3.2.51 \(\int \frac {(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(c i+d i x)^3} \, dx\) [151]

Optimal. Leaf size=382 \[ -\frac {3 B (b c-a d) g^3 n (a+b x)^2}{4 d^2 i^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^3 n (a+b x)}{d^3 i^3 (c+d x)}+\frac {b (b c-a d) g^3 (3 A+B n) (a+b x)}{d^3 i^3 (c+d x)}+\frac {3 b B (b c-a d) g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3 (c+d x)}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d i^3 (c+d x)^2}+\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d^2 i^3 (c+d x)^2}+\frac {b^2 (b c-a d) g^3 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^3}+\frac {3 b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2561, 2384, 45, 2393, 2332, 2341, 2354, 2438} \begin {gather*} \frac {3 b^2 B g^3 n (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}+\frac {b^2 g^3 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{d^4 i^3}+\frac {b g^3 (a+b x) (3 A+B n) (b c-a d)}{d^3 i^3 (c+d x)}+\frac {g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d^2 i^3 (c+d x)^2}+\frac {g^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i^3 (c+d x)^2}+\frac {3 b B g^3 (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3 (c+d x)}-\frac {3 b B g^3 n (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}-\frac {3 B g^3 n (a+b x)^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2354

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

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 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 {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(151 c+151 d x)^3} \, dx &=\int \left (\frac {b^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^3}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^3 (c+d x)^3}+\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^3 (c+d x)^2}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3442951 d^3}-\frac {\left (3 b^2 (b c-a d) g^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3442951 d^3}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3442951 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3442951 d^3}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6885902 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}+\frac {\left (b^3 B g^3\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{3442951 d^3}+\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3442951 d^4}+\frac {\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3442951 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{6885902 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6885902 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}+\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3442951 d^4}-\frac {\left (b^2 B (b c-a d) g^3 n\right ) \int \frac {1}{c+d x} \, dx}{3442951 d^3}+\frac {\left (3 b B (b c-a d)^3 g^3 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3442951 d^4}-\frac {\left (B (b c-a d)^4 g^3 n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{6885902 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6885902 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^4 (c+d x)}-\frac {b^2 B (b c-a d) g^3 n \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}+\frac {\left (3 b^3 B (b c-a d) g^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3442951 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3442951 d^3}+\frac {\left (3 b B (b c-a d)^3 g^3 n\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3442951 d^4}-\frac {\left (B (b c-a d)^4 g^3 n\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{6885902 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}-\frac {B (b c-a d)^3 g^3 n}{13771804 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n}{6885902 d^4 (c+d x)}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x)}{6885902 d^4}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6885902 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^4 (c+d x)}-\frac {7 b^2 B (b c-a d) g^3 n \log (c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3442951 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3442951 d^3}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}-\frac {B (b c-a d)^3 g^3 n}{13771804 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n}{6885902 d^4 (c+d x)}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x)}{6885902 d^4}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6885902 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^4 (c+d x)}-\frac {7 b^2 B (b c-a d) g^3 n \log (c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{6885902 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3442951 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}-\frac {B (b c-a d)^3 g^3 n}{13771804 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n}{6885902 d^4 (c+d x)}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x)}{6885902 d^4}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6885902 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^4 (c+d x)}-\frac {7 b^2 B (b c-a d) g^3 n \log (c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3442951 d^4}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 334, normalized size = 0.87 \begin {gather*} \frac {g^3 \left (4 A b^3 d x-\frac {B (b c-a d)^3 n}{(c+d x)^2}+\frac {10 b B (b c-a d)^2 n}{c+d x}+10 b^2 B (b c-a d) n \log (a+b x)+4 b^2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 )}{(c+d x)^2}-\frac {12 b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}-14 b^2 B (b c-a d) n \log (c+d x)-12 b^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+6 b^2 B (b c-a d) n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{4 d^4 i^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*(4*A*b^3*d*x - (B*(b*c - a*d)^3*n)/(c + d*x)^2 + (10*b*B*(b*c - a*d)^2*n)/(c + d*x) + 10*b^2*B*(b*c - a*d
)*n*Log[a + b*x] + 4*b^2*B*d*(a + b*x)*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]))/(c + d*x)^2 - (12*b*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - 14*b^
2*B*(b*c - a*d)*n*Log[c + d*x] - 12*b^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 6*b^
2*B*(b*c - a*d)*n*((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)])))/(4*d^4*i^3)

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (d i x +c i \right )^{3}}\, dx\]

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))^n))/(d*i*x+c*i)^3,x)

[Out]

int((b*g*x+a*g)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^3,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. 2731 vs. \(2 (360) = 720\).
time = 0.74, size = 2731, normalized size = 7.15 \begin {gather*} \text {Too large to display} \end {gather*}

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))^n))/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

