3.30 \(\int \frac{(c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x}))}{(a g+b g x)^7} \, dx\)

Optimal. Leaf size=281 \[ -\frac{b^2 i^3 (c+d x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 g^7 (a+b x)^6 (b c-a d)^3}-\frac{d^2 i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g^7 (a+b x)^4 (b c-a d)^3}+\frac{2 b d i^3 (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 g^7 (a+b x)^5 (b c-a d)^3}-\frac{b^2 B i^3 (c+d x)^6}{36 g^7 (a+b x)^6 (b c-a d)^3}-\frac{B d^2 i^3 (c+d x)^4}{16 g^7 (a+b x)^4 (b c-a d)^3}+\frac{2 b B d i^3 (c+d x)^5}{25 g^7 (a+b x)^5 (b c-a d)^3} \]

[Out]

-(B*d^2*i^3*(c + d*x)^4)/(16*(b*c - a*d)^3*g^7*(a + b*x)^4) + (2*b*B*d*i^3*(c + d*x)^5)/(25*(b*c - a*d)^3*g^7*
(a + b*x)^5) - (b^2*B*i^3*(c + d*x)^6)/(36*(b*c - a*d)^3*g^7*(a + b*x)^6) - (d^2*i^3*(c + d*x)^4*(A + B*Log[(e
*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^3*g^7*(a + b*x)^4) + (2*b*d*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(
c + d*x)]))/(5*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*
(b*c - a*d)^3*g^7*(a + b*x)^6)

________________________________________________________________________________________

Rubi [A]  time = 0.975718, antiderivative size = 445, normalized size of antiderivative = 1.58, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^3 i^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^4 g^7 (a+b x)^3}-\frac{3 d^2 i^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^4 g^7 (a+b x)^4}-\frac{3 d i^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^4 g^7 (a+b x)^5}-\frac{i^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 b^4 g^7 (a+b x)^6}-\frac{B d^5 i^3}{60 b^4 g^7 (a+b x) (b c-a d)^2}+\frac{B d^4 i^3}{120 b^4 g^7 (a+b x)^2 (b c-a d)}-\frac{19 B d^2 i^3 (b c-a d)}{240 b^4 g^7 (a+b x)^4}-\frac{B d^6 i^3 \log (a+b x)}{60 b^4 g^7 (b c-a d)^3}+\frac{B d^6 i^3 \log (c+d x)}{60 b^4 g^7 (b c-a d)^3}-\frac{13 B d i^3 (b c-a d)^2}{150 b^4 g^7 (a+b x)^5}-\frac{B i^3 (b c-a d)^3}{36 b^4 g^7 (a+b x)^6}-\frac{B d^3 i^3}{180 b^4 g^7 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^7,x]

[Out]

