\(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{c i+d i x} \, dx\) [31]

Optimal result

Integrand size = 40, antiderivative size = 252 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^2 i}+\frac {(b c-a d)^2 g^3 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3 i}+\frac {(b c-a d)^3 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4 i}+\frac {B (b c-a d)^3 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]

[Out]

1/3*g^3*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i-1/6*(-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+1/6*(-a*d+b*c)^2*g^3*(b*x+a)*(6*A+5*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^3/i+1/6*(-a*d+b*c)^3*g^3*ln((-a*d+
b*c)/b/(d*x+c))*(6*A+11*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^4/i+B*(-a*d+b*c)^3*g^3*polylog(2,d*(b*x+a)/b/(d*x+c))/d
^4/i

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 2384, 2354, 2438} \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^3 (b c-a d)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{6 d^4 i}+\frac {g^3 (a+b x) (b c-a d)^2 \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{6 d^3 i}-\frac {g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{6 d^2 i}+\frac {g^3 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d i}+\frac {B g^3 (b c-a d)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]

[In]

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

[Out]

(g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d*i) - ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B + 3*B*L
og[(e*(a + b*x))/(c + d*x)]))/(6*d^2*i) + ((b*c - a*d)^2*g^3*(a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c +
d*x)]))/(6*d^3*i) + ((b*c - a*d)^3*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c
+ d*x)]))/(6*d^4*i) + (B*(b*c - a*d)^3*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i)

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 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*} \text {integral}& = \frac {\left ((b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {x^3 (A+B \log (e x))}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{i} \\ & = \frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\frac {\left ((b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {x^2 (3 A+B+3 B \log (e x))}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 d i} \\ & = \frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\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 )}{6 d^2 i}+\frac {\left ((b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {x (3 B+2 (3 A+B)+6 B \log (e x))}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 d^2 i} \\ & = \frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\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 )}{6 d^2 i}+\frac {(b c-a d)^2 g^3 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3 i}-\frac {\left ((b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {9 B+2 (3 A+B)+6 B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 d^3 i} \\ & = \frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\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 )}{6 d^2 i}+\frac {(b c-a d)^2 g^3 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3 i}+\frac {(b c-a d)^3 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4 i}-\frac {\left (B (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^4 i} \\ & = \frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\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 )}{6 d^2 i}+\frac {(b c-a d)^2 g^3 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3 i}+\frac {(b c-a d)^3 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4 i}+\frac {B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.40 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^3 \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (i (c+d x))+3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (i (c+d x))\right ) \log (i (c+d x))+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{6 d^4 i} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1104\) vs. \(2(244)=488\).

Time = 1.65 (sec) , antiderivative size = 1105, normalized size of antiderivative = 4.38

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1105\)
default	\(\text {Expression too large to display}\)	\(1105\)
parts	\(\text {Expression too large to display}\)	\(1116\)
risch	\(\text {Expression too large to display}\)	\(3731\)

[In]

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

[Out]

-1/d^2*e*(a*d-b*c)*(A*d^2*g^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/e/i*(-3/2*b^2/d^4*e^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x
+c))*d)^2+1/d^4*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)+1/3*b^3*e^3/d^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^
3+3*b*e/d^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d))+B*d^2*g^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/e/i*(3*b^2/d^3*e^2*(-
1/2/b^2/e^2/d*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)+1/2/b/e/d/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)-1/2*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)/b^2/e^2/(b*e
-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2)+1/d^3*(dilog(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d+ln(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))*d-b*e)/b/e)/d)+b^3*e^3/d^3*(1/3/b^3/e^3/d*ln(b*e-(b*e/d+(
a*d-b*c)*e/d/(d*x+c))*d)-1/3/b^2/e^2/d/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)-1/6/b/e/d/(b*e-(b*e/d+(a*d-b*c)*e
/d/(d*x+c))*d)^2+1/3*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))*(3*e^2*b^2-3*(b*e/d+(a*d-b*
c)*e/d/(d*x+c))*d*b*e+d^2*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)/b^3/e^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3)+3*
b*e/d^3*(1/b/e/d*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)+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-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d))))

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

integrate((b*g*x+a*g)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (243) = 486\).

