3.74 \(\int (a g+b g x)^3 (c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x}))^2 \, dx\)

Optimal. Leaf size=1089 \[ \text{result too large to display} \]

[Out]

(5*B^2*(b*c - a*d)^6*g^3*i^3*x)/(84*b^3*d^3) + (B^2*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4)/(140*b^4) - (29*B^2*(b*
c - a*d)^5*g^3*i^3*(c + d*x)^2)/(840*b^2*d^4) + (47*B^2*(b*c - a*d)^4*g^3*i^3*(c + d*x)^3)/(1260*b*d^4) - (13*
B^2*(b*c - a*d)^3*g^3*i^3*(c + d*x)^4)/(420*d^4) + (b*B^2*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5)/(105*d^4) - (B^2*
(b*c - a*d)^7*g^3*i^3*Log[(a + b*x)/(c + d*x)])/(210*b^4*d^4) - (B*(b*c - a*d)^4*g^3*i^3*(a + b*x)^3*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)]))/(210*b^4*d) - (3*B*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(140*b^4) - (B*(b*c - a*d)^2*g^3*i^3*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(35
*b^3) + (2*B*(b*c - a*d)^4*g^3*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(21*b*d^4) - (3*B*(b*c -
a*d)^3*g^3*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(14*d^4) + (6*b*B*(b*c - a*d)^2*g^3*i^3*(c +
d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(35*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[(e*(
a + b*x))/(c + d*x)]))/(21*d^4) + ((b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(
140*b^4) + ((b*c - a*d)^2*g^3*i^3*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(35*b^3) + ((b
*c - a*d)*g^3*i^3*(a + b*x)^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(14*b^2) + (g^3*i^3*(a + b*x
)^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(7*b) + (B*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2*(3*A + B
+ 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(420*b^4*d^2) - (B*(b*c - a*d)^6*g^3*i^3*(a + b*x)*(6*A + 5*B + 6*B*Log[(
e*(a + b*x))/(c + d*x)]))/(420*b^4*d^3) - (B*(b*c - a*d)^7*g^3*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B
+ 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(420*b^4*d^4) - (11*B^2*(b*c - a*d)^7*g^3*i^3*Log[c + d*x])/(420*b^4*d^4)
- (B^2*(b*c - a*d)^7*g^3*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(70*b^4*d^4)

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Rubi [A]  time = 4.27025, antiderivative size = 896, normalized size of antiderivative = 0.82, number of steps used = 122, number of rules used = 13, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B^2 g^3 i^3 \log ^2(c+d x) (b c-a d)^7}{140 b^4 d^4}-\frac{B^2 g^3 i^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^7}{70 b^4 d^4}+\frac{B g^3 i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) (b c-a d)^7}{70 b^4 d^4}-\frac{B^2 g^3 i^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^7}{70 b^4 d^4}+\frac{B^2 g^3 i^3 x (b c-a d)^6}{70 b^3 d^3}-\frac{A B g^3 i^3 x (b c-a d)^6}{70 b^3 d^3}-\frac{B^2 g^3 i^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right ) (b c-a d)^6}{70 b^4 d^3}-\frac{3 B^2 g^3 i^3 (a+b x)^2 (b c-a d)^5}{280 b^4 d^2}+\frac{B g^3 i^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^5}{140 b^4 d^2}+\frac{11 B^2 g^3 i^3 (a+b x)^3 (b c-a d)^4}{1260 b^4 d}-\frac{B g^3 i^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4}{210 b^4 d}+\frac{B^2 g^3 i^3 (a+b x)^4 (b c-a d)^3}{42 b^4}+\frac{g^3 i^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^3}{4 b^4}-\frac{17 B g^3 i^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3}{140 b^4}+\frac{B^2 d g^3 i^3 (a+b x)^5 (b c-a d)^2}{105 b^4}+\frac{3 d g^3 i^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2}{5 b^4}-\frac{B d g^3 i^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2}{7 b^4}+\frac{d^2 g^3 i^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)}{2 b^4}-\frac{B d^2 g^3 i^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{21 b^4}+\frac{d^3 g^3 i^3 (a+b x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2,x]

[Out]

-(A*B*(b*c - a*d)^6*g^3*i^3*x)/(70*b^3*d^3) + (B^2*(b*c - a*d)^6*g^3*i^3*x)/(70*b^3*d^3) - (3*B^2*(b*c - a*d)^
5*g^3*i^3*(a + b*x)^2)/(280*b^4*d^2) + (11*B^2*(b*c - a*d)^4*g^3*i^3*(a + b*x)^3)/(1260*b^4*d) + (B^2*(b*c - a
*d)^3*g^3*i^3*(a + b*x)^4)/(42*b^4) + (B^2*d*(b*c - a*d)^2*g^3*i^3*(a + b*x)^5)/(105*b^4) - (B^2*(b*c - a*d)^6
*g^3*i^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(70*b^4*d^3) + (B*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)]))/(140*b^4*d^2) - (B*(b*c - a*d)^4*g^3*i^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(210*b^4*d) - (17*B*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(140*b^4
) - (B*d*(b*c - a*d)^2*g^3*i^3*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(7*b^4) - (B*d^2*(b*c - a*d)*
g^3*i^3*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(21*b^4) + ((b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B
*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b^4) + (3*d*(b*c - a*d)^2*g^3*i^3*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c
+ d*x)])^2)/(5*b^4) + (d^2*(b*c - a*d)*g^3*i^3*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^4) +
(d^3*g^3*i^3*(a + b*x)^7*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(7*b^4) - (B^2*(b*c - a*d)^7*g^3*i^3*Log[-((d
*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(70*b^4*d^4) + (B*(b*c - a*d)^7*g^3*i^3*(A + B*Log[(e*(a + b*x))/(c +
d*x)])*Log[c + d*x])/(70*b^4*d^4) + (B^2*(b*c - a*d)^7*g^3*i^3*Log[c + d*x]^2)/(140*b^4*d^4) - (B^2*(b*c - a*d
)^7*g^3*i^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(70*b^4*d^4)

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 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