-(B*(b*c - a*d)^3*i^3)/(36*b^4*g^7*(a + b*x)^6) - (13*B*d*(b*c - a*d)^2*i^3)/(150*b^4*g^7*(a + b*x)^5) - (19*B
*d^2*(b*c - a*d)*i^3)/(240*b^4*g^7*(a + b*x)^4) - (B*d^3*i^3)/(180*b^4*g^7*(a + b*x)^3) + (B*d^4*i^3)/(120*b^4
*(b*c - a*d)*g^7*(a + b*x)^2) - (B*d^5*i^3)/(60*b^4*(b*c - a*d)^2*g^7*(a + b*x)) - (B*d^6*i^3*Log[a + b*x])/(6
0*b^4*(b*c - a*d)^3*g^7) - ((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^4*g^7*(a + b*x)^6) -
(3*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^4*g^7*(a + b*x)^5) - (3*d^2*(b*c - a*d)*i^3*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^4*g^7*(a + b*x)^4) - (d^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/(3*b^4*g^7*(a + b*x)^3) + (B*d^6*i^3*Log[c + d*x])/(60*b^4*(b*c - a*d)^3*g^7)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(30 c+30 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx &=\int \left (\frac{27000 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^7}+\frac{81000 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^6}+\frac{81000 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^5}+\frac{27000 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^4}\right ) \, dx\\ &=\frac{\left (27000 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^7}+\frac{\left (81000 d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^3 g^7}+\frac{\left (81000 d (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^6} \, dx}{b^3 g^7}+\frac{\left (27000 (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^7} \, dx}{b^3 g^7}\\ &=-\frac{4500 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac{16200 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac{20250 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac{9000 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac{\left (9000 B d^3\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^7}+\frac{\left (20250 B d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^7}+\frac{\left (16200 B d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^6 (c+d x)} \, dx}{b^4 g^7}+\frac{\left (4500 B (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^7 (c+d x)} \, dx}{b^4 g^7}\\ &=-\frac{4500 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac{16200 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac{20250 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac{9000 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac{\left (9000 B d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^7}+\frac{\left (20250 B d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^7}+\frac{\left (16200 B d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^6 (c+d x)} \, dx}{b^4 g^7}+\frac{\left (4500 B (b c-a d)^4\right ) \int \frac{1}{(a+b x)^7 (c+d x)} \, dx}{b^4 g^7}\\ &=-\frac{4500 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac{16200 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac{20250 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac{9000 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac{\left (9000 B d^3 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^4 g^7}+\frac{\left (20250 B d^2 (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^4 g^7}+\frac{\left (16200 B d (b c-a d)^3\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^6}-\frac{b d}{(b c-a d)^2 (a+b x)^5}+\frac{b d^2}{(b c-a d)^3 (a+b x)^4}-\frac{b d^3}{(b c-a d)^4 (a+b x)^3}+\frac{b d^4}{(b c-a d)^5 (a+b x)^2}-\frac{b d^5}{(b c-a d)^6 (a+b x)}+\frac{d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{b^4 g^7}+\frac{\left (4500 B (b c-a d)^4\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^7}-\frac{b d}{(b c-a d)^2 (a+b x)^6}+\frac{b d^2}{(b c-a d)^3 (a+b x)^5}-\frac{b d^3}{(b c-a d)^4 (a+b x)^4}+\frac{b d^4}{(b c-a d)^5 (a+b x)^3}-\frac{b d^5}{(b c-a d)^6 (a+b x)^2}+\frac{b d^6}{(b c-a d)^7 (a+b x)}-\frac{d^7}{(b c-a d)^7 (c+d x)}\right ) \, dx}{b^4 g^7}\\ &=-\frac{750 B (b c-a d)^3}{b^4 g^7 (a+b x)^6}-\frac{2340 B d (b c-a d)^2}{b^4 g^7 (a+b x)^5}-\frac{4275 B d^2 (b c-a d)}{2 b^4 g^7 (a+b x)^4}-\frac{150 B d^3}{b^4 g^7 (a+b x)^3}+\frac{225 B d^4}{b^4 (b c-a d) g^7 (a+b x)^2}-\frac{450 B d^5}{b^4 (b c-a d)^2 g^7 (a+b x)}-\frac{450 B d^6 \log (a+b x)}{b^4 (b c-a d)^3 g^7}-\frac{4500 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac{16200 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac{20250 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac{9000 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac{450 B d^6 \log (c+d x)}{b^4 (b c-a d)^3 g^7}\\ \end{align*}

Mathematica [B]  time = 1.09176, size = 642, normalized size = 2.28 \[ \frac{i^3 \left (1200 d^3 (a+b x)^3 (a d-b c)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2700 d^2 (a+b x)^2 (b c-a d)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2160 d (a+b x) (a d-b c)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-600 (b c-a d)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2100 B d^5 (a+b x)^5 (b c-a d)+2160 a B d^5 (a+b x)^4 (a d-b c)-2160 b B d^5 x (a+b x)^4 (b c-a d)-1050 B d^4 (a+b x)^4 (b c-a d)^2+1080 a B d^4 (a+b x)^3 (b c-a d)^2+1080 b B d^4 x (a+b x)^3 (b c-a d)^2+700 B d^3 (a+b x)^3 (b c-a d)^3+720 a B d^3 (a+b x)^2 (a d-b c)^3-720 b B d^3 x (a+b x)^2 (b c-a d)^3-825 B d^2 (a+b x)^2 (b c-a d)^4+540 a B d^2 (a+b x) (b c-a d)^4+540 b B d^2 x (a+b x) (b c-a d)^4-2100 B d^6 (a+b x)^6 \log (c+d x)+2160 a B d^6 (a+b x)^5 \log (c+d x)+2160 b B d^6 x (a+b x)^5 \log (c+d x)-432 b B d x (b c-a d)^5+120 B d (a+b x) (b c-a d)^5+432 a B d (a d-b c)^5-100 B (b c-a d)^6+2100 B d^6 (a+b x)^6 \log (a+b x)-2160 a B d^6 (a+b x)^5 \log (a+b x)-2160 b B d^6 x (a+b x)^5 \log (a+b x)\right )}{3600 b^4 g^7 (a+b x)^6 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^7,x]

[Out]