Time = 0.26 (sec) , antiderivative size = 790, normalized size of antiderivative = 3.13 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=3 \, A a^{2} b g^{3} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} - \frac {1}{6} \, A b^{3} g^{3} {\left (\frac {6 \, c^{3} \log \left (d x + c\right )}{d^{4} i} - \frac {2 \, d^{2} x^{3} - 3 \, c d x^{2} + 6 \, c^{2} x}{d^{3} i}\right )} + \frac {3}{2} \, A a b^{2} g^{3} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A a^{3} g^{3} \log \left (d i x + c i\right )}{d i} - \frac {{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} 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} i} + \frac {{\left (6 \, a^{3} d^{3} g^{3} \log \left (e\right ) - {\left (6 \, g^{3} \log \left (e\right ) + 11 \, g^{3}\right )} b^{3} c^{3} + 9 \, {\left (2 \, g^{3} \log \left (e\right ) + 3 \, g^{3}\right )} a b^{2} c^{2} d - 18 \, {\left (g^{3} \log \left (e\right ) + g^{3}\right )} a^{2} b c d^{2}\right )} B \log \left (d x + c\right )}{6 \, d^{4} i} + \frac {2 \, B b^{3} d^{3} g^{3} x^{3} \log \left (e\right ) - {\left ({\left (3 \, g^{3} \log \left (e\right ) + g^{3}\right )} b^{3} c d^{2} - {\left (9 \, g^{3} \log \left (e\right ) + g^{3}\right )} a b^{2} d^{3}\right )} B x^{2} + 3 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} B \log \left (d x + c\right )^{2} + {\left ({\left (6 \, g^{3} \log \left (e\right ) + 5 \, g^{3}\right )} b^{3} c^{2} d - 6 \, {\left (3 \, g^{3} \log \left (e\right ) + 2 \, g^{3}\right )} a b^{2} c d^{2} + {\left (18 \, g^{3} \log \left (e\right ) + 7 \, g^{3}\right )} a^{2} b d^{3}\right )} B x + {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x + {\left (6 \, a b^{2} c^{2} d g^{3} - 15 \, a^{2} b c d^{2} g^{3} + 11 \, a^{3} d^{3} g^{3}\right )} B\right )} \log \left (b x + a\right ) - {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x\right )} \log \left (d x + c\right )}{6 \, d^{4} i} \]

[In]

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

[Out]

3*A*a^2*b*g^3*(x/(d*i) - c*log(d*x + c)/(d^2*i)) - 1/6*A*b^3*g^3*(6*c^3*log(d*x + c)/(d^4*i) - (2*d^2*x^3 - 3*
c*d*x^2 + 6*c^2*x)/(d^3*i)) + 3/2*A*a*b^2*g^3*(2*c^2*log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A*a^3*g
^3*log(d*i*x + c*i)/(d*i) - (b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*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*i) + 1/6*(6*a^3*d^3*g^3*log(e)
- (6*g^3*log(e) + 11*g^3)*b^3*c^3 + 9*(2*g^3*log(e) + 3*g^3)*a*b^2*c^2*d - 18*(g^3*log(e) + g^3)*a^2*b*c*d^2)*
B*log(d*x + c)/(d^4*i) + 1/6*(2*B*b^3*d^3*g^3*x^3*log(e) - ((3*g^3*log(e) + g^3)*b^3*c*d^2 - (9*g^3*log(e) + g
^3)*a*b^2*d^3)*B*x^2 + 3*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*B*log(d*x + c)^2
+ ((6*g^3*log(e) + 5*g^3)*b^3*c^2*d - 6*(3*g^3*log(e) + 2*g^3)*a*b^2*c*d^2 + (18*g^3*log(e) + 7*g^3)*a^2*b*d^3
)*B*x + (2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^
3 + 3*a^2*b*d^3*g^3)*B*x + (6*a*b^2*c^2*d*g^3 - 15*a^2*b*c*d^2*g^3 + 11*a^3*d^3*g^3)*B)*log(b*x + a) - (2*B*b^
3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3 + 3*a^2*b*d^3
*g^3)*B*x)*log(d*x + c))/(d^4*i)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3552 vs. \(2 (243) = 486\).

