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

Optimal. Leaf size=252 \[ \frac{B g^3 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^4 i}-\frac{g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (\frac{e (a+b x)}{c+d x}\right )+3 A+B\right )}{6 d^2 i}+\frac{g^3 (a+b x) (b c-a d)^2 \left (6 B \log \left (\frac{e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{6 d^3 i}+\frac{g^3 (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac{e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{6 d^4 i}+\frac{g^3 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 d i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.625548, antiderivative size = 408, normalized size of antiderivative = 1.62, number of steps used = 23, number of rules used = 13, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.325, Rules used = {2528, 2486, 31, 2525, 12, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B g^3 (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^4 i}-\frac{g^3 (a+b x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i}-\frac{g^3 (b c-a d)^3 \log (c i+d i x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^4 i}+\frac{g^3 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 d i}+\frac{A b g^3 x (b c-a d)^2}{d^3 i}+\frac{B g^3 (a+b x) (b c-a d)^2 \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^3 i}-\frac{B g^3 (a+b x)^2 (b c-a d)}{6 d^2 i}+\frac{5 b B g^3 x (b c-a d)^2}{6 d^3 i}-\frac{B g^3 (b c-a d)^3 \log ^2(i (c+d x))}{2 d^4 i}-\frac{11 B g^3 (b c-a d)^3 \log (c+d x)}{6 d^4 i}+\frac{B g^3 (b c-a d)^3 \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^4 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 \frac{(a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{31 c+31 d x} \, dx &=\int \left (\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{31 d^3}+\frac{(-b c+a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^3 (31 c+31 d x)}-\frac{b (b c-a d) g^2 (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{31 d^2}+\frac{b g (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{31 d}\right ) \, dx\\ &=\frac{(b g) \int (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{31 d}-\frac{\left (b (b c-a d) g^2\right ) \int (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{31 d^2}+\frac{\left (b (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{31 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 c+31 d x} \, dx}{d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac{B \int \frac{(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{93 d}+\frac{(B (b c-a d) g) \int \frac{(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{62 d^2}+\frac{\left (b B (b c-a d)^2 g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{31 d^3}+\frac{\left (B (b c-a d)^3 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 (31 c+31 d x)}{e (a+b x)} \, dx}{31 d^4}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac{\left (B (b c-a d) g^3\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{93 d}+\frac{\left (B (b c-a d)^2 g^3\right ) \int \frac{a+b x}{c+d x} \, dx}{62 d^2}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{1}{c+d x} \, dx}{31 d^3}+\frac{\left (B (b c-a d)^3 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 (31 c+31 d x)}{a+b x} \, dx}{31 d^4 e}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{B (b c-a d)^3 g^3 \log (c+d x)}{31 d^4}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac{\left (B (b c-a d) g^3\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{93 d}+\frac{\left (B (b c-a d)^2 g^3\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{62 d^2}+\frac{\left (B (b c-a d)^3 g^3\right ) \int \left (\frac{b e \log (31 c+31 d x)}{a+b x}-\frac{d e \log (31 c+31 d x)}{c+d x}\right ) \, dx}{31 d^4 e}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}+\frac{5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac{B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}+\frac{\left (b B (b c-a d)^3 g^3\right ) \int \frac{\log (31 c+31 d x)}{a+b x} \, dx}{31 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{\log (31 c+31 d x)}{c+d x} \, dx}{31 d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}+\frac{5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac{B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}+\frac{B (b c-a d)^3 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (31 c+31 d x)}{31 d^4}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \operatorname{Subst}\left (\int \frac{31 \log (x)}{x} \, dx,x,31 c+31 d x\right )}{961 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{\log \left (\frac{31 d (a+b x)}{-31 b c+31 a d}\right )}{31 c+31 d x} \, dx}{d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}+\frac{5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac{B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}+\frac{B (b c-a d)^3 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (31 c+31 d x)}{31 d^4}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,31 c+31 d x\right )}{31 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-31 b c+31 a d}\right )}{x} \, dx,x,31 c+31 d x\right )}{31 d^4}\\ &=\frac{A b (b c-a d)^2 g^3 x}{31 d^3}+\frac{5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac{B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{31 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac{11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}-\frac{B (b c-a d)^3 g^3 \log ^2(31 (c+d x))}{62 d^4}+\frac{B (b c-a d)^3 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (31 c+31 d x)}{31 d^4}-\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}+\frac{B (b c-a d)^3 g^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{31 d^4}\\ \end{align*}