(i^3*(-100*B*(b*c - a*d)^6 + 432*a*B*d*(-(b*c) + a*d)^5 - 432*b*B*d*(b*c - a*d)^5*x + 540*a*B*d^2*(b*c - a*d)^
4*(a + b*x) + 120*B*d*(b*c - a*d)^5*(a + b*x) + 540*b*B*d^2*(b*c - a*d)^4*x*(a + b*x) - 825*B*d^2*(b*c - a*d)^
4*(a + b*x)^2 + 720*a*B*d^3*(-(b*c) + a*d)^3*(a + b*x)^2 - 720*b*B*d^3*(b*c - a*d)^3*x*(a + b*x)^2 + 1080*a*B*
d^4*(b*c - a*d)^2*(a + b*x)^3 + 700*B*d^3*(b*c - a*d)^3*(a + b*x)^3 + 1080*b*B*d^4*(b*c - a*d)^2*x*(a + b*x)^3
- 1050*B*d^4*(b*c - a*d)^2*(a + b*x)^4 + 2160*a*B*d^5*(-(b*c) + a*d)*(a + b*x)^4 - 2160*b*B*d^5*(b*c - a*d)*x
*(a + b*x)^4 + 2100*B*d^5*(b*c - a*d)*(a + b*x)^5 - 2160*a*B*d^6*(a + b*x)^5*Log[a + b*x] - 2160*b*B*d^6*x*(a
+ b*x)^5*Log[a + b*x] + 2100*B*d^6*(a + b*x)^6*Log[a + b*x] - 600*(b*c - a*d)^6*(A + B*Log[(e*(a + b*x))/(c +
d*x)]) + 2160*d*(-(b*c) + a*d)^5*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2700*d^2*(b*c - a*d)^4*(a +
b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 1200*d^3*(-(b*c) + a*d)^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(
c + d*x)]) + 2160*a*B*d^6*(a + b*x)^5*Log[c + d*x] + 2160*b*B*d^6*x*(a + b*x)^5*Log[c + d*x] - 2100*B*d^6*(a +
b*x)^6*Log[c + d*x]))/(3600*b^4*(b*c - a*d)^3*g^7*(a + b*x)^6)

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 1262, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x)

[Out]

1/4*e^4*d^3*i^3/(a*d-b*c)^4/g^7*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/4*e^4*d^2*i^3/(a*d-b*c)^4/g^7*A/(b
*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c-2/5*e^5*d^2*i^3/(a*d-b*c)^4/g^7*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c
)^5*a+2/5*e^5*d*i^3/(a*d-b*c)^4/g^7*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*c+1/6*e^6*d*i^3/(a*d-b*c)^4/g^
7*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^6*a-1/6*e^6*i^3/(a*d-b*c)^4/g^7*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+
c)*b*c)^6*c+1/4*e^4*d^3*i^3/(a*d-b*c)^4/g^7*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*
x+c))*a-1/4*e^4*d^2*i^3/(a*d-b*c)^4/g^7*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)*b*c+1/16*e^4*d^3*i^3/(a*d-b*c)^4/g^7*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/16*e^4*d^2*i^3/(a*d-b*c)^4/
g^7*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c-2/5*e^5*d^2*i^3/(a*d-b*c)^4/g^7*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*
x+c)*b*c)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+2/5*e^5*d*i^3/(a*d-b*c)^4/g^7*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+
c)*b*c)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-2/25*e^5*d^2*i^3/(a*d-b*c)^4/g^7*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c
)*b*c)^5*a+2/25*e^5*d*i^3/(a*d-b*c)^4/g^7*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*c+1/6*e^6*d*i^3/(a*d-b*c
)^4/g^7*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/6*e^6*i^3/(a*d-b*c)^4/
g^7*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+1/36*e^6*d*i^3/(a*d-b*c)^4/g
^7*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^6*a-1/36*e^6*i^3/(a*d-b*c)^4/g^7*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*
x+c)*b*c)^6*c

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Maxima [B]  time = 3.32622, size = 7457, normalized size = 26.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x, algorithm="maxima")

[Out]