Time = 66.53 (sec) , antiderivative size = 3552, normalized size of antiderivative = 14.10 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/120*(6*(B*b^9*c^6*e^6*g^3 - 6*B*a*b^8*c^5*d*e^6*g^3 + 15*B*a^2*b^7*c^4*d^2*e^6*g^3 - 20*B*a^3*b^6*c^3*d^3*e
^6*g^3 + 15*B*a^4*b^5*c^2*d^4*e^6*g^3 - 6*B*a^5*b^4*c*d^5*e^6*g^3 + B*a^6*b^3*d^6*e^6*g^3 - 5*(b*e*x + a*e)*B*
b^8*c^6*d*e^5*g^3/(d*x + c) + 30*(b*e*x + a*e)*B*a*b^7*c^5*d^2*e^5*g^3/(d*x + c) - 75*(b*e*x + a*e)*B*a^2*b^6*
c^4*d^3*e^5*g^3/(d*x + c) + 100*(b*e*x + a*e)*B*a^3*b^5*c^3*d^4*e^5*g^3/(d*x + c) - 75*(b*e*x + a*e)*B*a^4*b^4
*c^2*d^5*e^5*g^3/(d*x + c) + 30*(b*e*x + a*e)*B*a^5*b^3*c*d^6*e^5*g^3/(d*x + c) - 5*(b*e*x + a*e)*B*a^6*b^2*d^
7*e^5*g^3/(d*x + c) + 10*(b*e*x + a*e)^2*B*b^7*c^6*d^2*e^4*g^3/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a*b^6*c^5*d^
3*e^4*g^3/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^2*b^5*c^4*d^4*e^4*g^3/(d*x + c)^2 - 200*(b*e*x + a*e)^2*B*a^3*
b^4*c^3*d^5*e^4*g^3/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^4*b^3*c^2*d^6*e^4*g^3/(d*x + c)^2 - 60*(b*e*x + a*e)
^2*B*a^5*b^2*c*d^7*e^4*g^3/(d*x + c)^2 + 10*(b*e*x + a*e)^2*B*a^6*b*d^8*e^4*g^3/(d*x + c)^2 - 10*(b*e*x + a*e)
^3*B*b^6*c^6*d^3*e^3*g^3/(d*x + c)^3 + 60*(b*e*x + a*e)^3*B*a*b^5*c^5*d^4*e^3*g^3/(d*x + c)^3 - 150*(b*e*x + a
*e)^3*B*a^2*b^4*c^4*d^5*e^3*g^3/(d*x + c)^3 + 200*(b*e*x + a*e)^3*B*a^3*b^3*c^3*d^6*e^3*g^3/(d*x + c)^3 - 150*
(b*e*x + a*e)^3*B*a^4*b^2*c^2*d^7*e^3*g^3/(d*x + c)^3 + 60*(b*e*x + a*e)^3*B*a^5*b*c*d^8*e^3*g^3/(d*x + c)^3 -
10*(b*e*x + a*e)^3*B*a^6*d^9*e^3*g^3/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))/(b^5*d^4*e^5*i - 5*(b*e*x + a*
e)*b^4*d^5*e^4*i/(d*x + c) + 10*(b*e*x + a*e)^2*b^3*d^6*e^3*i/(d*x + c)^2 - 10*(b*e*x + a*e)^3*b^2*d^7*e^2*i/(
d*x + c)^3 + 5*(b*e*x + a*e)^4*b*d^8*e*i/(d*x + c)^4 - (b*e*x + a*e)^5*d^9*i/(d*x + c)^5) + (6*A*b^10*c^6*e^6*
g^3 + 5*B*b^10*c^6*e^6*g^3 - 36*A*a*b^9*c^5*d*e^6*g^3 - 30*B*a*b^9*c^5*d*e^6*g^3 + 90*A*a^2*b^8*c^4*d^2*e^6*g^
