3.1.39 \(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(c i+d i x)^2} \, dx\) [39]

Optimal. Leaf size=341 \[ \frac {3 B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {(6 A+5 B) (b c-a d)^2 g^3 (a+b x)}{2 d^3 i^2 (c+d x)}-\frac {3 B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d i^2 (c+d x)}-\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^2 i^2 (c+d x)}-\frac {b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^4 i^2}-\frac {3 b B (b c-a d)^2 g^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \]

[Out]

3*B*(-a*d+b*c)^2*g^3*(b*x+a)/d^3/i^2/(d*x+c)-1/2*(6*A+5*B)*(-a*d+b*c)^2*g^3*(b*x+a)/d^3/i^2/(d*x+c)-3*B*(-a*d+
b*c)^2*g^3*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/d^3/i^2/(d*x+c)+1/2*g^3*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i^2/(
d*x+c)-1/2*(-a*d+b*c)*g^3*(b*x+a)^2*(3*A+B+3*B*ln(e*(b*x+a)/(d*x+c)))/d^2/i^2/(d*x+c)-1/2*b*(-a*d+b*c)^2*g^3*l
n((-a*d+b*c)/b/(d*x+c))*(6*A+5*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^4/i^2-3*b*B*(-a*d+b*c)^2*g^3*polylog(2,d*(b*x+a)
/b/(d*x+c))/d^4/i^2

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Rubi [A]
time = 0.27, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {2562, 2384, 45, 2393, 2332, 2354, 2438} \begin {gather*} -\frac {3 b B g^3 (b c-a d)^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b g^3 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{2 d^4 i^2}-\frac {g^3 (6 A+5 B) (a+b x) (b c-a d)^2}{2 d^3 i^2 (c+d x)}-\frac {g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d^2 i^2 (c+d x)}+\frac {g^3 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d i^2 (c+d x)}-\frac {3 B g^3 (a+b x) (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}+\frac {3 B g^3 (a+b x) (b c-a d)^2}{d^3 i^2 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(39 c+39 d x)^2} \, dx &=\int \left (-\frac {b^2 (2 b c-3 a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^3}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^2}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^3 (c+d x)^2}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{507 d^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{1521 d^2}-\frac {\left (b^2 (2 b c-3 a d) g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{1521 d^3}+\frac {\left (b (b c-a d)^2 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{507 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1521 d^3}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac {\left (b^3 B g^3\right ) \int \frac {(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{3042 d^2}-\frac {\left (b^2 B (2 b c-3 a d) g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{1521 d^3}-\frac {\left (b B (b c-a d)^2 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{507 d^4}-\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1521 d^4}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac {b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac {\left (b^3 B (b c-a d) g^3\right ) \int \frac {x^2}{(a+b x) (c+d x)} \, dx}{3042 d^2}+\frac {\left (b B (2 b c-3 a d) (b c-a d) g^3\right ) \int \frac {1}{c+d x} \, dx}{1521 d^3}-\frac {\left (B (b c-a d)^4 g^3\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1521 d^4}-\frac {\left (b B (b c-a d)^2 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{507 d^4 e}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac {b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac {\left (b^3 B (b c-a d) g^3\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}{3042 d^2}-\frac {\left (B (b c-a d)^4 g^3\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}{1521 d^4}-\frac {\left (b B (b c-a d)^2 g^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{507 d^4 e}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac {b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac {B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac {a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac {b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac {b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac {b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac {b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac {\left (b^2 B (b c-a d)^2 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{507 d^4}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{507 d^3}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac {b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac {B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac {a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac {b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac {b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac {b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac {b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}-\frac {b B (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{507 d^4}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}+\frac {\left (b B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{507 d^4}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{507 d^3}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac {b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac {B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac {a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac {b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac {b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac {b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac {b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}-\frac {b B (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{507 d^4}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}+\frac {b B (b c-a d)^2 g^3 \log ^2(c+d x)}{1014 d^4}+\frac {\left (b B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{507 d^4}\\ &=-\frac {A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac {b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac {B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac {a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac {b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac {b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac {b^3 g^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac {b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac {b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac {b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}-\frac {b B (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{507 d^4}+\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}+\frac {b B (b c-a d)^2 g^3 \log ^2(c+d x)}{1014 d^4}-\frac {b B (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{507 d^4}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 359, normalized size = 1.05 \begin {gather*} \frac {g^3 \left (-2 A b^2 d (2 b c-3 a d) x-2 b B d (2 b c-3 a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^3 d^2 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {2 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+2 b B (2 b c-3 a d) (b c-a d) \log (c+d x)+6 b (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B (b c-a d)^2 \left (\frac {b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )+b B \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-3 b B (b c-a d)^2 \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 )}{2 d^4 i^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1788\) vs. \(2(333)=666\).
time = 1.52, size = 1789, normalized size = 5.25