-1/3600*B*d^3*i^3*(60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^10
*g^7*x^6 + 6*a*b^9*g^7*x^5 + 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x +
a^6*b^4*g^7) + (57*a^3*b^5*c^5 - 405*a^4*b^4*c^4*d + 1470*a^5*b^3*c^3*d^2 - 730*a^6*b^2*c^2*d^3 + 245*a^7*b*c*
d^4 - 37*a^8*d^5 + 60*(20*b^8*c^3*d^2 - 15*a*b^7*c^2*d^3 + 6*a^2*b^6*c*d^4 - a^3*b^5*d^5)*x^5 - 30*(20*b^8*c^4
*d - 235*a*b^7*c^3*d^2 + 171*a^2*b^6*c^2*d^3 - 67*a^3*b^5*c*d^4 + 11*a^4*b^4*d^5)*x^4 + 20*(20*b^8*c^5 - 175*a
*b^7*c^4*d + 866*a^2*b^6*c^3*d^2 - 604*a^3*b^5*c^2*d^3 + 230*a^4*b^4*c*d^4 - 37*a^5*b^3*d^5)*x^3 + 15*(35*a*b^
7*c^5 - 271*a^2*b^6*c^4*d + 1128*a^3*b^5*c^3*d^2 - 700*a^4*b^4*c^2*d^3 + 245*a^5*b^3*c*d^4 - 37*a^6*b^2*d^5)*x
^2 + 6*(47*a^2*b^6*c^5 - 345*a^3*b^5*c^4*d + 1320*a^4*b^4*c^3*d^2 - 730*a^5*b^3*c^2*d^3 + 245*a^6*b^2*c*d^4 -
37*a^7*b*d^5)*x)/((b^15*c^5 - 5*a*b^14*c^4*d + 10*a^2*b^13*c^3*d^2 - 10*a^3*b^12*c^2*d^3 + 5*a^4*b^11*c*d^4 -
a^5*b^10*d^5)*g^7*x^6 + 6*(a*b^14*c^5 - 5*a^2*b^13*c^4*d + 10*a^3*b^12*c^3*d^2 - 10*a^4*b^11*c^2*d^3 + 5*a^5*b
^10*c*d^4 - a^6*b^9*d^5)*g^7*x^5 + 15*(a^2*b^13*c^5 - 5*a^3*b^12*c^4*d + 10*a^4*b^11*c^3*d^2 - 10*a^5*b^10*c^2
*d^3 + 5*a^6*b^9*c*d^4 - a^7*b^8*d^5)*g^7*x^4 + 20*(a^3*b^12*c^5 - 5*a^4*b^11*c^4*d + 10*a^5*b^10*c^3*d^2 - 10
*a^6*b^9*c^2*d^3 + 5*a^7*b^8*c*d^4 - a^8*b^7*d^5)*g^7*x^3 + 15*(a^4*b^11*c^5 - 5*a^5*b^10*c^4*d + 10*a^6*b^9*c
^3*d^2 - 10*a^7*b^8*c^2*d^3 + 5*a^8*b^7*c*d^4 - a^9*b^6*d^5)*g^7*x^2 + 6*(a^5*b^10*c^5 - 5*a^6*b^9*c^4*d + 10*
a^7*b^8*c^3*d^2 - 10*a^8*b^7*c^2*d^3 + 5*a^9*b^6*c*d^4 - a^10*b^5*d^5)*g^7*x + (a^6*b^9*c^5 - 5*a^7*b^8*c^4*d
+ 10*a^8*b^7*c^3*d^2 - 10*a^9*b^6*c^2*d^3 + 5*a^10*b^5*c*d^4 - a^11*b^4*d^5)*g^7) + 60*(20*b^3*c^3*d^3 - 15*a*
b^2*c^2*d^4 + 6*a^2*b*c*d^5 - a^3*d^6)*log(b*x + a)/((b^10*c^6 - 6*a*b^9*c^5*d + 15*a^2*b^8*c^4*d^2 - 20*a^3*b
^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7) - 60*(20*b^3*c^3*d^3 - 15*a*b^2*c^2*d^4
+ 6*a^2*b*c*d^5 - a^3*d^6)*log(d*x + c)/((b^10*c^6 - 6*a*b^9*c^5*d + 15*a^2*b^8*c^4*d^2 - 20*a^3*b^7*c^3*d^3 +
15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7)) - 1/1200*B*c*d^2*i^3*(60*(15*b^2*x^2 + 6*a*b*x + a^
2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^9*g^7*x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x
^3 + 15*a^4*b^5*g^7*x^2 + 6*a^5*b^4*g^7*x + a^6*b^3*g^7) + (37*a^2*b^5*c^5 - 245*a^3*b^4*c^4*d + 730*a^4*b^3*c
^3*d^2 - 1470*a^5*b^2*c^2*d^3 + 405*a^6*b*c*d^4 - 57*a^7*d^5 - 60*(15*b^7*c^2*d^3 - 6*a*b^6*c*d^4 + a^2*b^5*d^
5)*x^5 + 30*(15*b^7*c^3*d^2 - 171*a*b^6*c^2*d^3 + 67*a^2*b^5*c*d^4 - 11*a^3*b^4*d^5)*x^4 - 20*(15*b^7*c^4*d -
126*a*b^6*c^3*d^2 + 604*a^2*b^5*c^2*d^3 - 230*a^3*b^4*c*d^4 + 37*a^4*b^3*d^5)*x^3 + 15*(15*b^7*c^5 - 111*a*b^6
*c^4*d + 388*a^2*b^5*c^3*d^2 - 1000*a^3*b^4*c^2*d^3 + 365*a^4*b^3*c*d^4 - 57*a^5*b^2*d^5)*x^2 + 6*(27*a*b^6*c^