-(b^2*log(b*x + a)/(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 + 2*I*a^2*d^3) - b^2*log(d*x + c)/(2*I*b^2*c^2*d - 4*I*a*b*c
*d^2 + 2*I*a^2*d^3) - (2*b*d*x + 3*b*c - a*d)/(-4*I*b*c^3*d + 4*I*a*c^2*d^2 - 4*(I*b*c*d^3 - I*a*d^4)*x^2 - 8*
(I*b*c^2*d^2 - I*a*c*d^3)*x))*B*a^3*g^3*n - 3*B*a^2*b*g^3*n*((b^2*c - 2*a*b*d)*log(b*x + a)/(2*I*b^2*c^2*d^2 -
4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (b^2*c - 2*a*b*d)*log(d*x + c)/(2*I*b^2*c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4)
- (b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/(-4*I*b*c^3*d^2 + 4*I*a*c^2*d^3 - 4*(I*b*c*d^4 - I*a*d^5)*x^2 - 8
*(I*b*c^2*d^3 - I*a*c*d^4)*x)) + A*b^3*g^3*((6*c^2*d*x + 5*c^3)/(2*I*d^6*x^2 + 4*I*c*d^5*x + 2*I*c^2*d^4) + I*
x/d^3 - 3*I*c*log(d*x + c)/d^4) - 3*A*a*b^2*g^3*((4*c*d*x + 3*c^2)/(2*I*d^5*x^2 + 4*I*c*d^4*x + 2*I*c^2*d^3) -
I*log(d*x + c)/d^3) + 3*(2*d*x + c)*B*a^2*b*g^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(2*I*d^4*x^2 + 4*I*c*d
^3*x + 2*I*c^2*d^2) + 3*(2*d*x + c)*A*a^2*b*g^3/(2*I*d^4*x^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) + B*a^3*g^3*log((b*x
/(d*x + c) + a/(d*x + c))^n*e)/(2*I*d^3*x^2 + 4*I*c*d^2*x + 2*I*c^2*d) + A*a^3*g^3/(2*I*d^3*x^2 + 4*I*c*d^2*x
+ 2*I*c^2*d) + 1/2*(a*b^4*c^2*d*g^3*(19*I*n + 18*I) - 2*a^2*b^3*c*d^2*g^3*(7*I*n + 9*I) + b^5*c^3*g^3*(-7*I*n
- 6*I) + 6*I*a^3*b^2*d^3*g^3)*B*log(d*x + c)/(b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6) + 1/4*(4*(I*b^5*c^2*d^3*g^3
- 2*I*a*b^4*c*d^4*g^3 + I*a^2*b^3*d^5*g^3)*B*x^3 + 8*(I*b^5*c^3*d^2*g^3 - 2*I*a*b^4*c^2*d^3*g^3 + I*a^2*b^3*c
*d^4*g^3)*B*x^2 + 2*(a^2*b^3*c^2*d^3*g^3*(27*I*n - 28*I) + b^5*c^4*d*g^3*(5*I*n - 4*I) + 20*a*b^4*c^3*d^2*g^3*
(-I*n + I) + 12*a^3*b^2*c*d^4*g^3*(-I*n + I))*B*x + 12*((I*b^5*c^3*d^2*g^3*n - 3*I*a*b^4*c^2*d^3*g^3*n + 3*I*a
^2*b^3*c*d^4*g^3*n - I*a^3*b^2*d^5*g^3*n)*B*x^2 + 2*(I*b^5*c^4*d*g^3*n - 3*I*a*b^4*c^3*d^2*g^3*n + 3*I*a^2*b^3
*c^2*d^3*g^3*n - I*a^3*b^2*c*d^4*g^3*n)*B*x + (I*b^5*c^5*g^3*n - 3*I*a*b^4*c^4*d*g^3*n + 3*I*a^2*b^3*c^3*d^2*g
^3*n - I*a^3*b^2*c^2*d^3*g^3*n)*B)*log(b*x + a)*log(d*x + c) + 6*((-I*b^5*c^3*d^2*g^3*n + 3*I*a*b^4*c^2*d^3*g^
3*n - 3*I*a^2*b^3*c*d^4*g^3*n + I*a^3*b^2*d^5*g^3*n)*B*x^2 + 2*(-I*b^5*c^4*d*g^3*n + 3*I*a*b^4*c^3*d^2*g^3*n -
3*I*a^2*b^3*c^2*d^3*g^3*n + I*a^3*b^2*c*d^4*g^3*n)*B*x + (-I*b^5*c^5*g^3*n + 3*I*a*b^4*c^4*d*g^3*n - 3*I*a^2*
b^3*c^3*d^2*g^3*n + I*a^3*b^2*c^2*d^3*g^3*n)*B)*log(d*x + c)^2 - (a*b^4*c^4*d*g^3*(35*I*n - 38*I) - 3*a^3*b^2*
c^2*d^3*g^3*(-7*I*n + 6*I) + b^5*c^5*g^3*(-9*I*n + 10*I) + a^2*b^3*c^3*d^2*g^3*(-47*I*n + 46*I))*B + 2*((5*I*b
^5*c^3*d^2*g^3*n - 13*I*a*b^4*c^2*d^3*g^3*n + 8*I*a^2*b^3*c*d^4*g^3*n + 2*I*a^3*b^2*d^5*g^3*n)*B*x^2 + 2*(5*I*
