3.2.32 \(\int \frac {(c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^2} \, dx\) [132]

Optimal. Leaf size=390 \[ -\frac {B d^2 (b c-a d) i^3 n x}{2 b^3 g^2}-\frac {B (b c-a d)^2 i^3 n (c+d x)}{b^3 g^2 (a+b x)}+\frac {2 d^2 (b c-a d) i^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac {(b c-a d)^2 i^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {d i^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^2}-\frac {B d (b c-a d)^2 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{2 b^4 g^2}-\frac {5 B d (b c-a d)^2 i^3 n \log (c+d x)}{2 b^4 g^2}-\frac {3 d (b c-a d)^2 i^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {3 B d (b c-a d)^2 i^3 n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.29, antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {2561, 46, 2393, 2341, 2356, 2351, 31, 2379, 2438} \begin {gather*} \frac {3 B d i^3 n (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {2 d^2 i^3 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}-\frac {3 d i^3 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}-\frac {i^3 (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2 (a+b x)}+\frac {d i^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^2}-\frac {B d i^3 n (b c-a d)^2 \log \left (\frac {a+b x}{c+d x}\right )}{2 b^4 g^2}-\frac {5 B d i^3 n (b c-a d)^2 \log (c+d x)}{2 b^4 g^2}-\frac {B d^2 i^3 n x (b c-a d)}{2 b^3 g^2}-\frac {B i^3 n (c+d x) (b c-a d)^2}{b^3 g^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2351

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2438

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 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {(132 c+132 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac {2299968 d^2 (3 b c-2 a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {2299968 d^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)^2}+\frac {6899904 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}\right ) \, dx\\ &=\frac {\left (2299968 d^3\right ) \int x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g^2}+\frac {\left (2299968 d^2 (3 b c-2 a d)\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 g^2}+\frac {\left (6899904 d (b c-a d)^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^2}+\frac {\left (2299968 (b c-a d)^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^3 g^2}\\ &=\frac {2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac {1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac {6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac {\left (2299968 B d^2 (3 b c-2 a d)\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 g^2}-\frac {\left (1149984 B d^3 n\right ) \int \frac {(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {\left (6899904 B d (b c-a d)^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 g^2}+\frac {\left (2299968 B (b c-a d)^3 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac {2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac {2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac {1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac {6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac {\left (1149984 B d^3 (b c-a d) n\right ) \int \frac {x^2}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {\left (2299968 B d^2 (3 b c-2 a d) (b c-a d) n\right ) \int \frac {1}{c+d x} \, dx}{b^4 g^2}-\frac {\left (6899904 B d (b c-a d)^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^4 g^2}+\frac {\left (2299968 B (b c-a d)^4 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac {2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac {2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac {1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac {6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac {2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}-\frac {\left (1149984 B d^3 (b c-a d) n\right ) \int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{b^2 g^2}-\frac {\left (6899904 B d (b c-a d)^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^2}+\frac {\left (6899904 B d^2 (b c-a d)^2 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac {\left (2299968 B (b c-a d)^4 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^2}\\ &=\frac {2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac {1149984 B d^2 (b c-a d) n x}{b^3 g^2}-\frac {2299968 B (b c-a d)^3 n}{b^4 g^2 (a+b x)}-\frac {1149984 a^2 B d^3 n \log (a+b x)}{b^4 g^2}-\frac {2299968 B d (b c-a d)^2 n \log (a+b x)}{b^4 g^2}+\frac {2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac {1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac {6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac {1149984 B c^2 d n \log (c+d x)}{b^2 g^2}-\frac {2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}+\frac {2299968 B d (b c-a d)^2 n \log (c+d x)}{b^4 g^2}+\frac {6899904 B d (b c-a d)^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac {\left (6899904 B d (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}-\frac {\left (6899904 B d (b c-a d)^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}\\ &=\frac {2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac {1149984 B d^2 (b c-a d) n x}{b^3 g^2}-\frac {2299968 B (b c-a d)^3 n}{b^4 g^2 (a+b x)}-\frac {1149984 a^2 B d^3 n \log (a+b x)}{b^4 g^2}-\frac {2299968 B d (b c-a d)^2 n \log (a+b x)}{b^4 g^2}-\frac {3449952 B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}+\frac {2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac {1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac {6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac {1149984 B c^2 d n \log (c+d x)}{b^2 g^2}-\frac {2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}+\frac {2299968 B d (b c-a d)^2 n \log (c+d x)}{b^4 g^2}+\frac {6899904 B d (b c-a d)^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac {\left (6899904 B d (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}\\ &=\frac {2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac {1149984 B d^2 (b c-a d) n x}{b^3 g^2}-\frac {2299968 B (b c-a d)^3 n}{b^4 g^2 (a+b x)}-\frac {1149984 a^2 B d^3 n \log (a+b x)}{b^4 g^2}-\frac {2299968 B d (b c-a d)^2 n \log (a+b x)}{b^4 g^2}-\frac {3449952 B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}+\frac {2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac {1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac {6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac {1149984 B c^2 d n \log (c+d x)}{b^2 g^2}-\frac {2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}+\frac {2299968 B d (b c-a d)^2 n \log (c+d x)}{b^4 g^2}+\frac {6899904 B d (b c-a d)^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {6899904 B d (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^2}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 394, normalized size = 1.01 \begin {gather*} \frac {i^3 \left (2 A b d^2 (3 b c-2 a d) x-b B d^2 (b c-a d) n x-\frac {2 B (b c-a d)^3 n}{a+b x}-a^2 B d^3 n \log (a+b x)-2 B d (b c-a d)^2 n \log (a+b x)+2 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 d^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {2 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+6 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+b^2 B c^2 d n \log (c+d x)+2 B d (b c-a d)^2 n \log (c+d x)-2 B d (-b c+a d) (-3 b c+2 a d) n \log (c+d x)-3 B d (b c-a d)^2 n \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 {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{2 b^4 g^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (b g x +a g \right )^{2}}\, dx\]