5 - 185*a^2*b^5*c^4*d + 580*a^3*b^4*c^3*d^2 - 1270*a^4*b^3*c^2*d^3 + 405*a^5*b^2*c*d^4 - 57*a^6*b*d^5)*x)/((b^
14*c^5 - 5*a*b^13*c^4*d + 10*a^2*b^12*c^3*d^2 - 10*a^3*b^11*c^2*d^3 + 5*a^4*b^10*c*d^4 - a^5*b^9*d^5)*g^7*x^6
+ 6*(a*b^13*c^5 - 5*a^2*b^12*c^4*d + 10*a^3*b^11*c^3*d^2 - 10*a^4*b^10*c^2*d^3 + 5*a^5*b^9*c*d^4 - a^6*b^8*d^5
)*g^7*x^5 + 15*(a^2*b^12*c^5 - 5*a^3*b^11*c^4*d + 10*a^4*b^10*c^3*d^2 - 10*a^5*b^9*c^2*d^3 + 5*a^6*b^8*c*d^4 -
a^7*b^7*d^5)*g^7*x^4 + 20*(a^3*b^11*c^5 - 5*a^4*b^10*c^4*d + 10*a^5*b^9*c^3*d^2 - 10*a^6*b^8*c^2*d^3 + 5*a^7*
b^7*c*d^4 - a^8*b^6*d^5)*g^7*x^3 + 15*(a^4*b^10*c^5 - 5*a^5*b^9*c^4*d + 10*a^6*b^8*c^3*d^2 - 10*a^7*b^7*c^2*d^
3 + 5*a^8*b^6*c*d^4 - a^9*b^5*d^5)*g^7*x^2 + 6*(a^5*b^9*c^5 - 5*a^6*b^8*c^4*d + 10*a^7*b^7*c^3*d^2 - 10*a^8*b^
6*c^2*d^3 + 5*a^9*b^5*c*d^4 - a^10*b^4*d^5)*g^7*x + (a^6*b^8*c^5 - 5*a^7*b^7*c^4*d + 10*a^8*b^6*c^3*d^2 - 10*a
^9*b^5*c^2*d^3 + 5*a^10*b^4*c*d^4 - a^11*b^3*d^5)*g^7) - 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*log(b*x +
a)/((b^9*c^6 - 6*a*b^8*c^5*d + 15*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*b^4*c*d^5
+ a^6*b^3*d^6)*g^7) + 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*log(d*x + c)/((b^9*c^6 - 6*a*b^8*c^5*d + 15
*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*b^4*c*d^5 + a^6*b^3*d^6)*g^7)) - 1/600*B*c^
2*d*i^3*(60*(6*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x
^4 + 20*a^3*b^5*g^7*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) + (22*a*b^5*c^5 - 140*a^2*b^4*c^
4*d + 385*a^3*b^3*c^3*d^2 - 615*a^4*b^2*c^2*d^3 + 735*a^5*b*c*d^4 - 87*a^6*d^5 + 60*(6*b^6*c*d^4 - a*b^5*d^5)*
x^5 - 30*(6*b^6*c^2*d^3 - 67*a*b^5*c*d^4 + 11*a^2*b^4*d^5)*x^4 + 20*(6*b^6*c^3*d^2 - 49*a*b^5*c^2*d^3 + 230*a^
2*b^4*c*d^4 - 37*a^3*b^3*d^5)*x^3 - 15*(6*b^6*c^4*d - 43*a*b^5*c^3*d^2 + 145*a^2*b^4*c^2*d^3 - 365*a^3*b^3*c*d
^4 + 57*a^4*b^2*d^5)*x^2 + 6*(12*b^6*c^5 - 80*a*b^5*c^4*d + 235*a^2*b^4*c^3*d^2 - 415*a^3*b^3*c^2*d^3 + 585*a^
4*b^2*c*d^4 - 87*a^5*b*d^5)*x)/((b^13*c^5 - 5*a*b^12*c^4*d + 10*a^2*b^11*c^3*d^2 - 10*a^3*b^10*c^2*d^3 + 5*a^4
*b^9*c*d^4 - a^5*b^8*d^5)*g^7*x^6 + 6*(a*b^12*c^5 - 5*a^2*b^11*c^4*d + 10*a^3*b^10*c^3*d^2 - 10*a^4*b^9*c^2*d^
3 + 5*a^5*b^8*c*d^4 - a^6*b^7*d^5)*g^7*x^5 + 15*(a^2*b^11*c^5 - 5*a^3*b^10*c^4*d + 10*a^4*b^9*c^3*d^2 - 10*a^5
*b^8*c^2*d^3 + 5*a^6*b^7*c*d^4 - a^7*b^6*d^5)*g^7*x^4 + 20*(a^3*b^10*c^5 - 5*a^4*b^9*c^4*d + 10*a^5*b^8*c^3*d^
2 - 10*a^6*b^7*c^2*d^3 + 5*a^7*b^6*c*d^4 - a^8*b^5*d^5)*g^7*x^3 + 15*(a^4*b^9*c^5 - 5*a^5*b^8*c^4*d + 10*a^6*b
^7*c^3*d^2 - 10*a^7*b^6*c^2*d^3 + 5*a^8*b^5*c*d^4 - a^9*b^4*d^5)*g^7*x^2 + 6*(a^5*b^8*c^5 - 5*a^6*b^7*c^4*d +
10*a^7*b^6*c^3*d^2 - 10*a^8*b^5*c^2*d^3 + 5*a^9*b^4*c*d^4 - a^10*b^3*d^5)*g^7*x + (a^6*b^7*c^5 - 5*a^7*b^6*c^4