3 + 75*B*a^2*b^8*c^4*d^2*e^6*g^3 - 120*A*a^3*b^7*c^3*d^3*e^6*g^3 - 100*B*a^3*b^7*c^3*d^3*e^6*g^3 + 90*A*a^4*b^
6*c^2*d^4*e^6*g^3 + 75*B*a^4*b^6*c^2*d^4*e^6*g^3 - 36*A*a^5*b^5*c*d^5*e^6*g^3 - 30*B*a^5*b^5*c*d^5*e^6*g^3 + 6
*A*a^6*b^4*d^6*e^6*g^3 + 5*B*a^6*b^4*d^6*e^6*g^3 - 30*(b*e*x + a*e)*A*b^9*c^6*d*e^5*g^3/(d*x + c) - 19*(b*e*x
+ a*e)*B*b^9*c^6*d*e^5*g^3/(d*x + c) + 180*(b*e*x + a*e)*A*a*b^8*c^5*d^2*e^5*g^3/(d*x + c) + 114*(b*e*x + a*e)
*B*a*b^8*c^5*d^2*e^5*g^3/(d*x + c) - 450*(b*e*x + a*e)*A*a^2*b^7*c^4*d^3*e^5*g^3/(d*x + c) - 285*(b*e*x + a*e)
*B*a^2*b^7*c^4*d^3*e^5*g^3/(d*x + c) + 600*(b*e*x + a*e)*A*a^3*b^6*c^3*d^4*e^5*g^3/(d*x + c) + 380*(b*e*x + a*
e)*B*a^3*b^6*c^3*d^4*e^5*g^3/(d*x + c) - 450*(b*e*x + a*e)*A*a^4*b^5*c^2*d^5*e^5*g^3/(d*x + c) - 285*(b*e*x +
a*e)*B*a^4*b^5*c^2*d^5*e^5*g^3/(d*x + c) + 180*(b*e*x + a*e)*A*a^5*b^4*c*d^6*e^5*g^3/(d*x + c) + 114*(b*e*x +
a*e)*B*a^5*b^4*c*d^6*e^5*g^3/(d*x + c) - 30*(b*e*x + a*e)*A*a^6*b^3*d^7*e^5*g^3/(d*x + c) - 19*(b*e*x + a*e)*B
*a^6*b^3*d^7*e^5*g^3/(d*x + c) + 60*(b*e*x + a*e)^2*A*b^8*c^6*d^2*e^4*g^3/(d*x + c)^2 + 23*(b*e*x + a*e)^2*B*b
^8*c^6*d^2*e^4*g^3/(d*x + c)^2 - 360*(b*e*x + a*e)^2*A*a*b^7*c^5*d^3*e^4*g^3/(d*x + c)^2 - 138*(b*e*x + a*e)^2
*B*a*b^7*c^5*d^3*e^4*g^3/(d*x + c)^2 + 900*(b*e*x + a*e)^2*A*a^2*b^6*c^4*d^4*e^4*g^3/(d*x + c)^2 + 345*(b*e*x
+ a*e)^2*B*a^2*b^6*c^4*d^4*e^4*g^3/(d*x + c)^2 - 1200*(b*e*x + a*e)^2*A*a^3*b^5*c^3*d^5*e^4*g^3/(d*x + c)^2 -
460*(b*e*x + a*e)^2*B*a^3*b^5*c^3*d^5*e^4*g^3/(d*x + c)^2 + 900*(b*e*x + a*e)^2*A*a^4*b^4*c^2*d^6*e^4*g^3/(d*x
+ c)^2 + 345*(b*e*x + a*e)^2*B*a^4*b^4*c^2*d^6*e^4*g^3/(d*x + c)^2 - 360*(b*e*x + a*e)^2*A*a^5*b^3*c*d^7*e^4*
g^3/(d*x + c)^2 - 138*(b*e*x + a*e)^2*B*a^5*b^3*c*d^7*e^4*g^3/(d*x + c)^2 + 60*(b*e*x + a*e)^2*A*a^6*b^2*d^8*e
^4*g^3/(d*x + c)^2 + 23*(b*e*x + a*e)^2*B*a^6*b^2*d^8*e^4*g^3/(d*x + c)^2 - 60*(b*e*x + a*e)^3*A*b^7*c^6*d^3*e
^3*g^3/(d*x + c)^3 - 3*(b*e*x + a*e)^3*B*b^7*c^6*d^3*e^3*g^3/(d*x + c)^3 + 360*(b*e*x + a*e)^3*A*a*b^6*c^5*d^4