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 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int (74 c+74 d x)^3 (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac{(-b c+a d)^3 g^3 (74 c+74 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac{3 b (b c-a d)^2 g^3 (74 c+74 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{74 d^3}-\frac{3 b^2 (b c-a d) g^3 (74 c+74 d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5476 d^3}+\frac{b^3 g^3 (74 c+74 d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{405224 d^3}\right ) \, dx\\ &=\frac{\left (b^3 g^3\right ) \int (74 c+74 d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{405224 d^3}-\frac{\left (3 b^2 (b c-a d) g^3\right ) \int (74 c+74 d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{5476 d^3}+\frac{\left (3 b (b c-a d)^2 g^3\right ) \int (74 c+74 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{74 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int (74 c+74 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{d^3}\\ &=-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{\left (b^3 B g^3\right ) \int \frac{12151280273024 (b c-a d) (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{104953016 d^4}+\frac{\left (b^2 B (b c-a d) g^3\right ) \int \frac{164206490176 (b c-a d) (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{405224 d^4}-\frac{\left (3 b B (b c-a d)^2 g^3\right ) \int \frac{2219006624 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{13690 d^4}+\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{29986576 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{148 d^4}\\ &=-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{\left (810448 b^3 B (b c-a d) g^3\right ) \int \frac{(c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{7 d^4}+\frac{\left (405224 b^2 B (b c-a d)^2 g^3\right ) \int \frac{(c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^4}-\frac{\left (2431344 b B (b c-a d)^3 g^3\right ) \int \frac{(c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{5 d^4}+\frac{\left (202612 B (b c-a d)^4 g^3\right ) \int \frac{(c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^4}\\ &=-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{\left (810448 b^3 B (b c-a d) g^3\right ) \int \left (\frac{d (b c-a d)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^6}+\frac{(b c-a d)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^6 (a+b x)}+\frac{d (b c-a d)^4 (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^5}+\frac{d (b c-a d)^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac{d (b c-a d)^2 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{d (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac{d (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{7 d^4}+\frac{\left (405224 b^2 B (b c-a d)^2 g^3\right ) \int \left (\frac{d (b c-a d)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^5}+\frac{(b c-a d)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^5 (a+b x)}+\frac{d (b c-a d)^3 (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac{d (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{d (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac{d (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^4}-\frac{\left (2431344 b B (b c-a d)^3 g^3\right ) \int \left (\frac{d (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac{(b c-a d)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{d (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac{d (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{5 d^4}+\frac{\left (202612 B (b c-a d)^4 g^3\right ) \int \left (\frac{d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{(b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac{d (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^4}\\ &=-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{\left (810448 b^2 B (b c-a d) g^3\right ) \int (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{7 d^3}-\frac{\left (810448 b B (b c-a d)^2 g^3\right ) \int (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{7 d^3}+\frac{\left (405224 b B (b c-a d)^2 g^3\right ) \int (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d^3}-\frac{\left (810448 B (b c-a d)^3 g^3\right ) \int (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{7 d^3}+\frac{\left (405224 B (b c-a d)^3 g^3\right ) \int (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d^3}-\frac{\left (2431344 B (b c-a d)^3 g^3\right ) \int (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 d^3}-\frac{\left (810448 B (b c-a d)^4 g^3\right ) \int (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{7 b d^3}+\frac{\left (202612 B (b c-a d)^4 g^3\right ) \int (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^3}+\frac{\left (405224 B (b c-a d)^4 g^3\right ) \int (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^3}-\frac{\left (2431344 B (b c-a d)^4 g^3\right ) \int (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b d^3}-\frac{\left (810448 B (b c-a d)^5 g^3\right ) \int (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{7 b^2 d^3}+\frac{\left (202612 B (b c-a d)^5 g^3\right ) \int (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^3}+\frac{\left (405224 B (b c-a d)^5 g^3\right ) \int (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^3}-\frac{\left (2431344 B (b c-a d)^5 g^3\right ) \int (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d^3}-\frac{\left (810448 B (b c-a d)^6 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{7 b^3 d^3}+\frac{\left (202612 B (b c-a d)^6 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 d^3}+\frac{\left (405224 B (b c-a d)^6 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 d^3}-\frac{\left (2431344 B (b c-a d)^6 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^3 d^3}-\frac{\left (810448 B (b c-a d)^7 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{7 b^3 d^4}+\frac{\left (202612 B (b c-a d)^7 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 d^4}+\frac{\left (405224 B (b c-a d)^7 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 d^4}-\frac{\left (2431344 B (b c-a d)^7 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{5 b^3 d^4}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}+\frac{\left (405224 b^2 B^2 (b c-a d) g^3\right ) \int \frac{(b c-a d) (c+d x)^5}{a+b x} \, dx}{21 d^4}+\frac{\left (810448 b B^2 (b c-a d)^2 g^3\right ) \int \frac{(b c-a d) (c+d x)^4}{a+b x} \, dx}{35 d^4}-\frac{\left (405224 b B^2 (b c-a d)^2 g^3\right ) \int \frac{(b c-a d) (c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac{\left (202612 B^2 (b c-a d)^3 g^3\right ) \int \frac{(b c-a d) (c+d x)^3}{a+b x} \, dx}{7 d^4}-\frac{\left (101306 B^2 (b c-a d)^3 g^3\right ) \int \frac{(b c-a d) (c+d x)^3}{a+b x} \, dx}{d^4}+\frac{\left (607836 B^2 (b c-a d)^3 g^3\right ) \int \frac{(b c-a d) (c+d x)^3}{a+b x} \, dx}{5 d^4}+\frac{\left (810448 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{21 b d^4}-\frac{\left (202612 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^4}-\frac{\left (405224 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^4}+\frac{\left (810448 B^2 (b c-a d)^4 g^3\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{5 b d^4}+\frac{\left (405224 B^2 (b c-a d)^5 g^3\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{7 b^2 d^4}-\frac{\left (101306 B^2 (b c-a d)^5 g^3\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^4}-\frac{\left (202612 B^2 (b c-a d)^5 g^3\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac{\left (1215672 B^2 (b c-a d)^5 g^3\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^2 d^4}-\frac{\left (810448 B^2 (b c-a d)^6 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{7 b^3 d^3}+\frac{\left (202612 B^2 (b c-a d)^6 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b^3 d^3}+\frac{\left (405224 B^2 (b c-a d)^6 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b^3 d^3}-\frac{\left (2431344 B^2 (b c-a d)^6 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{5 b^3 d^3}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{7 b^4 d^4}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 d^4}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 d^4}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{5 b^4 d^4}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{202612 B^2 (b c-a d)^6 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{35 b^4 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}+\frac{\left (405224 b^2 B^2 (b c-a d)^2 g^3\right ) \int \frac{(c+d x)^5}{a+b x} \, dx}{21 d^4}+\frac{\left (810448 b B^2 (b c-a d)^3 g^3\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{35 d^4}-\frac{\left (405224 b B^2 (b c-a d)^3 g^3\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac{\left (202612 B^2 (b c-a d)^4 g^3\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{7 d^4}-\frac{\left (101306 B^2 (b c-a d)^4 g^3\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^4}+\frac{\left (607836 B^2 (b c-a d)^4 g^3\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{5 d^4}+\frac{\left (810448 B^2 (b c-a d)^5 g^3\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{21 b d^4}-\frac{\left (202612 B^2 (b c-a d)^5 g^3\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 b d^4}-\frac{\left (405224 B^2 (b c-a d)^5 g^3\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 b d^4}+\frac{\left (810448 B^2 (b c-a d)^5 g^3\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{5 b d^4}+\frac{\left (405224 B^2 (b c-a d)^6 g^3\right ) \int \frac{c+d x}{a+b x} \, dx}{7 b^2 d^4}-\frac{\left (101306 B^2 (b c-a d)^6 g^3\right ) \int \frac{c+d x}{a+b x} \, dx}{b^2 d^4}-\frac{\left (202612 B^2 (b c-a d)^6 g^3\right ) \int \frac{c+d x}{a+b x} \, dx}{b^2 d^4}+\frac{\left (1215672 B^2 (b c-a d)^6 g^3\right ) \int \frac{c+d x}{a+b x} \, dx}{5 b^2 d^4}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \frac{1}{c+d x} \, dx}{7 b^4 d^3}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \frac{1}{c+d x} \, dx}{b^4 d^3}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \frac{1}{c+d x} \, dx}{b^4 d^3}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \frac{1}{c+d x} \, dx}{5 b^4 d^3}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{7 b^4 d^4 e}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 d^4 e}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 d^4 e}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{5 b^4 d^4 e}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{202612 B^2 (b c-a d)^6 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{35 b^4 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{202612 B^2 (b c-a d)^7 g^3 \log (c+d x)}{35 b^4 d^4}+\frac{\left (405224 b^2 B^2 (b c-a d)^2 g^3\right ) \int \left (\frac{d (b c-a d)^4}{b^5}+\frac{(b c-a d)^5}{b^5 (a+b x)}+\frac{d (b c-a d)^3 (c+d x)}{b^4}+\frac{d (b c-a d)^2 (c+d x)^2}{b^3}+\frac{d (b c-a d) (c+d x)^3}{b^2}+\frac{d (c+d x)^4}{b}\right ) \, dx}{21 d^4}+\frac{\left (810448 b B^2 (b c-a d)^3 g^3\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{35 d^4}-\frac{\left (405224 b B^2 (b c-a d)^3 g^3\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^4}+\frac{\left (202612 B^2 (b c-a d)^4 g^3\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{7 d^4}-\frac{\left (101306 B^2 (b c-a d)^4 g^3\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{d^4}+\frac{\left (607836 B^2 (b c-a d)^4 g^3\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{5 d^4}+\frac{\left (810448 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{21 b d^4}-\frac{\left (202612 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 b d^4}-\frac{\left (405224 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 b d^4}+\frac{\left (810448 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{5 b d^4}+\frac{\left (405224 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{7 b^2 d^4}-\frac{\left (101306 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^4}-\frac{\left (202612 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^4}+\frac{\left (1215672 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{5 b^2 d^4}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{7 b^4 d^4 e}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 d^4 e}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 d^4 e}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{5 b^4 d^4 e}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{202612 B^2 (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{151959 B^2 (b c-a d)^5 g^3 (c+d x)^2}{35 b^2 d^4}+\frac{1114366 B^2 (b c-a d)^4 g^3 (c+d x)^3}{315 b d^4}-\frac{202612 B^2 (b c-a d)^3 g^3 (c+d x)^4}{21 d^4}+\frac{405224 b B^2 (b c-a d)^2 g^3 (c+d x)^5}{105 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^6 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{35 b^4 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{202612 B^2 (b c-a d)^7 g^3 \log (c+d x)}{35 b^4 d^4}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{7 b^3 d^4}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 d^4}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 d^4}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{5 b^3 d^4}-\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{7 b^4 d^3}+\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 d^3}+\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 d^3}-\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{5 b^4 d^3}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{202612 B^2 (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{151959 B^2 (b c-a d)^5 g^3 (c+d x)^2}{35 b^2 d^4}+\frac{1114366 B^2 (b c-a d)^4 g^3 (c+d x)^3}{315 b d^4}-\frac{202612 B^2 (b c-a d)^3 g^3 (c+d x)^4}{21 d^4}+\frac{405224 b B^2 (b c-a d)^2 g^3 (c+d x)^5}{105 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^6 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{35 b^4 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{202612 B^2 (b c-a d)^7 g^3 \log (c+d x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{7 b^4 d^4}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^4}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^4}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{5 b^4 d^4}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{7 b^3 d^4}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^4}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^4}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^3 d^4}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{202612 B^2 (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{151959 B^2 (b c-a d)^5 g^3 (c+d x)^2}{35 b^2 d^4}+\frac{1114366 B^2 (b c-a d)^4 g^3 (c+d x)^3}{315 b d^4}-\frac{202612 B^2 (b c-a d)^3 g^3 (c+d x)^4}{21 d^4}+\frac{405224 b B^2 (b c-a d)^2 g^3 (c+d x)^5}{105 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x)}{35 b^4 d^4}-\frac{101306 B^2 (b c-a d)^7 g^3 \log ^2(a+b x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^6 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{35 b^4 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{202612 B^2 (b c-a d)^7 g^3 \log (c+d x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac{\left (810448 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{7 b^4 d^4}-\frac{\left (202612 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 d^4}-\frac{\left (405224 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 d^4}+\frac{\left (2431344 B^2 (b c-a d)^7 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{5 b^4 d^4}\\ &=\frac{202612 A B (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{202612 B^2 (b c-a d)^6 g^3 x}{35 b^3 d^3}+\frac{151959 B^2 (b c-a d)^5 g^3 (c+d x)^2}{35 b^2 d^4}+\frac{1114366 B^2 (b c-a d)^4 g^3 (c+d x)^3}{315 b d^4}-\frac{202612 B^2 (b c-a d)^3 g^3 (c+d x)^4}{21 d^4}+\frac{405224 b B^2 (b c-a d)^2 g^3 (c+d x)^5}{105 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x)}{35 b^4 d^4}-\frac{101306 B^2 (b c-a d)^7 g^3 \log ^2(a+b x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^6 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{35 b^4 d^3}+\frac{101306 B (b c-a d)^5 g^3 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^2 d^4}+\frac{202612 B (b c-a d)^4 g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{105 b d^4}-\frac{1722202 B (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 d^4}+\frac{405224 b B (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac{405224 b^2 B (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^4}+\frac{202612 B (b c-a d)^7 g^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{35 b^4 d^4}-\frac{101306 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{1215672 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{5 d^4}-\frac{202612 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^4}+\frac{405224 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{7 d^4}-\frac{202612 B^2 (b c-a d)^7 g^3 \log (c+d x)}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac{202612 B^2 (b c-a d)^7 g^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{35 b^4 d^4}\\ \end{align*}