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1789\)
default	\(\text {Expression too large to display}\)	\(1789\)
risch	\(\text {Expression too large to display}\)	\(3868\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1089 vs. \(2 (315) = 630\).
time = 0.35, size = 1089, normalized size = 3.19 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, c^{3}}{d^{5} x + c d^{4}} + \frac {6 \, c^{2} \log \left (d x + c\right )}{d^{4}} + \frac {d x^{2} - 4 \, c x}{d^{3}}\right )} A b^{3} g^{3} + 3 \, A a b^{2} {\left (\frac {c^{2}}{d^{4} x + c d^{3}} - \frac {x}{d^{2}} + \frac {2 \, c \log \left (d x + c\right )}{d^{3}}\right )} g^{3} - B a^{3} g^{3} {\left (\frac {b \log \left (b x + a\right )}{b c d - a d^{2}} - \frac {b \log \left (d x + c\right )}{b c d - a d^{2}} - \frac {\log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} x + c d} + \frac {1}{d^{2} x + c d}\right )} - 3 \, A a^{2} b g^{3} {\left (\frac {c}{d^{3} x + c d^{2}} + \frac {\log \left (d x + c\right )}{d^{2}}\right )} + \frac {A a^{3} g^{3}}{d^{2} x + c d} - \frac {{\left (13 \, b^{4} c^{3} g^{3} - 35 \, a b^{3} c^{2} d g^{3} + 30 \, a^{2} b^{2} c d^{2} g^{3} - 6 \, a^{3} b d^{3} g^{3}\right )} B \log \left (d x + c\right )}{2 \, {\left (b c d^{4} - a d^{5}\right )}} - \frac {{\left (b^{4} c d^{3} g^{3} - a b^{3} d^{4} g^{3}\right )} B x^{3} - {\left (4 \, b^{4} c^{2} d^{2} g^{3} - 11 \, a b^{3} c d^{3} g^{3} + 7 \, a^{2} b^{2} d^{4} g^{3}\right )} B x^{2} - {\left (5 \, b^{4} c^{3} d g^{3} - 12 \, a b^{3} c^{2} d^{2} g^{3} + 7 \, a^{2} b^{2} c d^{3} g^{3}\right )} B x - 3 \, {\left ({\left (b^{4} c^{3} d g^{3} - 3 \, a b^{3} c^{2} d^{2} g^{3} + 3 \, a^{2} b^{2} c d^{3} g^{3} - a^{3} b d^{4} g^{3}\right )} B x + {\left (b^{4} c^{4} g^{3} - 3 \, a b^{3} c^{3} d g^{3} + 3 \, a^{2} b^{2} c^{2} d^{2} g^{3} - a^{3} b c d^{3} g^{3}\right )} B\right )} \log \left (d x + c\right )^{2} + {\left ({\left (b^{4} c d^{3} g^{3} - a b^{3} d^{4} g^{3}\right )} B x^{3} - 3 \, {\left (b^{4} c^{2} d^{2} g^{3} - 3 \, a b^{3} c d^{3} g^{3} + 2 \, a^{2} b^{2} d^{4} g^{3}\right )} B x^{2} - {\left (6 \, b^{4} c^{3} d g^{3} - 12 \, a b^{3} c^{2} d^{2} g^{3} + 3 \, a^{2} b^{2} c d^{3} g^{3} + 5 \, a^{3} b d^{4} g^{3}\right )} B x - {\left (6 \, a b^{3} c^{3} d g^{3} - 15 \, a^{2} b^{2} c^{2} d^{2} g^{3} + 11 \, a^{3} b c d^{3} g^{3}\right )} B\right )} \log \left (b x + a\right ) - {\left ({\left (b^{4} c d^{3} g^{3} - a b^{3} d^{4} g^{3}\right )} B x^{3} - 3 \, {\left (b^{4} c^{2} d^{2} g^{3} - 3 \, a b^{3} c d^{3} g^{3} + 2 \, a^{2} b^{2} d^{4} g^{3}\right )} B x^{2} - 2 \, {\left (2 \, b^{4} c^{3} d g^{3} - 5 \, a b^{3} c^{2} d^{2} g^{3} + 3 \, a^{2} b^{2} c d^{3} g^{3}\right )} B x + 2 \, {\left (b^{4} c^{4} g^{3} - 4 \, a b^{3} c^{3} d g^{3} + 6 \, a^{2} b^{2} c^{2} d^{2} g^{3} - 3 \, a^{3} b c d^{3} g^{3}\right )} B\right )} \log \left (d x + c\right )}{2 \, {\left (b c^{2} d^{4} - a c d^{5} + {\left (b c d^{5} - a d^{6}\right )} x\right )}} - \frac {3 \, {\left (b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}\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}{d^{4}} \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)))/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