*e^3*g^3/(d*x + c)^3 + 18*(b*e*x + a*e)^3*B*a*b^6*c^5*d^4*e^3*g^3/(d*x + c)^3 - 900*(b*e*x + a*e)^3*A*a^2*b^5*
c^4*d^5*e^3*g^3/(d*x + c)^3 - 45*(b*e*x + a*e)^3*B*a^2*b^5*c^4*d^5*e^3*g^3/(d*x + c)^3 + 1200*(b*e*x + a*e)^3*
A*a^3*b^4*c^3*d^6*e^3*g^3/(d*x + c)^3 + 60*(b*e*x + a*e)^3*B*a^3*b^4*c^3*d^6*e^3*g^3/(d*x + c)^3 - 900*(b*e*x
+ a*e)^3*A*a^4*b^3*c^2*d^7*e^3*g^3/(d*x + c)^3 - 45*(b*e*x + a*e)^3*B*a^4*b^3*c^2*d^7*e^3*g^3/(d*x + c)^3 + 36
0*(b*e*x + a*e)^3*A*a^5*b^2*c*d^8*e^3*g^3/(d*x + c)^3 + 18*(b*e*x + a*e)^3*B*a^5*b^2*c*d^8*e^3*g^3/(d*x + c)^3
- 60*(b*e*x + a*e)^3*A*a^6*b*d^9*e^3*g^3/(d*x + c)^3 - 3*(b*e*x + a*e)^3*B*a^6*b*d^9*e^3*g^3/(d*x + c)^3 - 6*
(b*e*x + a*e)^4*B*b^6*c^6*d^4*e^2*g^3/(d*x + c)^4 + 36*(b*e*x + a*e)^4*B*a*b^5*c^5*d^5*e^2*g^3/(d*x + c)^4 - 9
0*(b*e*x + a*e)^4*B*a^2*b^4*c^4*d^6*e^2*g^3/(d*x + c)^4 + 120*(b*e*x + a*e)^4*B*a^3*b^3*c^3*d^7*e^2*g^3/(d*x +
c)^4 - 90*(b*e*x + a*e)^4*B*a^4*b^2*c^2*d^8*e^2*g^3/(d*x + c)^4 + 36*(b*e*x + a*e)^4*B*a^5*b*c*d^9*e^2*g^3/(d
*x + c)^4 - 6*(b*e*x + a*e)^4*B*a^6*d^10*e^2*g^3/(d*x + c)^4)/(b^6*d^4*e^5*i - 5*(b*e*x + a*e)*b^5*d^5*e^4*i/(
d*x + c) + 10*(b*e*x + a*e)^2*b^4*d^6*e^3*i/(d*x + c)^2 - 10*(b*e*x + a*e)^3*b^3*d^7*e^2*i/(d*x + c)^3 + 5*(b*
e*x + a*e)^4*b^2*d^8*e*i/(d*x + c)^4 - (b*e*x + a*e)^5*b*d^9*i/(d*x + c)^5) + 6*(B*b^6*c^6*e*g^3 - 6*B*a*b^5*c
^5*d*e*g^3 + 15*B*a^2*b^4*c^4*d^2*e*g^3 - 20*B*a^3*b^3*c^3*d^3*e*g^3 + 15*B*a^4*b^2*c^2*d^4*e*g^3 - 6*B*a^5*b*
c*d^5*e*g^3 + B*a^6*d^6*e*g^3)*log(-b*e + (b*e*x + a*e)*d/(d*x + c))/(b^2*d^4*i) - 6*(B*b^6*c^6*e*g^3 - 6*B*a*
b^5*c^5*d*e*g^3 + 15*B*a^2*b^4*c^4*d^2*e*g^3 - 20*B*a^3*b^3*c^3*d^3*e*g^3 + 15*B*a^4*b^2*c^2*d^4*e*g^3 - 6*B*a
^5*b*c*d^5*e*g^3 + B*a^6*d^6*e*g^3)*log((b*e*x + a*e)/(d*x + c))/(b^2*d^4*i))*(b*c/((b*c*e - a*d*e)*(b*c - a*d
)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2

Mupad [F(-1)]

Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \]

[In]

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

[Out]

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