*d + 10*a^8*b^5*c^3*d^2 - 10*a^9*b^4*c^2*d^3 + 5*a^10*b^3*c*d^4 - a^11*b^2*d^5)*g^7) + 60*(6*b*c*d^5 - a*d^6)*
log(b*x + a)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*
b^3*c*d^5 + a^6*b^2*d^6)*g^7) - 60*(6*b*c*d^5 - a*d^6)*log(d*x + c)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4
*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*g^7)) + 1/360*B*c^3*i^3*((60*b
^5*d^5*x^5 - 10*b^5*c^5 + 62*a*b^4*c^4*d - 163*a^2*b^3*c^3*d^2 + 237*a^3*b^2*c^2*d^3 - 213*a^4*b*c*d^4 + 147*a
^5*d^5 - 30*(b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 8*a*b^4*c*d^4 + 37*a^2*b^3*d^5)*x^3 - 15*(b^5*c
^3*d^2 - 7*a*b^4*c^2*d^3 + 23*a^2*b^3*c*d^4 - 57*a^3*b^2*d^5)*x^2 + 6*(2*b^5*c^4*d - 13*a*b^4*c^3*d^2 + 37*a^2
*b^3*c^2*d^3 - 63*a^3*b^2*c*d^4 + 87*a^4*b*d^5)*x)/((b^12*c^5 - 5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2 - 10*a^3*
b^9*c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)*g^7*x^6 + 6*(a*b^11*c^5 - 5*a^2*b^10*c^4*d + 10*a^3*b^9*c^3*d^2 -
10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^6*b^6*d^5)*g^7*x^5 + 15*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a^4*b^8
*c^3*d^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^7*b^5*d^5)*g^7*x^4 + 20*(a^3*b^9*c^5 - 5*a^4*b^8*c^4*d + 1
0*a^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + 5*a^7*b^5*c*d^4 - a^8*b^4*d^5)*g^7*x^3 + 15*(a^4*b^8*c^5 - 5*a^5*b^7*
c^4*d + 10*a^6*b^6*c^3*d^2 - 10*a^7*b^5*c^2*d^3 + 5*a^8*b^4*c*d^4 - a^9*b^3*d^5)*g^7*x^2 + 6*(a^5*b^7*c^5 - 5*
a^6*b^6*c^4*d + 10*a^7*b^5*c^3*d^2 - 10*a^8*b^4*c^2*d^3 + 5*a^9*b^3*c*d^4 - a^10*b^2*d^5)*g^7*x + (a^6*b^6*c^5
- 5*a^7*b^5*c^4*d + 10*a^8*b^4*c^3*d^2 - 10*a^9*b^3*c^2*d^3 + 5*a^10*b^2*c*d^4 - a^11*b*d^5)*g^7) - 60*log(b*
e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 + 15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a
^4*b^3*g^7*x^2 + 6*a^5*b^2*g^7*x + a^6*b*g^7) + 60*d^6*log(b*x + a)/((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4
*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*g^7) - 60*d^6*log(d*x + c)/((b^7
*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*
d^6)*g^7)) - 1/10*(6*b*x + a)*A*c^2*d*i^3/(b^8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x^4 + 20*a^3*b^5*g^7
*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) - 1/20*(15*b^2*x^2 + 6*a*b*x + a^2)*A*c*d^2*i^3/(b^
9*g^7*x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x^3 + 15*a^4*b^5*g^7*x^2 + 6*a^5*b^4*g^7*x +
a^6*b^3*g^7) - 1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*A*d^3*i^3/(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5 +
15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^6*b^4*g^7) - 1/6*A*c^3*i^3
/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 + 15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a^4*b^3*g^7*x^2 + 6*a^5*b^2*g^7
*x + a^6*b*g^7)

