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

Optimal. Leaf size=420 \[ -\frac{B^2 i^3 (b c-a d)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{2 b^4 d}+\frac{B i^3 (b c-a d)^4 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 d}-\frac{B i^3 (a+b x) (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4}-\frac{B i^3 (c+d x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^2 d}-\frac{B i^3 (c+d x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 b d}+\frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 d}+\frac{5 B^2 i^3 x (b c-a d)^3}{12 b^3}+\frac{B^2 i^3 (c+d x)^2 (b c-a d)^2}{12 b^2 d}+\frac{5 B^2 i^3 (b c-a d)^4 \log \left (\frac{a+b x}{c+d x}\right )}{12 b^4 d}+\frac{11 B^2 i^3 (b c-a d)^4 \log (c+d x)}{12 b^4 d} \]

[Out]

(5*B^2*(b*c - a*d)^3*i^3*x)/(12*b^3) + (B^2*(b*c - a*d)^2*i^3*(c + d*x)^2)/(12*b^2*d) + (5*B^2*(b*c - a*d)^4*i
^3*Log[(a + b*x)/(c + d*x)])/(12*b^4*d) - (B*(b*c - a*d)^3*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/(2*b^4) - (B*(b*c - a*d)^2*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2*d) - (B*(b*c - a*d)*i
^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b*d) + (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c +
d*x)])^2)/(4*d) + (11*B^2*(b*c - a*d)^4*i^3*Log[c + d*x])/(12*b^4*d) + (B*(b*c - a*d)^4*i^3*(A + B*Log[(e*(a +
b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d) - (B^2*(b*c - a*d)^4*i^3*PolyLog[2, (b*(c +
d*x))/(d*(a + b*x))])/(2*b^4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.616502, antiderivative size = 503, normalized size of antiderivative = 1.2, number of steps used = 24, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ -\frac{B^2 i^3 (b c-a d)^4 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{2 b^4 d}-\frac{B i^3 (b c-a d)^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 d}-\frac{B i^3 (c+d x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^2 d}-\frac{A B i^3 x (b c-a d)^3}{2 b^3}-\frac{B i^3 (c+d x)^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 b d}+\frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 d}-\frac{B^2 i^3 (a+b x) (b c-a d)^3 \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}+\frac{5 B^2 i^3 x (b c-a d)^3}{12 b^3}+\frac{B^2 i^3 (c+d x)^2 (b c-a d)^2}{12 b^2 d}+\frac{B^2 i^3 (b c-a d)^4 \log ^2(a+b x)}{4 b^4 d}+\frac{5 B^2 i^3 (b c-a d)^4 \log (a+b x)}{12 b^4 d}+\frac{B^2 i^3 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac{B^2 i^3 (b c-a d)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d} \]

Antiderivative was successfully verified.

[In]

Int[(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)^3*i^3*x)/(2*b^3) + (5*B^2*(b*c - a*d)^3*i^3*x)/(12*b^3) + (B^2*(b*c - a*d)^2*i^3*(c + d*x)^2
)/(12*b^2*d) + (5*B^2*(b*c - a*d)^4*i^3*Log[a + b*x])/(12*b^4*d) + (B^2*(b*c - a*d)^4*i^3*Log[a + b*x]^2)/(4*b
^4*d) - (B^2*(b*c - a*d)^3*i^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(2*b^4) - (B*(b*c - a*d)^2*i^3*(c + d*x
)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2*d) - (B*(b*c - a*d)*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/
(c + d*x)]))/(6*b*d) - (B*(b*c - a*d)^4*i^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4*d) + (i^
3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*d) + (B^2*(b*c - a*d)^4*i^3*Log[c + d*x])/(2*b^4*d) -
(B^2*(b*c - a*d)^4*i^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*b^4*d) - (B^2*(b*c - a*d)^4*i^3*PolyLo
g[2, -((d*(a + b*x))/(b*c - a*d))])/(2*b^4*d)

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 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 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 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 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]

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 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])