Mathematica [B]  time = 3.14045, size = 2330, normalized size = 2.14 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2,x]

[Out]

(g^3*i^3*(35*(b*c - a*d)^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 84*d*(b*c - a*d)^2*(a + b*x)^5
*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 70*d^2*(b*c - a*d)*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])^
2 + 20*d^3*(a + b*x)^7*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - (35*B*(b*c - a*d)^4*(6*A*b*d*(b*c - a*d)^2*x +
6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(
a + b*x))/(c + d*x)]) + 2*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x
] - 6*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x - d
^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b*c - a*d)^2*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B
*(b*c - a*d)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))
/(b*c - a*d)])))/(3*d^4) + (7*B*(b*c - a*d)^3*(24*A*b*d*(b*c - a*d)^3*x + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[(
e*(a + b*x))/(c + d*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 8*d^3*(b*c -
a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*d^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]
) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 24*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 4*B
*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + B*(b*c - a*d)*(6*b*d*(
b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 12*B*(
b*c - a*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*(b*c - a*d)^4*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)]
- Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^4 - (7*B*(b*c - a*d)^2*(24*b^2*B*c
*d*(b*c - a*d)^3*x + 120*A*b*d*(b*c - a*d)^4*x + 130*b*B*d*(b*c - a*d)^4*x + 24*a*b*B*d^2*(-(b*c) + a*d)^3*x -
12*b*B*c*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*a*B*d^3*(b*c - a*d)^2*(a + b*x)^2 + 35*B*d^2*(-(b*c) + a*d)^3*(a
+ b*x)^2 + 8*b*B*c*d^3*(b*c - a*d)*(a + b*x)^3 + 10*B*d^3*(b*c - a*d)^2*(a + b*x)^3 + 8*a*B*d^4*(-(b*c) + a*d)
*(a + b*x)^3 - 6*b*B*c*d^4*(a + b*x)^4 + 6*a*B*d^5*(a + b*x)^4 + 120*B*d*(b*c - a*d)^4*(a + b*x)*Log[(e*(a + b
*x))/(c + d*x)] + 60*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 40*d^3*(b*c - a*d
)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 30*d^4*(-(b*c) + a*d)*(a + b*x)^4*(A + B*Log[(e*(a + b*
x))/(c + d*x)]) + 24*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 24*b*B*c*(b*c - a*d)^4*Log[c + d*x
] + 24*a*B*d*(b*c - a*d)^4*Log[c + d*x] - 250*B*(b*c - a*d)^5*Log[c + d*x] - 120*(b*c - a*d)^5*(A + B*Log[(e*(
a + b*x))/(c + d*x)])*Log[c + d*x] + 60*B*(b*c - a*d)^5*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*
Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(6*d^4) + (B*(b*c - a*d)*(60*b^2*B*c*d*(b*c - a*d)^4
*x - 60*a*b*B*d^2*(b*c - a*d)^4*x + 360*A*b*d*(b*c - a*d)^5*x + 462*b*B*d*(b*c - a*d)^5*x - 30*b*B*c*d^2*(b*c
- a*d)^3*(a + b*x)^2 + 30*a*B*d^3*(b*c - a*d)^3*(a + b*x)^2 - 141*B*d^2*(b*c - a*d)^4*(a + b*x)^2 + 20*b*B*c*d
^3*(b*c - a*d)^2*(a + b*x)^3 - 20*a*B*d^4*(b*c - a*d)^2*(a + b*x)^3 + 54*B*d^3*(b*c - a*d)^3*(a + b*x)^3 - 15*
b*B*c*d^4*(b*c - a*d)*(a + b*x)^4 + 15*a*B*d^5*(b*c - a*d)*(a + b*x)^4 - 18*B*d^4*(b*c - a*d)^2*(a + b*x)^4 +
12*b*B*c*d^5*(a + b*x)^5 - 12*a*B*d^6*(a + b*x)^5 + 360*B*d*(b*c - a*d)^5*(a + b*x)*Log[(e*(a + b*x))/(c + d*x
)] - 180*d^2*(b*c - a*d)^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 120*d^3*(b*c - a*d)^3*(a + b*x)^
3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])
+ 72*d^5*(b*c - a*d)*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 60*d^6*(a + b*x)^6*(A + B*Log[(e*(a +
b*x))/(c + d*x)]) - 60*b*B*c*(b*c - a*d)^5*Log[c + d*x] + 60*a*B*d*(b*c - a*d)^5*Log[c + d*x] - 822*B*(b*c -
a*d)^6*Log[c + d*x] - 360*(b*c - a*d)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 180*B*(b*c - a*d)^
6*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]
)))/(9*d^4)))/(140*b^4)