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Fricas [B]  time = 0.59535, size = 2057, normalized size = 7.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x, algorithm="fricas")

[Out]

-1/3600*(60*(B*b^6*c*d^5 - B*a*b^5*d^6)*i^3*x^5 - 30*(B*b^6*c^2*d^4 - 12*B*a*b^5*c*d^5 + 11*B*a^2*b^4*d^6)*i^3
*x^4 + 20*((60*A + B)*b^6*c^3*d^3 - 9*(20*A + B)*a*b^5*c^2*d^4 + 45*(4*A + B)*a^2*b^4*c*d^5 - (60*A + 37*B)*a^
3*b^3*d^6)*i^3*x^3 + 15*((180*A + 19*B)*b^6*c^4*d^2 - 24*(20*A + 3*B)*a*b^5*c^3*d^3 + 90*(4*A + B)*a^2*b^4*c^2
*d^4 - (60*A + 37*B)*a^4*b^2*d^6)*i^3*x^2 + 6*(4*(90*A + 13*B)*b^6*c^5*d - 15*(60*A + 11*B)*a*b^5*c^4*d^2 + 15
0*(4*A + B)*a^2*b^4*c^3*d^3 - (60*A + 37*B)*a^5*b*d^6)*i^3*x + (100*(6*A + B)*b^6*c^6 - 288*(5*A + B)*a*b^5*c^
5*d + 225*(4*A + B)*a^2*b^4*c^4*d^2 - (60*A + 37*B)*a^6*d^6)*i^3 + 60*(B*b^6*d^6*i^3*x^6 + 6*B*a*b^5*d^6*i^3*x
^5 + 15*B*a^2*b^4*d^6*i^3*x^4 + 20*(B*b^6*c^3*d^3 - 3*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*i^3*x^3 + 15*(3*B*b
^6*c^4*d^2 - 8*B*a*b^5*c^3*d^3 + 6*B*a^2*b^4*c^2*d^4)*i^3*x^2 + 6*(6*B*b^6*c^5*d - 15*B*a*b^5*c^4*d^2 + 10*B*a
^2*b^4*c^3*d^3)*i^3*x + (10*B*b^6*c^6 - 24*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2)*i^3)*log((b*e*x + a*e)/(d*x +
c)))/((b^13*c^3 - 3*a*b^12*c^2*d + 3*a^2*b^11*c*d^2 - a^3*b^10*d^3)*g^7*x^6 + 6*(a*b^12*c^3 - 3*a^2*b^11*c^2*
d + 3*a^3*b^10*c*d^2 - a^4*b^9*d^3)*g^7*x^5 + 15*(a^2*b^11*c^3 - 3*a^3*b^10*c^2*d + 3*a^4*b^9*c*d^2 - a^5*b^8*
d^3)*g^7*x^4 + 20*(a^3*b^10*c^3 - 3*a^4*b^9*c^2*d + 3*a^5*b^8*c*d^2 - a^6*b^7*d^3)*g^7*x^3 + 15*(a^4*b^9*c^3 -
3*a^5*b^8*c^2*d + 3*a^6*b^7*c*d^2 - a^7*b^6*d^3)*g^7*x^2 + 6*(a^5*b^8*c^3 - 3*a^6*b^7*c^2*d + 3*a^7*b^6*c*d^2
- a^8*b^5*d^3)*g^7*x + (a^6*b^7*c^3 - 3*a^7*b^6*c^2*d + 3*a^8*b^5*c*d^2 - a^9*b^4*d^3)*g^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.41452, size = 1678, normalized size = 5.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x, algorithm="giac")