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))^n))/(b*g*x+a*g)^2,x)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1458 vs. \(2 (358) = 716\).
time = 0.62, size = 1458, normalized size = 3.74 \begin {gather*} i \, B c^{3} n {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} + 3 i \, A {\left (\frac {a^{2}}{b^{4} g^{2} x + a b^{3} g^{2}} - \frac {x}{b^{2} g^{2}} + \frac {2 \, a \log \left (b x + a\right )}{b^{3} g^{2}}\right )} c d^{2} - \frac {1}{2} i \, {\left (\frac {2 \, a^{3}}{b^{5} g^{2} x + a b^{4} g^{2}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{4} g^{2}} + \frac {b x^{2} - 4 \, a x}{b^{3} g^{2}}\right )} A d^{3} - 3 i \, A c^{2} d {\left (\frac {a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac {\log \left (b x + a\right )}{b^{2} g^{2}}\right )} + \frac {i \, B c^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {i \, A c^{3}}{b^{2} g^{2} x + a b g^{2}} + \frac {{\left (5 i \, b^{3} c^{3} d n - 3 i \, a b^{2} c^{2} d^{2} n - 2 i \, a^{2} b c d^{3} n + 2 i \, a^{3} d^{4} n\right )} B \log \left (d x + c\right )}{2 \, {\left (b^{5} c g^{2} - a b^{4} d g^{2}\right )}} - \frac {{\left (i \, b^{4} c d^{3} - i \, a b^{3} d^{4}\right )} B x^{3} + {\left (a b^{3} c d^{3} {\left (2 i \, n - 9 i\right )} + b^{4} c^{2} d^{2} {\left (-i \, n + 6 i\right )} + a^{2} b^{2} d^{4} {\left (-i \, n + 3 i\right )}\right )} B x^{2} + {\left (a b^{3} c^{2} d^{2} {\left (-i \, n + 6 i\right )} - 2 \, a^{2} b^{2} c d^{3} {\left (-i \, n + 5 i\right )} + a^{3} b d^{4} {\left (-i \, n + 4 i\right )}\right )} B x - 3 \, {\left ({\left (i \, b^{4} c^{3} d n - 3 i \, a b^{3} c^{2} d^{2} n + 3 i \, a^{2} b^{2} c d^{3} n - i \, a^{3} b d^{4} n\right )} B x + {\left (i \, a b^{3} c^{3} d n - 3 i \, a^{2} b^{2} c^{2} d^{2} n + 3 i \, a^{3} b c d^{3} n - i \, a^{4} d^{4} n\right )} B\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (6 \, a^{2} b^{2} c^{2} d^{2} {\left (i \, n + i\right )} + a^{4} d^{4} {\left (i \, n + i\right )} + 3 \, a b^{3} c^{3} d {\left (-i \, n - i\right )} + 4 \, a^{3} b c d^{3} {\left (-i \, n - i\right )}\right )} B + {\left ({\left (a^{3} b