________________________________________________________________________________________

Maple [F]  time = 2.264, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{3} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

[Out]

int((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

________________________________________________________________________________________

Maxima [B]  time = 2.37413, size = 9343, normalized size = 8.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

1/7*A^2*b^3*d^3*g^3*i^3*x^7 + 1/2*A^2*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A^2*a*b^2*d^3*g^3*i^3*x^6 + 3/5*A^2*b^3*c^2*
d*g^3*i^3*x^5 + 9/5*A^2*a*b^2*c*d^2*g^3*i^3*x^5 + 3/5*A^2*a^2*b*d^3*g^3*i^3*x^5 + 1/4*A^2*b^3*c^3*g^3*i^3*x^4
+ 9/4*A^2*a*b^2*c^2*d*g^3*i^3*x^4 + 9/4*A^2*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A^2*a^3*d^3*g^3*i^3*x^4 + A^2*a*b^2*
c^3*g^3*i^3*x^3 + 3*A^2*a^2*b*c^2*d*g^3*i^3*x^3 + A^2*a^3*c*d^2*g^3*i^3*x^3 + 3/2*A^2*a^2*b*c^3*g^3*i^3*x^2 +
3/2*A^2*a^3*c^2*d*g^3*i^3*x^2 + 2*(x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/
d)*A*B*a^3*c^3*g^3*i^3 + 3*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)
/d^2 - (b*c - a*d)*x/(b*d))*A*B*a^2*b*c^3*g^3*i^3 + (2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*
x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a*b^2
*c^3*g^3*i^3 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/
d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B
*b^3*c^3*g^3*i^3 + 3*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 -
(b*c - a*d)*x/(b*d))*A*B*a^3*c^2*d*g^3*i^3 + 3*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x +
a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a^2*b*c^2
*d*g^3*i^3 + 3/4*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4
- (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*a*
b^2*c^2*d*g^3*i^3 + 1/10*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d
*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x
^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*b^3*c^2*d*g^3*i^3 + (2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))
+ 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d
^2))*A*B*a^3*c*d^2*g^3*i^3 + 3/4*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*
log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(
b^3*d^3))*A*B*a^2*b*c*d^2*g^3*i^3 + 3/10*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^
5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*
d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*a*b^2*c*d^2*g^3*i^3 + 1/60*(60*x^6*log(b*e*x/(d*
x + c) + a*e/(d*x + c)) - 60*a^6*log(b*x + a)/b^6 + 60*c^6*log(d*x + c)/d^6 - (12*(b^5*c*d^4 - a*b^4*d^5)*x^5
- 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 6
0*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5))*A*B*b^3*c*d^2*g^3*i^3 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) -
6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*
x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*a^3*d^3*g^3*i^3 + 1/10*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x +
c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^
2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*a^2*b*d^3*g^3*i^3 +
1/60*(60*x^6*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 60*a^6*log(b*x + a)/b^6 + 60*c^6*log(d*x + c)/d^6 - (12*(b
^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*
c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5))*A*B*a*b^2*d^3*g^3*i^3 + 1/210*(60*x^7*log(b*e*x/
(d*x + c) + a*e/(d*x + c)) + 60*a^7*log(b*x + a)/b^7 - 60*c^7*log(d*x + c)/d^7 - (10*(b^6*c*d^5 - a*b^5*d^6)*x
^6 - 12*(b^6*c^2*d^4 - a^2*b^4*d^6)*x^5 + 15*(b^6*c^3*d^3 - a^3*b^3*d^6)*x^4 - 20*(b^6*c^4*d^2 - a^4*b^2*d^6)*
x^3 + 30*(b^6*c^5*d - a^5*b*d^6)*x^2 - 60*(b^6*c^6 - a^6*d^6)*x)/(b^6*d^6))*A*B*b^3*d^3*g^3*i^3 + A^2*a^3*c^3*
g^3*i^3*x + 1/420*(6*b^6*c^7*g^3*i^3*log(e) - 107*a^4*b^2*c^3*d^4*g^3*i^3 + 39*a^5*b*c^2*d^5*g^3*i^3 - 6*a^6*c
*d^6*g^3*i^3 - 6*(7*g^3*i^3*log(e) - g^3*i^3)*a*b^5*c^6*d + 3*(42*g^3*i^3*log(e) - 13*g^3*i^3)*a^2*b^4*c^5*d^2
- (210*g^3*i^3*log(e) - 107*g^3*i^3)*a^3*b^3*c^4*d^3)*B^2*log(d*x + c)/(b^3*d^4) + 1/70*(b^7*c^7*g^3*i^3 - 7*