Mathematica [A]  time = 0.284686, size = 354, normalized size = 1.4 \[ \frac{g^3 \left (3 B (b c-a d)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (i (c+d x)) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )\right )+2 d^3 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-6 (b c-a d)^3 \log (i (c+d x)) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+6 A b d x (b c-a d)^2+B (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (\frac{e (a+b x)}{c+d x}\right )-6 B (b c-a d)^3 \log (c+d x)+3 B (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{6 d^4 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.187, size = 4297, normalized size = 17.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/6/d*B*g^3/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^3-1/d*B*g^3/i*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)-b*e)/b/e)*a^3-1/d*A*g^3/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^3+1/d^4*B*g^3/i*dilog(-(d*(b*e/d+(a*d-b*
c)*e/d/(d*x+c))-b*e)/b/e)*b^3*c^3-1/d*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x
+c))-b*e)/b/e)*a^3+11/6/d^4*B*g^3/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*b^3*c^3+1/d^4*A*g^3/i*ln(d*(b*e/d+
(a*d-b*c)*e/d/(d*x+c))-b*e)*b^3*c^3+3/2*e^2/d*B*g^3/i*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*
x+c)*b*c)^2*a^3-3*e/d^4*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*b^4*c^3+1/3*e^3*
d^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^6/(d*x+c)^3-3*e/d^4*A*g^3/i*b^4/
(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^3+7/6*e/d*B*g^3/i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3+11/2/d^2*B*g^3/i*ln(d*(b
*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^2*b*c+3*e*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*
b*c)*a^4/(d*x+c)+3/d^2*B*g^3/i*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a^2*b*c+1/3*e^3/d*A*g^3/i*b^3
/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3-3/d^3*B*g^3/i*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*b^2*c^2*a
+1/d^4*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*b^3*c^3-11/2/d^3
*B*g^3/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*b^2*c^2*a+5*e^3/d^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(
d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*b^4*c^4/(d*x+c)^3*a^2+18*e/d^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*
x+c)*a-e/(d*x+c)*b*c)*b^2*c^2/(d*x+c)*a^2-20/3*e^3/d*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/
(d*x+c)*b*c)^3*b^3*c^3/(d*x+c)^3*a^3-12*e/d*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b
*c)*a^3/(d*x+c)*b*c-12*e/d^3*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*b^3*c^3/(d*
x+c)*a+15*e^2/d^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^3*c^3/(d*x+c)^2*a^
2-2*e^3*d*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^5/(d*x+c)^3*b*c-15*e^2/d*B
*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3/(d*x+c)^2*b^2*c^2-2*e^3/d^3*B*g^3/i
*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*b^5*c^5/(d*x+c)^3*a-15/2*e^2/d^3*B*g^3/i*ln(b
*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^4*c^4/(d*x+c)^2*a-3/2*e^2/d^4*B*g^3/i*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^5*c^3-9/2*e^2/d^2*A*g^3/i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)
*b*c)^2*a^2*c+9/2*e^2/d^3*A*g^3/i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^2*a-9/2*e^2/d^2*B*g^3/i*b^3*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*c+9*e/d^3*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d
*e/(d*x+c)*a-e/(d*x+c)*b*c)*a*b^3*c^2+3*e/d^4*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)
*b*c)*b^4*c^4/(d*x+c)+1/3*e^3/d^4*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*b^6*
c^6/(d*x+c)^3+9/2*e^2/d^3*B*g^3/i*b^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a*c^2+3/
2*e^2/d^4*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^5*c^5/(d*x+c)^2-e^3/d^2*B*
g^3/i*b^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c+e^3/d^3*B*g^3/i*b^5*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a*c^2+15/2*e^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(
d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^4/(d*x+c)^2*b*c+5*e^3*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-
e/(d*x+c)*b*c)^3*a^4/(d*x+c)^3*b^2*c^2-9*e/d^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)*a^2*b^2*c-3/2*e^2*d*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^5/(d*x+c)
^2-1/3*e^3/d^4*B*g^3/i*b^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3+3*e/d*B*g^3/i*l
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3*b+3/d^2*B*g^3/i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c
))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a^2*b*c+1/2*e^2/d^3*B*g^3/i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c
)^2*c^2*a-7/2*e/d^2*B*g^3/i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c-1/2*e^2/d^2*B*g^3/i*b^3/(d*e/(d*x+c)*a-e/(
d*x+c)*b*c)^2*a^2*c+7/2*e/d^3*B*g^3/i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^2*a-3/d^3*B*g^3/i*ln(b*e/d+(a*d-b*c)
*e/d/(d*x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a*b^2*c^2+1/3*e^3/d*B*g^3/i*b^3*ln(b*e/d+(a*d-b*c
)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3+e^3/d^3*A*g^3/i*b^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^2*a-9
*e/d^2*A*g^3/i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c+9*e/d^3*A*g^3/i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a*c^2
-e^3/d^2*A*g^3/i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c-7/6*e/d^4*B*g^3/i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)
*c^3-3/d^3*A*g^3/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*b^2*c^2*a+3/d^2*A*g^3/i*ln(d*(b*e/d+(a*d-b*c)*e/d/(
d*x+c))-b*e)*a^2*b*c-3/2*e^2/d^4*A*g^3/i*b^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^3-1/3*e^3/d^4*A*g^3/i*b^6/(d*e/
(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3+3/2*e^2/d*A*g^3/i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3-1/6*e^2/d^4*B*g^3/i*b
^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^3+3*e/d*A*g^3/i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3+1/6*e^2/d*B*g^3/i*b^2
/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3