d^{4} {\left (7 i \, n - 6 i\right )} - 6 \, a b^{3} c^{2} d^{2} {\left (-2 i \, n + 3 i\right )} + a^{2} b^{2} c d^{3} {\left (-17 i \, n + 18 i\right )} + 6 i \, b^{4} c^{3} d\right )} B x + {\left (a^{4} d^{4} {\left (7 i \, n - 6 i\right )} - 6 \, a^{2} b^{2} c^{2} d^{2} {\left (-2 i \, n + 3 i\right )} + a^{3} b c d^{3} {\left (-17 i \, n + 18 i\right )} + 6 i \, a b^{3} c^{3} d\right )} B\right )} \log \left (b x + a\right ) + {\left ({\left (i \, b^{4} c d^{3} - i \, a b^{3} d^{4}\right )} B x^{3} - 3 \, {\left (-2 i \, b^{4} c^{2} d^{2} + 3 i \, a b^{3} c d^{3} - i \, a^{2} b^{2} d^{4}\right )} B x^{2} - 2 \, {\left (-3 i \, a b^{3} c^{2} d^{2} + 5 i \, a^{2} b^{2} c d^{3} - 2 i \, a^{3} b d^{4}\right )} B x - 2 \, {\left (-3 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} B - 6 \, {\left ({\left (-i \, b^{4} c^{3} d + 3 i \, a b^{3} c^{2} d^{2} - 3 i \, a^{2} b^{2} c d^{3} + i \, a^{3} b d^{4}\right )} B x + {\left (-i \, a b^{3} c^{3} d + 3 i \, a^{2} b^{2} c^{2} d^{2} - 3 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} B\right )} \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) + {\left ({\left (-i \, b^{4} c d^{3} + i \, a b^{3} d^{4}\right )} B x^{3} - 3 \, {\left (2 i \, b^{4} c^{2} d^{2} - 3 i \, a b^{3} c d^{3} + i \, a^{2} b^{2} d^{4}\right )} B x^{2} - 2 \, {\left (3 i \, a b^{3} c^{2} d^{2} - 5 i \, a^{2} b^{2} c d^{3} + 2 i \, a^{3} b d^{4}\right )} B x - 2 \, {\left (3 i \, a b^{3} c^{3} d - 6 i \, a^{2} b^{2} c^{2} d^{2} + 4 i \, a^{3} b c d^{3} - i \, a^{4} d^{4}\right )} B - 6 \, {\left ({\left (i \, b^{4} c^{3} d - 3 i \, a b^{3} c^{2} d^{2} + 3 i \, a^{2} b^{2} c d^{3} - i \, a^{3} b d^{4}\right )} B x + {\left (i \, a b^{3} c^{3} d - 3 i \, a^{2} b^{2} c^{2} d^{2} + 3 i \, a^{3} b c d^{3} - i \, a^{4} d^{4}\right )} B\right )} \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, {\left (a b^{5} c g^{2} - a^{2} b^{4} d g^{2} + {\left (b^{6} c g^{2} - a b^{5} d g^{2}\right )} x\right )}} + \frac {3 \, {\left (-i \, b^{2} c^{2} d n + 2 i \, a b c d^{2} n - i \, a^{2} d^{3} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{4} g^{2}} \end {gather*}

