3.1.26 \(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^3} \, dx\) [26]

Optimal. Leaf size=345 \[ -\frac {2 B d (b c-a d) i^3 (c+d x)}{b^3 g^3 (a+b x)}-\frac {B (b c-a d) i^3 (c+d x)^2}{4 b^2 g^3 (a+b x)^2}+\frac {d^3 i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac {2 d (b c-a d) i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {(b c-a d) i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 (b c-a d) i^3 \log (c+d x)}{b^4 g^3}-\frac {3 d^2 (b c-a d) i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3}+\frac {3 B d^2 (b c-a d) i^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3} \]

[Out]

-2*B*d*(-a*d+b*c)*i^3*(d*x+c)/b^3/g^3/(b*x+a)-1/4*B*(-a*d+b*c)*i^3*(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)))/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)))/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)))/b^2/g^3/(b*x+a)^2-B*d^2*(-a*d+b*c)*i^3*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)))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g^3+3*B*d^2*(-a*d+b*c)*i^3*po
lylog(2,b*(d*x+c)/d/(b*x+a))/b^4/g^3

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Rubi [A]
time = 0.26, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2562, 46, 2393, 2341, 2351, 31, 2379, 2438} \begin {gather*} \frac {3 B d^2 i^3 (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 (\frac {e (a+b x)}{c+d x}\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 (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac {2 d i^3 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\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 (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 i^3 (b c-a d) \log (c+d x)}{b^4 g^3}-\frac {2 B d i^3 (c+d x) (b c-a d)}{b^3 g^3 (a+b x)}-\frac {B i^3 (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)]))/(a*g + b*g*x)^3,x]

[Out]

(-2*B*d*(b*c - a*d)*i^3*(c + d*x))/(b^3*g^3*(a + b*x)) - (B*(b*c - a*d)*i^3*(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)]))/(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)]))/(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)]))/(2*b^2*g^3*(a + b*x)^2) - (B*d^2*(b*c - a*d)*i^3*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)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (3*B*d^2*(b*c - a*d)*i^
3*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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGtQ[n, 0] && 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 {(26 c+26 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {17576 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {17576 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^3}+\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}+\frac {52728 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (17576 d^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 g^3}+\frac {\left (52728 d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^3}+\frac {\left (52728 d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^3}+\frac {\left (17576 (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {\left (17576 B d^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 g^3}+\frac {\left (52728 B d (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac {\left (8788 B (b c-a d)^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac {\left (17576 B d^3 (b c-a d)\right ) \int \frac {1}{c+d x} \, dx}{b^4 g^3}+\frac {\left (52728 B d (b c-a d)^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac {\left (8788 B (b c-a d)^4\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 e g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac {17576 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {\left (52728 B d (b c-a d)^3\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 (8788 B (b c-a d)^4\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}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 e g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^3}+\frac {\left (52728 B d^3 (b c-a d)\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}-\frac {26364 B d^2 (b c-a d) \log ^2(a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\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 {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}-\frac {26364 B d^2 (b c-a d) \log ^2(a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \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 = 0.26, size = 314, normalized size = 0.91 \begin {gather*} \frac {i^3 \left (4 A b d^3 x-\frac {B (b c-a d)^3}{(a+b x)^2}-\frac {10 B d (b c-a d)^2}{a+b x}+10 B d^2 (-b c+a d) \log (a+b x)+4 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-\frac {2 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}-\frac {12 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+12 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 B d^2 (b c-a d) \log (c+d x)+6 B d^2 (-b c+a d) \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)]))/(a*g + b*g*x)^3,x]

[Out]