b^5*c^4*d*g^3*n - 13*I*a*b^4*c^3*d^2*g^3*n + 8*I*a^2*b^3*c^2*d^3*g^3*n + 2*I*a^3*b^2*c*d^4*g^3*n)*B*x + (5*I*b
^5*c^5*g^3*n - 13*I*a*b^4*c^4*d*g^3*n + 8*I*a^2*b^3*c^3*d^2*g^3*n + 2*I*a^3*b^2*c^2*d^3*g^3*n)*B)*log(b*x + a)
+ 2*(2*(I*b^5*c^2*d^3*g^3 - 2*I*a*b^4*c*d^4*g^3 + I*a^2*b^3*d^5*g^3)*B*x^3 + 4*(I*b^5*c^3*d^2*g^3 - 2*I*a*b^4
*c^2*d^3*g^3 + I*a^2*b^3*c*d^4*g^3)*B*x^2 + 4*(-I*b^5*c^4*d*g^3 + 5*I*a*b^4*c^3*d^2*g^3 - 7*I*a^2*b^3*c^2*d^3*
g^3 + 3*I*a^3*b^2*c*d^4*g^3)*B*x + (-5*I*b^5*c^5*g^3 + 19*I*a*b^4*c^4*d*g^3 - 23*I*a^2*b^3*c^3*d^2*g^3 + 9*I*a
^3*b^2*c^2*d^3*g^3)*B + 6*((-I*b^5*c^3*d^2*g^3 + 3*I*a*b^4*c^2*d^3*g^3 - 3*I*a^2*b^3*c*d^4*g^3 + I*a^3*b^2*d^5
*g^3)*B*x^2 + 2*(-I*b^5*c^4*d*g^3 + 3*I*a*b^4*c^3*d^2*g^3 - 3*I*a^2*b^3*c^2*d^3*g^3 + I*a^3*b^2*c*d^4*g^3)*B*x
+ (-I*b^5*c^5*g^3 + 3*I*a*b^4*c^4*d*g^3 - 3*I*a^2*b^3*c^3*d^2*g^3 + I*a^3*b^2*c^2*d^3*g^3)*B)*log(d*x + c))*l
og((b*x + a)^n) + 2*(2*(-I*b^5*c^2*d^3*g^3 + 2*I*a*b^4*c*d^4*g^3 - I*a^2*b^3*d^5*g^3)*B*x^3 + 4*(-I*b^5*c^3*d^
2*g^3 + 2*I*a*b^4*c^2*d^3*g^3 - I*a^2*b^3*c*d^4*g^3)*B*x^2 + 4*(I*b^5*c^4*d*g^3 - 5*I*a*b^4*c^3*d^2*g^3 + 7*I*
a^2*b^3*c^2*d^3*g^3 - 3*I*a^3*b^2*c*d^4*g^3)*B*x + (5*I*b^5*c^5*g^3 - 19*I*a*b^4*c^4*d*g^3 + 23*I*a^2*b^3*c^3*
d^2*g^3 - 9*I*a^3*b^2*c^2*d^3*g^3)*B + 6*((I*b^5*c^3*d^2*g^3 - 3*I*a*b^4*c^2*d^3*g^3 + 3*I*a^2*b^3*c*d^4*g^3 -
I*a^3*b^2*d^5*g^3)*B*x^2 + 2*(I*b^5*c^4*d*g^3 - 3*I*a*b^4*c^3*d^2*g^3 + 3*I*a^2*b^3*c^2*d^3*g^3 - I*a^3*b^2*c
*d^4*g^3)*B*x + (I*b^5*c^5*g^3 - 3*I*a*b^4*c^4*d*g^3 + 3*I*a^2*b^3*c^3*d^2*g^3 - I*a^3*b^2*c^2*d^3*g^3)*B)*log
(d*x + c))*log((d*x + c)^n))/(b^2*c^4*d^4 - 2*a*b*c^3*d^5 + a^2*c^2*d^6 + (b^2*c^2*d^6 - 2*a*b*c*d^7 + a^2*d^8
)*x^2 + 2*(b^2*c^3*d^5 - 2*a*b*c^2*d^6 + a^2*c*d^7)*x) - 3*(I*b^3*c*g^3*n - I*a*b^2*d*g^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/d^4

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {g^{3} \left (\int \frac {A a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {A b^{3} x^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B a^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 A a b^{2} x^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 A a^{2} b x}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B b^{3} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 B a b^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 B a^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx\right )}{i^{3}} \end {gather*}

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))**n))/(d*i*x+c*i)**3,x)

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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