3.2.33 \(\int \frac {(c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^3} \, dx\) [133]

Optimal. Leaf size=361 \[ -\frac {2 B d (b c-a d) i^3 n (c+d x)}{b^3 g^3 (a+b x)}-\frac {B (b c-a d) i^3 n (c+d x)^2}{4 b^2 g^3 (a+b x)^2}+\frac {d^3 i^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^3}-\frac {2 d (b c-a d) i^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}-\frac {(b c-a d) i^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 (b c-a d) i^3 n \log (c+d x)}{b^4 g^3}-\frac {3 d^2 (b c-a d) i^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3}+\frac {3 B d^2 (b c-a d) i^3 n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 361, 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, 46, 2393, 2341, 2351, 31, 2379, 2438} \begin {gather*} \frac {3 B d^2 i^3 n (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3}+\frac {d^3 i^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^3}-\frac {3 d^2 i^3 (b c-a d) \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^3}-\frac {2 d i^3 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^3 (a+b x)}-\frac {i^3 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 i^3 n (b c-a d) \log (c+d x)}{b^4 g^3}-\frac {2 B d i^3 n (c+d x) (b c-a d)}{b^3 g^3 (a+b x)}-\frac {B i^3 n (c+d x)^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2} \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)^3,x]

[Out]

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

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 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 {(133 c+133 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)^3} \, dx &=\int \left (\frac {2352637 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3}+\frac {2352637 (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^3 (a+b x)^3}+\frac {7057911 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^3 (a+b x)^2}+\frac {7057911 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (2352637 d^3\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 g^3}+\frac {\left (7057911 d^2 (b c-a d)\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^3}+\frac {\left (7057911 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)^2} \, dx}{b^3 g^3}+\frac {\left (2352637 (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)^3} \, dx}{b^3 g^3}\\ &=\frac {2352637 A d^3 x}{b^3 g^3}-\frac {2352637 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 g^3 (a+b x)^2}-\frac {7057911 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^4 g^3 (a+b x)}+\frac {7057911 d^2 (b c-a d) \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^3}+\frac {\left (2352637 B d^3\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 g^3}-\frac {\left (7057911 B d^2 (b c-a d) 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^3}+\frac {\left (7057911 B d (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac {\left (2352637 B (b c-a d)^3 n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^3}\\ &=\frac {2352637 A d^3 x}{b^3 g^3}+\frac {2352637 B d^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^3}-\frac {2352637 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 g^3 (a+b x)^2}-\frac {7057911 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^4 g^3 (a+b x)}+\frac {7057911 d^2 (b c-a d) \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^3}-\frac {\left (7057911 B d^2 (b c-a d) 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^3}-\frac {\left (2352637 B d^3 (b c-a d) n\right ) \int \frac {1}{c+d x} \, dx}{b^4 g^3}+\frac {\left (7057911 B d (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac {\left (2352637 B (b c-a d)^4 n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^3}\\ &=\frac {2352637 A d^3 x}{b^3 g^3}+\frac {2352637 B d^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^3}-\frac {2352637 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 g^3 (a+b x)^2}-\frac {7057911 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^4 g^3 (a+b x)}+\frac {7057911 d^2 (b c-a d) \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^3}-\frac {2352637 B d^2 (b c-a d) n \log (c+d x)}{b^4 g^3}-\frac {\left (7057911 B d^2 (b c-a d) n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^3}+\frac {\left (7057911 B d^3 (b c-a d) n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^3}+\frac {\left (7057911 B d (b c-a d)^3 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^3}+\frac {\left (2352637 B (b c-a d)^4 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^4 g^3}\\ &=\frac {2352637 A d^3 x}{b^3 g^3}-\frac {2352637 B (b c-a d)^3 n}{4 b^4 g^3 (a+b x)^2}-\frac {11763185 B d (b c-a d)^2 n}{2 b^4 g^3 (a+b x)}-\frac {11763185 B d^2 (b c-a d) n \log (a+b x)}{2 b^4 g^3}+\frac {2352637 B d^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^3}-\frac {2352637 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 g^3 (a+b x)^2}-\frac {7057911 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^4 g^3 (a+b x)}+\frac {7057911 d^2 (b c-a d) \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^3}+\frac {7057911 B d^2 (b c-a d) n \log (c+d x)}{2 b^4 g^3}+\frac {7057911 B d^2 (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {\left (7057911 B d^2 (b c-a d) n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^3}-\frac {\left (7057911 B d^2 (b c-a d) n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^3}\\ &=\frac {2352637 A d^3 x}{b^3 g^3}-\frac {2352637 B (b c-a d)^3 n}{4 b^4 g^3 (a+b x)^2}-\frac {11763185 B d (b c-a d)^2 n}{2 b^4 g^3 (a+b x)}-\frac {11763185 B d^2 (b c-a d) n \log (a+b x)}{2 b^4 g^3}-\frac {7057911 B d^2 (b c-a d) n \log ^2(a+b x)}{2 b^4 g^3}+\frac {2352637 B d^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^3}-\frac {2352637 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 g^3 (a+b x)^2}-\frac {7057911 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^4 g^3 (a+b x)}+\frac {7057911 d^2 (b c-a d) \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^3}+\frac {7057911 B d^2 (b c-a d) n \log (c+d x)}{2 b^4 g^3}+\frac {7057911 B d^2 (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {\left (7057911 B d^2 (b c-a d) 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^3}\\ &=\frac {2352637 A d^3 x}{b^3 g^3}-\frac {2352637 B (b c-a d)^3 n}{4 b^4 g^3 (a+b x)^2}-\frac {11763185 B d (b c-a d)^2 n}{2 b^4 g^3 (a+b x)}-\frac {11763185 B d^2 (b c-a d) n \log (a+b x)}{2 b^4 g^3}-\frac {7057911 B d^2 (b c-a d) n \log ^2(a+b x)}{2 b^4 g^3}+\frac {2352637 B d^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g^3}-\frac {2352637 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 g^3 (a+b x)^2}-\frac {7057911 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^4 g^3 (a+b x)}+\frac {7057911 d^2 (b c-a d) \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^3}+\frac {7057911 B d^2 (b c-a d) n \log (c+d x)}{2 b^4 g^3}+\frac {7057911 B d^2 (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}+\frac {7057911 B d^2 (b c-a d) n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^3}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 331, normalized size = 0.92 \begin {gather*} \frac {i^3 \left (4 A b d^3 x-\frac {B (b c-a d)^3 n}{(a+b x)^2}-\frac {10 B d (b c-a d)^2 n}{a+b x}+10 B d^2 (-b c+a d) n \log (a+b x)+4 B d^3 (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 )}{(a+b x)^2}-\frac {12 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+12 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 B d^2 (b c-a d) n \log (c+d x)+6 B d^2 (-b c+a d) 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 )}{4 b^4 g^3} \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)^3,x]

[Out]

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

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (b g x +a g \right )^{3}}\, dx\]

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)^3,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)^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. 2336 vs. \(2 (338) = 676\).
time = 0.72, size = 2336, normalized size = 6.47 \begin {gather*} \text {Too large to display} \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)^3,x, algorithm="maxima")

[Out]

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

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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)^3,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^3*g^3*x^3 + 3*a*b^2*g^
3*x^2 + 3*a^2*b*g^3*x + a^3*g^3), x)

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

[Out]

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

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