-1/2*(2*c^3/(d^5*x + c*d^4) + 6*c^2*log(d*x + c)/d^4 + (d*x^2 - 4*c*x)/d^3)*A*b^3*g^3 + 3*A*a*b^2*(c^2/(d^4*x
+ c*d^3) - x/d^2 + 2*c*log(d*x + c)/d^3)*g^3 - B*a^3*g^3*(b*log(b*x + a)/(b*c*d - a*d^2) - b*log(d*x + c)/(b*c
*d - a*d^2) - log(b*x*e/(d*x + c) + a*e/(d*x + c))/(d^2*x + c*d) + 1/(d^2*x + c*d)) - 3*A*a^2*b*g^3*(c/(d^3*x
+ c*d^2) + log(d*x + c)/d^2) + A*a^3*g^3/(d^2*x + c*d) - 1/2*(13*b^4*c^3*g^3 - 35*a*b^3*c^2*d*g^3 + 30*a^2*b^2
*c*d^2*g^3 - 6*a^3*b*d^3*g^3)*B*log(d*x + c)/(b*c*d^4 - a*d^5) - 1/2*((b^4*c*d^3*g^3 - a*b^3*d^4*g^3)*B*x^3 -
(4*b^4*c^2*d^2*g^3 - 11*a*b^3*c*d^3*g^3 + 7*a^2*b^2*d^4*g^3)*B*x^2 - (5*b^4*c^3*d*g^3 - 12*a*b^3*c^2*d^2*g^3 +
7*a^2*b^2*c*d^3*g^3)*B*x - 3*((b^4*c^3*d*g^3 - 3*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3 - a^3*b*d^4*g^3)*B*x
+ (b^4*c^4*g^3 - 3*a*b^3*c^3*d*g^3 + 3*a^2*b^2*c^2*d^2*g^3 - a^3*b*c*d^3*g^3)*B)*log(d*x + c)^2 + ((b^4*c*d^3
*g^3 - a*b^3*d^4*g^3)*B*x^3 - 3*(b^4*c^2*d^2*g^3 - 3*a*b^3*c*d^3*g^3 + 2*a^2*b^2*d^4*g^3)*B*x^2 - (6*b^4*c^3*d
*g^3 - 12*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3 + 5*a^3*b*d^4*g^3)*B*x - (6*a*b^3*c^3*d*g^3 - 15*a^2*b^2*c^2
*d^2*g^3 + 11*a^3*b*c*d^3*g^3)*B)*log(b*x + a) - ((b^4*c*d^3*g^3 - a*b^3*d^4*g^3)*B*x^3 - 3*(b^4*c^2*d^2*g^3 -
3*a*b^3*c*d^3*g^3 + 2*a^2*b^2*d^4*g^3)*B*x^2 - 2*(2*b^4*c^3*d*g^3 - 5*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3
)*B*x + 2*(b^4*c^4*g^3 - 4*a*b^3*c^3*d*g^3 + 6*a^2*b^2*c^2*d^2*g^3 - 3*a^3*b*c*d^3*g^3)*B)*log(d*x + c))/(b*c^
2*d^4 - a*c*d^5 + (b*c*d^5 - a*d^6)*x) - 3*(b^3*c^2*g^3 - 2*a*b^2*c*d*g^3 + a^2*b*d^2*g^3)*(log(b*x + a)*log((
b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/d^4

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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)))/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3797 vs. \(2 (315) = 630\).
time = 79.57, size = 3797, normalized size = 11.13 \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)))/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