[Out]

-1/60*B*d^6*log(b*x + a)/(b^7*c^3*g^7*i - 3*a*b^6*c^2*d*g^7*i + 3*a^2*b^5*c*d^2*g^7*i - a^3*b^4*d^3*g^7*i) + 1
/60*B*d^6*log(d*x + c)/(b^7*c^3*g^7*i - 3*a*b^6*c^2*d*g^7*i + 3*a^2*b^5*c*d^2*g^7*i - a^3*b^4*d^3*g^7*i) + 1/6
0*(20*B*b^3*d^3*i*x^3 + 45*B*b^3*c*d^2*i*x^2 + 15*B*a*b^2*d^3*i*x^2 + 36*B*b^3*c^2*d*i*x + 18*B*a*b^2*c*d^2*i*
x + 6*B*a^2*b*d^3*i*x + 10*B*b^3*c^3*i + 6*B*a*b^2*c^2*d*i + 3*B*a^2*b*c*d^2*i + B*a^3*d^3*i)*log((b*x + a)/(d
*x + c))/(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5 + 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^
5*b^5*g^7*x + a^6*b^4*g^7) - 1/3600*(60*B*b^5*d^5*x^5 - 30*B*b^5*c*d^4*x^4 + 330*B*a*b^4*d^5*x^4 + 1200*A*b^5*
c^2*d^3*x^3 + 1220*B*b^5*c^2*d^3*x^3 - 2400*A*a*b^4*c*d^4*x^3 - 2560*B*a*b^4*c*d^4*x^3 + 1200*A*a^2*b^3*d^5*x^
3 + 1940*B*a^2*b^3*d^5*x^3 + 2700*A*b^5*c^3*d^2*x^2 + 2985*B*b^5*c^3*d^2*x^2 - 4500*A*a*b^4*c^2*d^3*x^2 - 5295
*B*a*b^4*c^2*d^3*x^2 + 900*A*a^2*b^3*c*d^4*x^2 + 1455*B*a^2*b^3*c*d^4*x^2 + 900*A*a^3*b^2*d^5*x^2 + 1455*B*a^3
*b^2*d^5*x^2 + 2160*A*b^5*c^4*d*x + 2472*B*b^5*c^4*d*x - 3240*A*a*b^4*c^3*d^2*x - 3918*B*a*b^4*c^3*d^2*x + 360
*A*a^2*b^3*c^2*d^3*x + 582*B*a^2*b^3*c^2*d^3*x + 360*A*a^3*b^2*c*d^4*x + 582*B*a^3*b^2*c*d^4*x + 360*A*a^4*b*d
^5*x + 582*B*a^4*b*d^5*x + 600*A*b^5*c^5 + 700*B*b^5*c^5 - 840*A*a*b^4*c^4*d - 1028*B*a*b^4*c^4*d + 60*A*a^2*b
^3*c^3*d^2 + 97*B*a^2*b^3*c^3*d^2 + 60*A*a^3*b^2*c^2*d^3 + 97*B*a^3*b^2*c^2*d^3 + 60*A*a^4*b*c*d^4 + 97*B*a^4*
b*c*d^4 + 60*A*a^5*d^5 + 97*B*a^5*d^5)/(b^12*c^2*g^7*i*x^6 - 2*a*b^11*c*d*g^7*i*x^6 + a^2*b^10*d^2*g^7*i*x^6 +
6*a*b^11*c^2*g^7*i*x^5 - 12*a^2*b^10*c*d*g^7*i*x^5 + 6*a^3*b^9*d^2*g^7*i*x^5 + 15*a^2*b^10*c^2*g^7*i*x^4 - 30
*a^3*b^9*c*d*g^7*i*x^4 + 15*a^4*b^8*d^2*g^7*i*x^4 + 20*a^3*b^9*c^2*g^7*i*x^3 - 40*a^4*b^8*c*d*g^7*i*x^3 + 20*a
^5*b^7*d^2*g^7*i*x^3 + 15*a^4*b^8*c^2*g^7*i*x^2 - 30*a^5*b^7*c*d*g^7*i*x^2 + 15*a^6*b^6*d^2*g^7*i*x^2 + 6*a^5*
b^7*c^2*g^7*i*x - 12*a^6*b^6*c*d*g^7*i*x + 6*a^7*b^5*d^2*g^7*i*x + a^6*b^6*c^2*g^7*i - 2*a^7*b^5*c*d*g^7*i + a
^8*b^4*d^2*g^7*i)