\(\int \frac {(a g+b g x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{c i+d i x} \, dx\) [33]

Optimal result

Integrand size = 38, antiderivative size = 125 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {B (b c-a d) g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2562, 2384, 2354, 2438} \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d^2 i}+\frac {g (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i}+\frac {B g (b c-a d) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[In]

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

[Out]

(g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i) + ((b*c - a*d)*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A +
B + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i) + (B*(b*c - a*d)*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*
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 {((b c-a d) g) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{i} \\ & = \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}-\frac {((b c-a d) g) \text {Subst}\left (\int \frac {A+B+B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d i} \\ & = \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}-\frac {(B (b c-a d) g) \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^2 i} \\ & = \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {B (b c-a d) g \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.30 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g \left (2 A b d x+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B (b c-a d) \log (c+d x)-2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^2 i} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(125)=250\).

Time = 1.34 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.66

method	result	size
parts	\(\frac {g A \left (\frac {b x}{d}+\frac {\left (a d -c b \right ) \ln \left (d x +c \right )}{d^{2}}\right )}{i}-\frac {g B \left (b e \left (a d -c b \right ) \left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right )+\left (a d -c b \right ) \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )\right )}{i d}\)	\(332\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}\right )}{d^{2}}\)	\(380\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}\right )}{d^{2}}\)	\(380\)
risch	\(\frac {g A b x}{i d}+\frac {g A \ln \left (d x +c \right ) a}{i d}-\frac {g A \ln \left (d x +c \right ) c b}{i \,d^{2}}-\frac {g B \ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right ) a}{i d}+\frac {g B b \ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right ) c}{i \,d^{2}}+\frac {g B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{i d \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right )}-\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right )}+\frac {g B e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{i \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}-\frac {2 g B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c}{i d \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}+\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2}}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}-\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{i d}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c b}{i \,d^{2}}-\frac {g 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 {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{i d}+\frac {g 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 {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c b}{i \,d^{2}}\)	\(769\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=A b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A a g \log \left (d i x + c i\right )}{d i} - \frac {{\left (b c g - a d g\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{2} i} + \frac {{\left (a d g \log \left (e\right ) - {\left (g \log \left (e\right ) + g\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} - \frac {2 \, B b d g x \log \left (d x + c\right ) - 2 \, B b d g x \log \left (e\right ) - {\left (b c g - a d g\right )} B \log \left (d x + c\right )^{2} - 2 \, {\left (B b d g x + B a d g\right )} \log \left (b x + a\right )}{2 \, d^{2} i} \]

[In]

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

[Out]

A*b*g*(x/(d*i) - c*log(d*x + c)/(d^2*i)) + A*a*g*log(d*i*x + c*i)/(d*i) - (b*c*g - a*d*g)*(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^2*i) + (a*d*g*log(e) - (g*log(e) + g)*b
*c)*B*log(d*x + c)/(d^2*i) - 1/2*(2*B*b*d*g*x*log(d*x + c) - 2*B*b*d*g*x*log(e) - (b*c*g - a*d*g)*B*log(d*x +
c)^2 - 2*(B*b*d*g*x + B*a*d*g)*log(b*x + a))/(d^2*i)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (124) = 248\).

Time = 42.82 (sec) , antiderivative size = 1230, normalized size of antiderivative = 9.84 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*((B*b^5*c^4*e^4*g - 4*B*a*b^4*c^3*d*e^4*g + 6*B*a^2*b^3*c^2*d^2*e^4*g - 4*B*a^3*b^2*c*d^3*e^4*g + B*a^4*b
*d^4*e^4*g - 3*(b*e*x + a*e)*B*b^4*c^4*d*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^3*c^3*d^2*e^3*g/(d*x + c) -
18*(b*e*x + a*e)*B*a^2*b^2*c^2*d^3*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*B*a^3*b*c*d^4*e^3*g/(d*x + c) - 3*(b*e*x
+ a*e)*B*a^4*d^5*e^3*g/(d*x + c))*log((b*e*x + a*e)/(d*x + c))/(b^3*d^2*e^3*i - 3*(b*e*x + a*e)*b^2*d^3*e^2*i
/(d*x + c) + 3*(b*e*x + a*e)^2*b*d^4*e*i/(d*x + c)^2 - (b*e*x + a*e)^3*d^5*i/(d*x + c)^3) + (A*b^6*c^4*e^4*g -
4*A*a*b^5*c^3*d*e^4*g + 6*A*a^2*b^4*c^2*d^2*e^4*g - 4*A*a^3*b^3*c*d^3*e^4*g + A*a^4*b^2*d^4*e^4*g - 3*(b*e*x
+ a*e)*A*b^5*c^4*d*e^3*g/(d*x + c) + (b*e*x + a*e)*B*b^5*c^4*d*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*A*a*b^4*c^3*
d^2*e^3*g/(d*x + c) - 4*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3*g/(d*x + c) - 18*(b*e*x + a*e)*A*a^2*b^3*c^2*d^3*e^3
*g/(d*x + c) + 6*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*A*a^3*b^2*c*d^4*e^3*g/(d*x
+ c) - 4*(b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g/(d*x + c) - 3*(b*e*x + a*e)*A*a^4*b*d^5*e^3*g/(d*x + c) + (b*e*x
+ a*e)*B*a^4*b*d^5*e^3*g/(d*x + c) - (b*e*x + a*e)^2*B*b^4*c^4*d^2*e^2*g/(d*x + c)^2 + 4*(b*e*x + a*e)^2*B*a*
b^3*c^3*d^3*e^2*g/(d*x + c)^2 - 6*(b*e*x + a*e)^2*B*a^2*b^2*c^2*d^4*e^2*g/(d*x + c)^2 + 4*(b*e*x + a*e)^2*B*a^
3*b*c*d^5*e^2*g/(d*x + c)^2 - (b*e*x + a*e)^2*B*a^4*d^6*e^2*g/(d*x + c)^2)/(b^4*d^2*e^3*i - 3*(b*e*x + a*e)*b^
3*d^3*e^2*i/(d*x + c) + 3*(b*e*x + a*e)^2*b^2*d^4*e*i/(d*x + c)^2 - (b*e*x + a*e)^3*b*d^5*i/(d*x + c)^3) + (B*
b^4*c^4*e*g - 4*B*a*b^3*c^3*d*e*g + 6*B*a^2*b^2*c^2*d^2*e*g - 4*B*a^3*b*c*d^3*e*g + B*a^4*d^4*e*g)*log(-b*e +
(b*e*x + a*e)*d/(d*x + c))/(b^2*d^2*i) - (B*b^4*c^4*e*g - 4*B*a*b^3*c^3*d*e*g + 6*B*a^2*b^2*c^2*d^2*e*g - 4*B*
a^3*b*c*d^3*e*g + B*a^4*d^4*e*g)*log((b*e*x + a*e)/(d*x + c))/(b^2*d^2*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. \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \]

[In]

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

[Out]

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