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))^n))/(b*g*x+a*g)^2,x, algorithm="maxima")

[Out]

I*B*c^3*n*(1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*g^
2)) + 3*I*A*(a^2/(b^4*g^2*x + a*b^3*g^2) - x/(b^2*g^2) + 2*a*log(b*x + a)/(b^3*g^2))*c*d^2 - 1/2*I*(2*a^3/(b^5
*g^2*x + a*b^4*g^2) + 6*a^2*log(b*x + a)/(b^4*g^2) + (b*x^2 - 4*a*x)/(b^3*g^2))*A*d^3 - 3*I*A*c^2*d*(a/(b^3*g^
2*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) + I*B*c^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(b^2*g^2*x + a*b*g
^2) + I*A*c^3/(b^2*g^2*x + a*b*g^2) + 1/2*(5*I*b^3*c^3*d*n - 3*I*a*b^2*c^2*d^2*n - 2*I*a^2*b*c*d^3*n + 2*I*a^3
*d^4*n)*B*log(d*x + c)/(b^5*c*g^2 - a*b^4*d*g^2) - 1/2*((I*b^4*c*d^3 - I*a*b^3*d^4)*B*x^3 + (a*b^3*c*d^3*(2*I*
n - 9*I) + b^4*c^2*d^2*(-I*n + 6*I) + a^2*b^2*d^4*(-I*n + 3*I))*B*x^2 + (a*b^3*c^2*d^2*(-I*n + 6*I) - 2*a^2*b^
2*c*d^3*(-I*n + 5*I) + a^3*b*d^4*(-I*n + 4*I))*B*x - 3*((I*b^4*c^3*d*n - 3*I*a*b^3*c^2*d^2*n + 3*I*a^2*b^2*c*d
^3*n - I*a^3*b*d^4*n)*B*x + (I*a*b^3*c^3*d*n - 3*I*a^2*b^2*c^2*d^2*n + 3*I*a^3*b*c*d^3*n - I*a^4*d^4*n)*B)*log
(b*x + a)^2 - 2*(6*a^2*b^2*c^2*d^2*(I*n + I) + a^4*d^4*(I*n + I) + 3*a*b^3*c^3*d*(-I*n - I) + 4*a^3*b*c*d^3*(-
I*n - I))*B + ((a^3*b*d^4*(7*I*n - 6*I) - 6*a*b^3*c^2*d^2*(-2*I*n + 3*I) + a^2*b^2*c*d^3*(-17*I*n + 18*I) + 6*
I*b^4*c^3*d)*B*x + (a^4*d^4*(7*I*n - 6*I) - 6*a^2*b^2*c^2*d^2*(-2*I*n + 3*I) + a^3*b*c*d^3*(-17*I*n + 18*I) +
6*I*a*b^3*c^3*d)*B)*log(b*x + a) + ((I*b^4*c*d^3 - I*a*b^3*d^4)*B*x^3 - 3*(-2*I*b^4*c^2*d^2 + 3*I*a*b^3*c*d^3
- I*a^2*b^2*d^4)*B*x^2 - 2*(-3*I*a*b^3*c^2*d^2 + 5*I*a^2*b^2*c*d^3 - 2*I*a^3*b*d^4)*B*x - 2*(-3*I*a*b^3*c^3*d
+ 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*B - 6*((-I*b^4*c^3*d + 3*I*a*b^3*c^2*d^2 - 3*I*a^2*b^2*c*
d^3 + I*a^3*b*d^4)*B*x + (-I*a*b^3*c^3*d + 3*I*a^2*b^2*c^2*d^2 - 3*I*a^3*b*c*d^3 + I*a^4*d^4)*B)*log(b*x + a))
*log((b*x + a)^n) + ((-I*b^4*c*d^3 + I*a*b^3*d^4)*B*x^3 - 3*(2*I*b^4*c^2*d^2 - 3*I*a*b^3*c*d^3 + I*a^2*b^2*d^4
)*B*x^2 - 2*(3*I*a*b^3*c^2*d^2 - 5*I*a^2*b^2*c*d^3 + 2*I*a^3*b*d^4)*B*x - 2*(3*I*a*b^3*c^3*d - 6*I*a^2*b^2*c^2
*d^2 + 4*I*a^3*b*c*d^3 - I*a^4*d^4)*B - 6*((I*b^4*c^3*d - 3*I*a*b^3*c^2*d^2 + 3*I*a^2*b^2*c*d^3 - I*a^3*b*d^4)
*B*x + (I*a*b^3*c^3*d - 3*I*a^2*b^2*c^2*d^2 + 3*I*a^3*b*c*d^3 - I*a^4*d^4)*B)*log(b*x + a))*log((d*x + c)^n))/
(a*b^5*c*g^2 - a^2*b^4*d*g^2 + (b^6*c*g^2 - a*b^5*d*g^2)*x) + 3*(-I*b^2*c^2*d*n + 2*I*a*b*c*d^2*n - I*a^2*d^3*
n)*(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/(b^4*g^2)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^n))/(b*g*x+a*g)^2,x, algorithm="fricas")

[Out]

integral(((-I*A - I*B)*d^3*x^3 - 3*(I*A + I*B)*c*d^2*x^2 - 3*(I*A + I*B)*c^2*d*x + (-I*A - I*B)*c^3 + (-I*B*d^
3*n*x^3 - 3*I*B*c*d^2*n*x^2 - 3*I*B*c^2*d*n*x - I*B*c^3*n)*log((b*x + a)/(d*x + c)))/(b^2*g^2*x^2 + 2*a*b*g^2*
x + a^2*g^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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))**n))/(b*g*x+a*g)**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^n))/(b*g*x+a*g)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,i+d\,i\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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