a*b^6*c^6*d*g^3*i^3 + 21*a^2*b^5*c^5*d^2*g^3*i^3 - 35*a^3*b^4*c^4*d^3*g^3*i^3 + 35*a^4*b^3*c^3*d^4*g^3*i^3 - 2
1*a^5*b^2*c^2*d^5*g^3*i^3 + 7*a^6*b*c*d^6*g^3*i^3 - a^7*d^7*g^3*i^3)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*
d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b^4*d^4) + 1/2520*(360*B^2*b^7*d^7*g^3*i^3*x^7*log(e)^2 + 60
*((21*g^3*i^3*log(e)^2 - 2*g^3*i^3*log(e))*b^7*c*d^6 + (21*g^3*i^3*log(e)^2 + 2*g^3*i^3*log(e))*a*b^6*d^7)*B^2
*x^6 + 24*((63*g^3*i^3*log(e)^2 - 15*g^3*i^3*log(e) + g^3*i^3)*b^7*c^2*d^5 + (189*g^3*i^3*log(e)^2 - 2*g^3*i^3
)*a*b^6*c*d^6 + (63*g^3*i^3*log(e)^2 + 15*g^3*i^3*log(e) + g^3*i^3)*a^2*b^5*d^7)*B^2*x^5 + 6*((105*g^3*i^3*log
(e)^2 - 51*g^3*i^3*log(e) + 10*g^3*i^3)*b^7*c^3*d^4 + (945*g^3*i^3*log(e)^2 - 147*g^3*i^3*log(e) - 10*g^3*i^3)
*a*b^6*c^2*d^5 + (945*g^3*i^3*log(e)^2 + 147*g^3*i^3*log(e) - 10*g^3*i^3)*a^2*b^5*c*d^6 + (105*g^3*i^3*log(e)^
2 + 51*g^3*i^3*log(e) + 10*g^3*i^3)*a^3*b^4*d^7)*B^2*x^4 - 2*((6*g^3*i^3*log(e) - 11*g^3*i^3)*b^7*c^4*d^3 - 4*
(315*g^3*i^3*log(e)^2 - 147*g^3*i^3*log(e) + 19*g^3*i^3)*a*b^6*c^3*d^4 - 6*(630*g^3*i^3*log(e)^2 - 29*g^3*i^3)
*a^2*b^5*c^2*d^5 - 4*(315*g^3*i^3*log(e)^2 + 147*g^3*i^3*log(e) + 19*g^3*i^3)*a^3*b^4*c*d^6 - (6*g^3*i^3*log(e
) + 11*g^3*i^3)*a^4*b^3*d^7)*B^2*x^3 + 3*(3*(2*g^3*i^3*log(e) - 3*g^3*i^3)*b^7*c^5*d^2 - (42*g^3*i^3*log(e) -
67*g^3*i^3)*a*b^6*c^4*d^3 + 2*(630*g^3*i^3*log(e)^2 - 252*g^3*i^3*log(e) - 29*g^3*i^3)*a^2*b^5*c^3*d^4 + 2*(63
0*g^3*i^3*log(e)^2 + 252*g^3*i^3*log(e) - 29*g^3*i^3)*a^3*b^4*c^2*d^5 + (42*g^3*i^3*log(e) + 67*g^3*i^3)*a^4*b
^3*c*d^6 - 3*(2*g^3*i^3*log(e) + 3*g^3*i^3)*a^5*b^2*d^7)*B^2*x^2 - 6*(6*(g^3*i^3*log(e) - g^3*i^3)*b^7*c^6*d -
3*(14*g^3*i^3*log(e) - 15*g^3*i^3)*a*b^6*c^5*d^2 + 2*(63*g^3*i^3*log(e) - 73*g^3*i^3)*a^2*b^5*c^4*d^3 - 2*(21
0*g^3*i^3*log(e)^2 - 107*g^3*i^3)*a^3*b^4*c^3*d^4 - 2*(63*g^3*i^3*log(e) + 73*g^3*i^3)*a^4*b^3*c^2*d^5 + 3*(14
*g^3*i^3*log(e) + 15*g^3*i^3)*a^5*b^2*c*d^6 - 6*(g^3*i^3*log(e) + g^3*i^3)*a^6*b*d^7)*B^2*x + 18*(20*B^2*b^7*d
^7*g^3*i^3*x^7 + 140*B^2*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(b^7*c*d^6*g^3*i^3 + a*b^6*d^7*g^3*i^3)*B^2*x^6 + 84*(
b^7*c^2*d^5*g^3*i^3 + 3*a*b^6*c*d^6*g^3*i^3 + a^2*b^5*d^7*g^3*i^3)*B^2*x^5 + 35*(b^7*c^3*d^4*g^3*i^3 + 9*a*b^6
*c^2*d^5*g^3*i^3 + 9*a^2*b^5*c*d^6*g^3*i^3 + a^3*b^4*d^7*g^3*i^3)*B^2*x^4 + 140*(a*b^6*c^3*d^4*g^3*i^3 + 3*a^2
*b^5*c^2*d^5*g^3*i^3 + a^3*b^4*c*d^6*g^3*i^3)*B^2*x^3 + 210*(a^2*b^5*c^3*d^4*g^3*i^3 + a^3*b^4*c^2*d^5*g^3*i^3
)*B^2*x^2 + (35*a^4*b^3*c^3*d^4*g^3*i^3 - 21*a^5*b^2*c^2*d^5*g^3*i^3 + 7*a^6*b*c*d^6*g^3*i^3 - a^7*d^7*g^3*i^3
)*B^2)*log(b*x + a)^2 + 18*(20*B^2*b^7*d^7*g^3*i^3*x^7 + 140*B^2*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(b^7*c*d^6*g^3
*i^3 + a*b^6*d^7*g^3*i^3)*B^2*x^6 + 84*(b^7*c^2*d^5*g^3*i^3 + 3*a*b^6*c*d^6*g^3*i^3 + a^2*b^5*d^7*g^3*i^3)*B^2
*x^5 + 35*(b^7*c^3*d^4*g^3*i^3 + 9*a*b^6*c^2*d^5*g^3*i^3 + 9*a^2*b^5*c*d^6*g^3*i^3 + a^3*b^4*d^7*g^3*i^3)*B^2*
x^4 + 140*(a*b^6*c^3*d^4*g^3*i^3 + 3*a^2*b^5*c^2*d^5*g^3*i^3 + a^3*b^4*c*d^6*g^3*i^3)*B^2*x^3 + 210*(a^2*b^5*c
^3*d^4*g^3*i^3 + a^3*b^4*c^2*d^5*g^3*i^3)*B^2*x^2 - (b^7*c^7*g^3*i^3 - 7*a*b^6*c^6*d*g^3*i^3 + 21*a^2*b^5*c^5*
d^2*g^3*i^3 - 35*a^3*b^4*c^4*d^3*g^3*i^3)*B^2)*log(d*x + c)^2 + 6*(120*B^2*b^7*d^7*g^3*i^3*x^7*log(e) + 20*((2
1*g^3*i^3*log(e) - g^3*i^3)*b^7*c*d^6 + (21*g^3*i^3*log(e) + g^3*i^3)*a*b^6*d^7)*B^2*x^6 + 12*(126*a*b^6*c*d^6
*g^3*i^3*log(e) + (42*g^3*i^3*log(e) - 5*g^3*i^3)*b^7*c^2*d^5 + (42*g^3*i^3*log(e) + 5*g^3*i^3)*a^2*b^5*d^7)*B
^2*x^5 + 3*((70*g^3*i^3*log(e) - 17*g^3*i^3)*b^7*c^3*d^4 + 7*(90*g^3*i^3*log(e) - 7*g^3*i^3)*a*b^6*c^2*d^5 + 7
*(90*g^3*i^3*log(e) + 7*g^3*i^3)*a^2*b^5*c*d^6 + (70*g^3*i^3*log(e) + 17*g^3*i^3)*a^3*b^4*d^7)*B^2*x^4 + 2*(12
60*a^2*b^5*c^2*d^5*g^3*i^3*log(e) - b^7*c^4*d^3*g^3*i^3 + a^4*b^3*d^7*g^3*i^3 + 14*(30*g^3*i^3*log(e) - 7*g^3*
i^3)*a*b^6*c^3*d^4 + 14*(30*g^3*i^3*log(e) + 7*g^3*i^3)*a^3*b^4*c*d^6)*B^2*x^3 + 3*(b^7*c^5*d^2*g^3*i^3 - 7*a*
b^6*c^4*d^3*g^3*i^3 + 7*a^4*b^3*c*d^6*g^3*i^3 - a^5*b^2*d^7*g^3*i^3 + 84*(5*g^3*i^3*log(e) - g^3*i^3)*a^2*b^5*