Rubi steps

\begin{align*} \int (77 c+77 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}-\frac{B \int \frac{35153041 (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}{154 d}\\ &=\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}-\frac{(456533 B (b c-a d)) \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}{2 d}\\ &=\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}-\frac{(456533 B (b c-a d)) \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}{2 d}\\ &=\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}-\frac{(456533 B (b c-a d)) \int (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{2 b}-\frac{\left (456533 B (b c-a d)^2\right ) \int (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{2 b^2}-\frac{\left (456533 B (b c-a d)^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{2 b^3}-\frac{\left (456533 B (b c-a d)^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{\left (456533 B^2 (b c-a d)\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{6 b d}+\frac{\left (456533 B^2 (b c-a d)^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{4 b^2 d}-\frac{\left (456533 B^2 (b c-a d)^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{2 b^3}+\frac{\left (456533 B^2 (b c-a d)^4\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}{2 b^4 d}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}-\frac{456533 B^2 (b c-a d)^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{\left (456533 B^2 (b c-a d)^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{6 b d}+\frac{\left (456533 B^2 (b c-a d)^3\right ) \int \frac{c+d x}{a+b x} \, dx}{4 b^2 d}+\frac{\left (456533 B^2 (b c-a d)^4\right ) \int \frac{1}{c+d x} \, dx}{2 b^4}+\frac{\left (456533 B^2 (b c-a d)^4\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}{2 b^4 d e}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}-\frac{456533 B^2 (b c-a d)^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{456533 B^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}+\frac{\left (456533 B^2 (b c-a d)^2\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}{6 b d}+\frac{\left (456533 B^2 (b c-a d)^3\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{4 b^2 d}+\frac{\left (456533 B^2 (b c-a d)^4\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}{2 b^4 d e}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}+\frac{2282665 B^2 (b c-a d)^3 x}{12 b^3}+\frac{456533 B^2 (b c-a d)^2 (c+d x)^2}{12 b^2 d}+\frac{2282665 B^2 (b c-a d)^4 \log (a+b x)}{12 b^4 d}-\frac{456533 B^2 (b c-a d)^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{456533 B^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac{\left (456533 B^2 (b c-a d)^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b^4}+\frac{\left (456533 B^2 (b c-a d)^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}+\frac{2282665 B^2 (b c-a d)^3 x}{12 b^3}+\frac{456533 B^2 (b c-a d)^2 (c+d x)^2}{12 b^2 d}+\frac{2282665 B^2 (b c-a d)^4 \log (a+b x)}{12 b^4 d}-\frac{456533 B^2 (b c-a d)^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{456533 B^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac{456533 B^2 (b c-a d)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d}+\frac{\left (456533 B^2 (b c-a d)^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 d}+\frac{\left (456533 B^2 (b c-a d)^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}+\frac{2282665 B^2 (b c-a d)^3 x}{12 b^3}+\frac{456533 B^2 (b c-a d)^2 (c+d x)^2}{12 b^2 d}+\frac{2282665 B^2 (b c-a d)^4 \log (a+b x)}{12 b^4 d}+\frac{456533 B^2 (b c-a d)^4 \log ^2(a+b x)}{4 b^4 d}-\frac{456533 B^2 (b c-a d)^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{456533 B^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac{456533 B^2 (b c-a d)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d}+\frac{\left (456533 B^2 (b c-a d)^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2 b^4 d}\\ &=-\frac{456533 A B (b c-a d)^3 x}{2 b^3}+\frac{2282665 B^2 (b c-a d)^3 x}{12 b^3}+\frac{456533 B^2 (b c-a d)^2 (c+d x)^2}{12 b^2 d}+\frac{2282665 B^2 (b c-a d)^4 \log (a+b x)}{12 b^4 d}+\frac{456533 B^2 (b c-a d)^4 \log ^2(a+b x)}{4 b^4 d}-\frac{456533 B^2 (b c-a d)^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{2 b^4}-\frac{456533 B (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 d}-\frac{456533 B (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac{456533 B (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 d}+\frac{456533 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 d}+\frac{456533 B^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac{456533 B^2 (b c-a d)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 d}-\frac{456533 B^2 (b c-a d)^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b^4 d}\\ \end{align*}

Mathematica [A]  time = 0.303566, size = 389, normalized size = 0.93 \[ \frac{i^3 \left ((c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2-\frac{B (b c-a d) \left (-3 B (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+3 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 b^3 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+6 (b c-a d)^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+6 A b d x (b c-a d)^2-B (b c-a d) \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (\frac{e (a+b x)}{c+d x}\right )-6 B (b c-a d)^3 \log (c+d x)-3 B (b c-a d)^2 ((b c-a d) \log (a+b x)+b d x)\right )}{3 b^4}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(i^3*((c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - (B*(b*c - a*d)*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c
- a*d)^2*(b*d*x + (b*c - a*d)*Log[a + b*x]) - B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c -
a*d)^2*Log[a + b*x]) + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*b^2*(b*c - a*d)*(c + d*x
)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*(b*c - a
*d)^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 3*B*(b*c - a*d)^3*(
Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))
/(3*b^4)))/(4*d)

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Maple [F]  time = 2.146, size = 0, normalized size = 0. \begin{align*} \int \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((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

[Out]

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

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Maxima [B]  time = 1.87103, size = 2415, normalized size = 5.75 \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)))^2,x, algorithm="maxima")

[Out]