1/24*(6*B*b^9*c^5*g^3*e^5*log(b*e - (b*x*e + a*e)*d/(d*x + c)) - 30*B*a*b^8*c^4*d*g^3*e^5*log(b*e - (b*x*e + a
*e)*d/(d*x + c)) + 60*B*a^2*b^7*c^3*d^2*g^3*e^5*log(b*e - (b*x*e + a*e)*d/(d*x + c)) - 60*B*a^3*b^6*c^2*d^3*g^
3*e^5*log(b*e - (b*x*e + a*e)*d/(d*x + c)) + 30*B*a^4*b^5*c*d^4*g^3*e^5*log(b*e - (b*x*e + a*e)*d/(d*x + c)) -
6*B*a^5*b^4*d^5*g^3*e^5*log(b*e - (b*x*e + a*e)*d/(d*x + c)) - 24*(b*x*e + a*e)*B*b^8*c^5*d*g^3*e^4*log(b*e -
(b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 120*(b*x*e + a*e)*B*a*b^7*c^4*d^2*g^3*e^4*log(b*e - (b*x*e + a*e)*d/(d
*x + c))/(d*x + c) - 240*(b*x*e + a*e)*B*a^2*b^6*c^3*d^3*g^3*e^4*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c
) + 240*(b*x*e + a*e)*B*a^3*b^5*c^2*d^4*g^3*e^4*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 120*(b*x*e +
a*e)*B*a^4*b^4*c*d^5*g^3*e^4*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 24*(b*x*e + a*e)*B*a^5*b^3*d^6*g
^3*e^4*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 36*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g^3*e^3*log(b*e - (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 180*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*g^3*e^3*log(b*e - (b*x*e + a*e)*d/(d
*x + c))/(d*x + c)^2 + 360*(b*x*e + a*e)^2*B*a^2*b^5*c^3*d^4*g^3*e^3*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^2 - 360*(b*x*e + a*e)^2*B*a^3*b^4*c^2*d^5*g^3*e^3*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 180
*(b*x*e + a*e)^2*B*a^4*b^3*c*d^6*g^3*e^3*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 36*(b*x*e + a*e)^2
*B*a^5*b^2*d^7*g^3*e^3*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 24*(b*x*e + a*e)^3*B*b^6*c^5*d^3*g^3
*e^2*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 120*(b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^3*e^2*log(b*e -
(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 240*(b*x*e + a*e)^3*B*a^2*b^4*c^3*d^5*g^3*e^2*log(b*e - (b*x*e + a*e)
*d/(d*x + c))/(d*x + c)^3 + 240*(b*x*e + a*e)^3*B*a^3*b^3*c^2*d^6*g^3*e^2*log(b*e - (b*x*e + a*e)*d/(d*x + c))
/(d*x + c)^3 - 120*(b*x*e + a*e)^3*B*a^4*b^2*c*d^7*g^3*e^2*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 +
24*(b*x*e + a*e)^3*B*a^5*b*d^8*g^3*e^2*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 6*(b*x*e + a*e)^4*B*
b^5*c^5*d^4*g^3*e*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 30*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^3*e*
log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 60*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*g^3*e*log(b*e - (b*x*e
+ a*e)*d/(d*x + c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^3*e*log(b*e - (b*x*e + a*e)*d/(d*x +
c))/(d*x + c)^4 + 30*(b*x*e + a*e)^4*B*a^4*b*c*d^8*g^3*e*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 6
*(b*x*e + a*e)^4*B*a^5*d^9*g^3*e*log(b*e - (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 6*(b*x*e + a*e)^4*B*b^5*c^