c^3*d^4 + 84*(5*g^3*i^3*log(e) + g^3*i^3)*a^3*b^4*c^2*d^5)*B^2*x^2 + 6*(140*a^3*b^4*c^3*d^4*g^3*i^3*log(e) - b
^7*c^6*d*g^3*i^3 + 7*a*b^6*c^5*d^2*g^3*i^3 - 21*a^2*b^5*c^4*d^3*g^3*i^3 + 21*a^4*b^3*c^2*d^5*g^3*i^3 - 7*a^5*b
^2*c*d^6*g^3*i^3 + a^6*b*d^7*g^3*i^3)*B^2*x - (6*a^7*d^7*g^3*i^3*log(e) + 6*a*b^6*c^6*d*g^3*i^3 - 39*a^2*b^5*c
^5*d^2*g^3*i^3 + 107*a^3*b^4*c^4*d^3*g^3*i^3 - (210*g^3*i^3*log(e) + 107*g^3*i^3)*a^4*b^3*c^3*d^4 + 3*(42*g^3*
i^3*log(e) + 13*g^3*i^3)*a^5*b^2*c^2*d^5 - 6*(7*g^3*i^3*log(e) + g^3*i^3)*a^6*b*c*d^6)*B^2)*log(b*x + a) - 6*(
120*B^2*b^7*d^7*g^3*i^3*x^7*log(e) + 20*((21*g^3*i^3*log(e) - g^3*i^3)*b^7*c*d^6 + (21*g^3*i^3*log(e) + g^3*i^
3)*a*b^6*d^7)*B^2*x^6 + 12*(126*a*b^6*c*d^6*g^3*i^3*log(e) + (42*g^3*i^3*log(e) - 5*g^3*i^3)*b^7*c^2*d^5 + (42
*g^3*i^3*log(e) + 5*g^3*i^3)*a^2*b^5*d^7)*B^2*x^5 + 3*((70*g^3*i^3*log(e) - 17*g^3*i^3)*b^7*c^3*d^4 + 7*(90*g^
3*i^3*log(e) - 7*g^3*i^3)*a*b^6*c^2*d^5 + 7*(90*g^3*i^3*log(e) + 7*g^3*i^3)*a^2*b^5*c*d^6 + (70*g^3*i^3*log(e)
+ 17*g^3*i^3)*a^3*b^4*d^7)*B^2*x^4 + 2*(1260*a^2*b^5*c^2*d^5*g^3*i^3*log(e) - b^7*c^4*d^3*g^3*i^3 + a^4*b^3*d
^7*g^3*i^3 + 14*(30*g^3*i^3*log(e) - 7*g^3*i^3)*a*b^6*c^3*d^4 + 14*(30*g^3*i^3*log(e) + 7*g^3*i^3)*a^3*b^4*c*d
^6)*B^2*x^3 + 3*(b^7*c^5*d^2*g^3*i^3 - 7*a*b^6*c^4*d^3*g^3*i^3 + 7*a^4*b^3*c*d^6*g^3*i^3 - a^5*b^2*d^7*g^3*i^3
+ 84*(5*g^3*i^3*log(e) - g^3*i^3)*a^2*b^5*c^3*d^4 + 84*(5*g^3*i^3*log(e) + g^3*i^3)*a^3*b^4*c^2*d^5)*B^2*x^2
+ 6*(140*a^3*b^4*c^3*d^4*g^3*i^3*log(e) - b^7*c^6*d*g^3*i^3 + 7*a*b^6*c^5*d^2*g^3*i^3 - 21*a^2*b^5*c^4*d^3*g^3
*i^3 + 21*a^4*b^3*c^2*d^5*g^3*i^3 - 7*a^5*b^2*c*d^6*g^3*i^3 + a^6*b*d^7*g^3*i^3)*B^2*x + 6*(20*B^2*b^7*d^7*g^3
*i^3*x^7 + 140*B^2*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(b^7*c*d^6*g^3*i^3 + a*b^6*d^7*g^3*i^3)*B^2*x^6 + 84*(b^7*c^
2*d^5*g^3*i^3 + 3*a*b^6*c*d^6*g^3*i^3 + a^2*b^5*d^7*g^3*i^3)*B^2*x^5 + 35*(b^7*c^3*d^4*g^3*i^3 + 9*a*b^6*c^2*d
^5*g^3*i^3 + 9*a^2*b^5*c*d^6*g^3*i^3 + a^3*b^4*d^7*g^3*i^3)*B^2*x^4 + 140*(a*b^6*c^3*d^4*g^3*i^3 + 3*a^2*b^5*c
^2*d^5*g^3*i^3 + a^3*b^4*c*d^6*g^3*i^3)*B^2*x^3 + 210*(a^2*b^5*c^3*d^4*g^3*i^3 + a^3*b^4*c^2*d^5*g^3*i^3)*B^2*
x^2 + (35*a^4*b^3*c^3*d^4*g^3*i^3 - 21*a^5*b^2*c^2*d^5*g^3*i^3 + 7*a^6*b*c*d^6*g^3*i^3 - a^7*d^7*g^3*i^3)*B^2)
*log(b*x + a))*log(d*x + c))/(b^4*d^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{3} d^{3} g^{3} i^{3} x^{6} + A^{2} a^{3} c^{3} g^{3} i^{3} + 3 \,{\left (A^{2} b^{3} c d^{2} + A^{2} a b^{2} d^{3}\right )} g^{3} i^{3} x^{5} + 3 \,{\left (A^{2} b^{3} c^{2} d + 3 \, A^{2} a b^{2} c d^{2} + A^{2} a^{2} b d^{3}\right )} g^{3} i^{3} x^{4} +{\left (A^{2} b^{3} c^{3} + 9 \, A^{2} a b^{2} c^{2} d + 9 \, A^{2} a^{2} b c d^{2} + A^{2} a^{3} d^{3}\right )} g^{3} i^{3} x^{3} + 3 \,{\left (A^{2} a b^{2} c^{3} + 3 \, A^{2} a^{2} b c^{2} d + A^{2} a^{3} c d^{2}\right )} g^{3} i^{3} x^{2} + 3 \,{\left (A^{2} a^{2} b c^{3} + A^{2} a^{3} c^{2} d\right )} g^{3} i^{3} x +{\left (B^{2} b^{3} d^{3} g^{3} i^{3} x^{6} + B^{2} a^{3} c^{3} g^{3} i^{3} + 3 \,{\left (B^{2} b^{3} c d^{2} + B^{2} a b^{2} d^{3}\right )} g^{3} i^{3} x^{5} + 3 \,{\left (B^{2} b^{3} c^{2} d + 3 \, B^{2} a b^{2} c d^{2} + B^{2} a^{2} b d^{3}\right )} g^{3} i^{3} x^{4} +{\left (B^{2} b^{3} c^{3} + 9 \, B^{2} a b^{2} c^{2} d + 9 \, B^{2} a^{2} b c d^{2} + B^{2} a^{3} d^{3}\right )} g^{3} i^{3} x^{3} + 3 \,{\left (B^{2} a b^{2} c^{3} + 3 \, B^{2} a^{2} b c^{2} d + B^{2} a^{3} c d^{2}\right )} g^{3} i^{3} x^{2} + 3 \,{\left (B^{2} a^{2} b c^{3} + B^{2} a^{3} c^{2} d\right )} g^{3} i^{3} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B b^{3} d^{3} g^{3} i^{3} x^{6} + A B a^{3} c^{3} g^{3} i^{3} + 3 \,{\left (A B b^{3} c d^{2} + A B a b^{2} d^{3}\right )} g^{3} i^{3} x^{5} + 3 \,{\left (A B b^{3} c^{2} d + 3 \, A B a b^{2} c d^{2} + A B a^{2} b d^{3}\right )} g^{3} i^{3} x^{4} +{\left (A B b^{3} c^{3} + 9 \, A B a b^{2} c^{2} d + 9 \, A B a^{2} b c d^{2} + A B a^{3} d^{3}\right )} g^{3} i^{3} x^{3} + 3 \,{\left (A B a b^{2} c^{3} + 3 \, A B a^{2} b c^{2} d + A B a^{3} c d^{2}\right )} g^{3} i^{3} x^{2} + 3 \,{\left (A B a^{2} b c^{3} + A B a^{3} c^{2} d\right )} g^{3} i^{3} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