1/4*A^2*d^3*i^3*x^4 + A^2*c*d^2*i^3*x^3 + 3/2*A^2*c^2*d*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*c^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*c^2*d*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*c*d^2*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*d^3*i^3 + A^2*c^3*i^3*x - 1/12*(26*a*b^2*c^3*d*i^3 - 21*a^2*b*c^2*d^2*i^3 + 6*a^3*c*d^3*i^3 + (6*i^
3*log(e) - 11*i^3)*b^3*c^4)*B^2*log(d*x + c)/(b^3*d) - 1/2*(b^4*c^4*i^3 - 4*a*b^3*c^3*d*i^3 + 6*a^2*b^2*c^2*d^
2*i^3 - 4*a^3*b*c*d^3*i^3 + a^4*d^4*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) + 1/12*(3*B^2*b^4*d^4*i^3*x^4*log(e)^2 + 2*(a*b^3*d^4*i^3*log(e) + (6*i^3*log(e)^
2 - i^3*log(e))*b^4*c*d^3)*B^2*x^3 + ((18*i^3*log(e)^2 - 9*i^3*log(e) + i^3)*b^4*c^2*d^2 + 2*(6*i^3*log(e) - i
^3)*a*b^3*c*d^3 - (3*i^3*log(e) - i^3)*a^2*b^2*d^4)*B^2*x^2 + ((12*i^3*log(e)^2 - 18*i^3*log(e) + 7*i^3)*b^4*c
^3*d + (36*i^3*log(e) - 19*i^3)*a*b^3*c^2*d^2 - (24*i^3*log(e) - 17*i^3)*a^2*b^2*c*d^3 + (6*i^3*log(e) - 5*i^3
)*a^3*b*d^4)*B^2*x + 3*(B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*
c^3*d*i^3*x + (4*a*b^3*c^3*d*i^3 - 6*a^2*b^2*c^2*d^2*i^3 + 4*a^3*b*c*d^3*i^3 - a^4*d^4*i^3)*B^2)*log(b*x + a)^
2 + 3*(B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x + B^2
*b^4*c^4*i^3)*log(d*x + c)^2 + (6*B^2*b^4*d^4*i^3*x^4*log(e) + 2*(a*b^3*d^4*i^3 + (12*i^3*log(e) - i^3)*b^4*c*
d^3)*B^2*x^3 + 3*(4*a*b^3*c*d^3*i^3 - a^2*b^2*d^4*i^3 + 3*(4*i^3*log(e) - i^3)*b^4*c^2*d^2)*B^2*x^2 + 6*(6*a*b
^3*c^2*d^2*i^3 - 4*a^2*b^2*c*d^3*i^3 + a^3*b*d^4*i^3 + (4*i^3*log(e) - 3*i^3)*b^4*c^3*d)*B^2*x + (6*(4*i^3*log
(e) - 3*i^3)*a*b^3*c^3*d - 9*(4*i^3*log(e) - 5*i^3)*a^2*b^2*c^2*d^2 + 2*(12*i^3*log(e) - 19*i^3)*a^3*b*c*d^3 -
(6*i^3*log(e) - 11*i^3)*a^4*d^4)*B^2)*log(b*x + a) - (6*B^2*b^4*d^4*i^3*x^4*log(e) + 2*(a*b^3*d^4*i^3 + (12*i
^3*log(e) - i^3)*b^4*c*d^3)*B^2*x^3 + 3*(4*a*b^3*c*d^3*i^3 - a^2*b^2*d^4*i^3 + 3*(4*i^3*log(e) - i^3)*b^4*c^2*
d^2)*B^2*x^2 + 6*(6*a*b^3*c^2*d^2*i^3 - 4*a^2*b^2*c*d^3*i^3 + a^3*b*d^4*i^3 + (4*i^3*log(e) - 3*i^3)*b^4*c^3*d
)*B^2*x + 6*(B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x
+ (4*a*b^3*c^3*d*i^3 - 6*a^2*b^2*c^2*d^2*i^3 + 4*a^3*b*c*d^3*i^3 - a^4*d^4*i^3)*B^2)*log(b*x + a))*log(d*x +
c))/(b^4*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} d^{3} i^{3} x^{3} + 3 \, A^{2} c d^{2} i^{3} x^{2} + 3 \, A^{2} c^{2} d i^{3} x + A^{2} c^{3} i^{3} +{\left (B^{2} d^{3} i^{3} x^{3} + 3 \, B^{2} c d^{2} i^{3} x^{2} + 3 \, B^{2} c^{2} d i^{3} x + B^{2} c^{3} i^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B d^{3} i^{3} x^{3} + 3 \, A B c d^{2} i^{3} x^{2} + 3 \, A B c^{2} d i^{3} x + A B c^{3} i^{3}\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((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

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

[Out]

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