5*d^4*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 30*(b*x*e + a*e)^4*B*a*b^4*c^4*d^5*g^3*e*log((b*x*e + a
*e)/(d*x + c))/(d*x + c)^4 - 60*(b*x*e + a*e)^4*B*a^2*b^3*c^3*d^6*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)
^4 + 60*(b*x*e + a*e)^4*B*a^3*b^2*c^2*d^7*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 30*(b*x*e + a*e)^4*
B*a^4*b*c*d^8*g^3*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 6*(b*x*e + a*e)^4*B*a^5*d^9*g^3*e*log((b*x*e +
a*e)/(d*x + c))/(d*x + c)^4 + 6*A*b^9*c^5*g^3*e^5 + 11*B*b^9*c^5*g^3*e^5 - 30*A*a*b^8*c^4*d*g^3*e^5 - 55*B*a*b
^8*c^4*d*g^3*e^5 + 60*A*a^2*b^7*c^3*d^2*g^3*e^5 + 110*B*a^2*b^7*c^3*d^2*g^3*e^5 - 60*A*a^3*b^6*c^2*d^3*g^3*e^5
- 110*B*a^3*b^6*c^2*d^3*g^3*e^5 + 30*A*a^4*b^5*c*d^4*g^3*e^5 + 55*B*a^4*b^5*c*d^4*g^3*e^5 - 6*A*a^5*b^4*d^5*g
^3*e^5 - 11*B*a^5*b^4*d^5*g^3*e^5 - 24*(b*x*e + a*e)*A*b^8*c^5*d*g^3*e^4/(d*x + c) - 38*(b*x*e + a*e)*B*b^8*c^
5*d*g^3*e^4/(d*x + c) + 120*(b*x*e + a*e)*A*a*b^7*c^4*d^2*g^3*e^4/(d*x + c) + 190*(b*x*e + a*e)*B*a*b^7*c^4*d^
2*g^3*e^4/(d*x + c) - 240*(b*x*e + a*e)*A*a^2*b^6*c^3*d^3*g^3*e^4/(d*x + c) - 380*(b*x*e + a*e)*B*a^2*b^6*c^3*
d^3*g^3*e^4/(d*x + c) + 240*(b*x*e + a*e)*A*a^3*b^5*c^2*d^4*g^3*e^4/(d*x + c) + 380*(b*x*e + a*e)*B*a^3*b^5*c^
2*d^4*g^3*e^4/(d*x + c) - 120*(b*x*e + a*e)*A*a^4*b^4*c*d^5*g^3*e^4/(d*x + c) - 190*(b*x*e + a*e)*B*a^4*b^4*c*
d^5*g^3*e^4/(d*x + c) + 24*(b*x*e + a*e)*A*a^5*b^3*d^6*g^3*e^4/(d*x + c) + 38*(b*x*e + a*e)*B*a^5*b^3*d^6*g^3*
e^4/(d*x + c) + 36*(b*x*e + a*e)^2*A*b^7*c^5*d^2*g^3*e^3/(d*x + c)^2 + 45*(b*x*e + a*e)^2*B*b^7*c^5*d^2*g^3*e^
3/(d*x + c)^2 - 180*(b*x*e + a*e)^2*A*a*b^6*c^4*d^3*g^3*e^3/(d*x + c)^2 - 225*(b*x*e + a*e)^2*B*a*b^6*c^4*d^3*
g^3*e^3/(d*x + c)^2 + 360*(b*x*e + a*e)^2*A*a^2*b^5*c^3*d^4*g^3*e^3/(d*x + c)^2 + 450*(b*x*e + a*e)^2*B*a^2*b^
5*c^3*d^4*g^3*e^3/(d*x + c)^2 - 360*(b*x*e + a*e)^2*A*a^3*b^4*c^2*d^5*g^3*e^3/(d*x + c)^2 - 450*(b*x*e + a*e)^
2*B*a^3*b^4*c^2*d^5*g^3*e^3/(d*x + c)^2 + 180*(b*x*e + a*e)^2*A*a^4*b^3*c*d^6*g^3*e^3/(d*x + c)^2 + 225*(b*x*e
+ a*e)^2*B*a^4*b^3*c*d^6*g^3*e^3/(d*x + c)^2 - 36*(b*x*e + a*e)^2*A*a^5*b^2*d^7*g^3*e^3/(d*x + c)^2 - 45*(b*x
*e + a*e)^2*B*a^5*b^2*d^7*g^3*e^3/(d*x + c)^2 - 24*(b*x*e + a*e)^3*A*b^6*c^5*d^3*g^3*e^2/(d*x + c)^3 - 18*(b*x
*e + a*e)^3*B*b^6*c^5*d^3*g^3*e^2/(d*x + c)^3 + 120*(b*x*e + a*e)^3*A*a*b^5*c^4*d^4*g^3*e^2/(d*x + c)^3 + 90*(
b*x*e + a*e)^3*B*a*b^5*c^4*d^4*g^3*e^2/(d*x + c...

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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 (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (c\,i+d\,i\,x\right )}^2} \,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))))/(c*i + d*i*x)^2,x)

[Out]

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

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