integral(A^2*b^3*d^3*g^3*i^3*x^6 + A^2*a^3*c^3*g^3*i^3 + 3*(A^2*b^3*c*d^2 + A^2*a*b^2*d^3)*g^3*i^3*x^5 + 3*(A^
2*b^3*c^2*d + 3*A^2*a*b^2*c*d^2 + A^2*a^2*b*d^3)*g^3*i^3*x^4 + (A^2*b^3*c^3 + 9*A^2*a*b^2*c^2*d + 9*A^2*a^2*b*
c*d^2 + A^2*a^3*d^3)*g^3*i^3*x^3 + 3*(A^2*a*b^2*c^3 + 3*A^2*a^2*b*c^2*d + A^2*a^3*c*d^2)*g^3*i^3*x^2 + 3*(A^2*
a^2*b*c^3 + A^2*a^3*c^2*d)*g^3*i^3*x + (B^2*b^3*d^3*g^3*i^3*x^6 + B^2*a^3*c^3*g^3*i^3 + 3*(B^2*b^3*c*d^2 + B^2
*a*b^2*d^3)*g^3*i^3*x^5 + 3*(B^2*b^3*c^2*d + 3*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*g^3*i^3*x^4 + (B^2*b^3*c^3 + 9
*B^2*a*b^2*c^2*d + 9*B^2*a^2*b*c*d^2 + B^2*a^3*d^3)*g^3*i^3*x^3 + 3*(B^2*a*b^2*c^3 + 3*B^2*a^2*b*c^2*d + B^2*a
^3*c*d^2)*g^3*i^3*x^2 + 3*(B^2*a^2*b*c^3 + B^2*a^3*c^2*d)*g^3*i^3*x)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B*b
^3*d^3*g^3*i^3*x^6 + A*B*a^3*c^3*g^3*i^3 + 3*(A*B*b^3*c*d^2 + A*B*a*b^2*d^3)*g^3*i^3*x^5 + 3*(A*B*b^3*c^2*d +
3*A*B*a*b^2*c*d^2 + A*B*a^2*b*d^3)*g^3*i^3*x^4 + (A*B*b^3*c^3 + 9*A*B*a*b^2*c^2*d + 9*A*B*a^2*b*c*d^2 + A*B*a^
3*d^3)*g^3*i^3*x^3 + 3*(A*B*a*b^2*c^3 + 3*A*B*a^2*b*c^2*d + A*B*a^3*c*d^2)*g^3*i^3*x^2 + 3*(A*B*a^2*b*c^3 + A*
B*a^3*c^2*d)*g^3*i^3*x)*log((b*e*x + a*e)/(d*x + c)), x)

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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((b*g*x+a*g)**3*(d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}^{3}{\left (d i x + c i\right )}^{3}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="giac")

[Out]

integrate((b*g*x + a*g)^3*(d*i*x + c*i)^3*(B*log((b*x + a)*e/(d*x + c)) + A)^2, x)