(i^3*(4*A*b*d^3*x - (B*(b*c - a*d)^3)/(a + b*x)^2 - (10*B*d*(b*c - a*d)^2)/(a + b*x) + 10*B*d^2*(-(b*c) + a*d)
*Log[a + b*x] + 4*B*d^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - (2*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(a + b*x)^2 - (12*d*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + 12*d^2*(b*c - a*
d)*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*B*d^2*(b*c - a*d)*Log[c + d*x] + 6*B*d^2*(-(b*c) + a*
d)*(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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(845\) vs. \(2(341)=682\).
time = 1.38, size = 846, normalized size = 2.45

method	result	size
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{3} d^{2} e A}{2 g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 i^{3} d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,g^{3} b^{4}}-\frac {2 i^{3} d^{3} A}{g^{3} b^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {i^{3} d^{4} A}{g^{3} b^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {3 i^{3} d^{4} A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e \,g^{3} b^{4}}-\frac {2 i^{3} d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} b^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 i^{3} d^{3} B}{g^{3} b^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 i^{3} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,g^{3} b^{4}}-\frac {3 i^{3} d^{4} B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \,g^{3} b^{4}}-\frac {3 i^{3} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \,g^{3} b^{4}}+\frac {i^{3} d^{4} B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e \,g^{3} b^{4}}+\frac {i^{3} d^{5} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,g^{3} b^{4} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {i^{3} d^{2} e B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i^{3} d^{2} e B}{4 g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{d^{2}}\)	\(846\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{3} d^{2} e A}{2 g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 i^{3} d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,g^{3} b^{4}}-\frac {2 i^{3} d^{3} A}{g^{3} b^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {i^{3} d^{4} A}{g^{3} b^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {3 i^{3} d^{4} A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e \,g^{3} b^{4}}-\frac {2 i^{3} d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} b^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 i^{3} d^{3} B}{g^{3} b^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 i^{3} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,g^{3} b^{4}}-\frac {3 i^{3} d^{4} B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \,g^{3} b^{4}}-\frac {3 i^{3} d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e \,g^{3} b^{4}}+\frac {i^{3} d^{4} B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e \,g^{3} b^{4}}+\frac {i^{3} d^{5} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,g^{3} b^{4} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {i^{3} d^{2} e B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i^{3} d^{2} e B}{4 g^{3} b^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{d^{2}}\)	\(846\)
risch	\(\text {Expression too large to display}\)	\(4194\)

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)))/(b*g*x+a*g)^3,x,method=_RETURNVERBOSE)

[Out]

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

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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. 1901 vs. \(2 (322) = 644\).
time = 0.47, size = 1901, normalized size = 5.51 \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)))/(b*g*x+a*g)^3,x, algorithm="maxima")

[Out]

3/4*I*B*c^2*d*(2*(2*b*x + a)*log(b*x*e/(d*x + c) + a*e/(d*x + c))/(b^4*g^3*x^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3)
+ (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)) + 1/2*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)) - 1/4*I*B*c^3*((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*log(
b*x*e/(d*x + c) + a*e/(d*x + c))/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*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)) + 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*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 + 8*I*a*b^2*c^2*d^3 - 13*I*a^2*b*c*d^4 + 5*I*a^3*d^5)*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*(-24*I*a*b^4*c^3*d^2 + 55*I*a^2*b^3*c^2*d^3 - 40*I*a^
3*b^2*c*d^4 + 9*I*a^4*b*d^5)*B*x + 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)^2 - (39*I*a^2*b^3*c^3*d^2 - 93*I*a^3*b^
2*c^2*d^3 + 73*I*a^4*b*c*d^4 - 19*I*a^5*d^5)*B + 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
+ (-6*I*b^5*c^3*d^2 + 9*I*a^2*b^3*c*d^4 - 5*I*a^3*b^2*d^5)*B*x^2 + 2*(-12*I*a*b^4*c^3*d^2 + 18*I*a^2*b^3*c^2*
d^3 - 9*I*a^3*b^2*c*d^4 + I*a^4*b*d^5)*B*x + (-15*I*a^2*b^3*c^3*d^2 + 27*I*a^3*b^2*c^2*d^3 - 18*I*a^4*b*c*d^4
+ 4*I*a^5*d^5)*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(d*x + c))/(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 - I*a*d^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/
(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)))/(b*g*x+a*g)^3,x, algorithm="fricas")

[Out]

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

[Out]

integrate((I*d*x + I*c)^3*(B*log((b*x + a)*e/(d*x + c)) + 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 (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\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))))/(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))))/(a*g + b*g*x)^3, x)

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