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Maxima [B]  time = 1.5426, size = 1067, normalized size = 4.23 \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*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorithm="maxima")

[Out]

3*A*a^2*b*g^3*(x/(d*i) - c*log(d*x + c)/(d^2*i)) - 1/6*A*b^3*g^3*(6*c^3*log(d*x + c)/(d^4*i) - (2*d^2*x^3 - 3*
c*d*x^2 + 6*c^2*x)/(d^3*i)) + 3/2*A*a*b^2*g^3*(2*c^2*log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A*a^3*g
^3*log(d*i*x + c*i)/(d*i) - (b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^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/(d^4*i) + 1/6*(6*a^3*d^3*g^3*log(e)
- (6*g^3*log(e) + 11*g^3)*b^3*c^3 + 9*(2*g^3*log(e) + 3*g^3)*a*b^2*c^2*d - 18*(g^3*log(e) + g^3)*a^2*b*c*d^2)*
B*log(d*x + c)/(d^4*i) + 1/6*(2*B*b^3*d^3*g^3*x^3*log(e) - ((3*g^3*log(e) + g^3)*b^3*c*d^2 - (9*g^3*log(e) + g
^3)*a*b^2*d^3)*B*x^2 + 3*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*B*log(d*x + c)^2
+ ((6*g^3*log(e) + 5*g^3)*b^3*c^2*d - 6*(3*g^3*log(e) + 2*g^3)*a*b^2*c*d^2 + (18*g^3*log(e) + 7*g^3)*a^2*b*d^3
)*B*x + (2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^
3 + 3*a^2*b*d^3*g^3)*B*x + (6*a*b^2*c^2*d*g^3 - 15*a^2*b*c*d^2*g^3 + 11*a^3*d^3*g^3)*B)*log(b*x + a) - (2*B*b^
3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3 + 3*a^2*b*d^3
*g^3)*B*x)*log(d*x + c))/(d^4*i)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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