*c*d^3*i^2)/(60*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x^2*(60*A*a^2*b^2*d^4*i^2 + 47*B*a^2*b^2*d^4*i^2 + 60
*A*b^4*c^2*d^2*i^2 + 2*B*b^4*c^2*d^2*i^2 - 120*A*a*b^3*c*d^3*i^2 - 13*B*a*b^3*c*d^3*i^2))/(6*(a^2*d^2 + b^2*c^
2 - 2*a*b*c*d)) + (x*(60*A*a^3*b*d^4*i^2 + 47*B*a^3*b*d^4*i^2 + 180*A*b^4*c^3*d*i^2 + 27*B*b^4*c^3*d*i^2 - 300
*A*a*b^3*c^2*d^2*i^2 + 60*A*a^2*b^2*c*d^3*i^2 - 73*B*a*b^3*c^2*d^2*i^2 + 47*B*a^2*b^2*c*d^3*i^2))/(12*(a^2*d^2
+ b^2*c^2 - 2*a*b*c*d)) + (d*x^3*(9*B*a*b^3*d^3*i^2 - B*b^4*c*d^2*i^2))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) +
(B*b^4*d^4*i^2*x^4)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(30*a^5*b^3*g^6 + 30*b^8*g^6*x^5 + 150*a^4*b^4*g^6*x + 1
50*a*b^7*g^6*x^4 + 300*a^3*b^5*g^6*x^2 + 300*a^2*b^6*g^6*x^3)