\(\int \frac {(a g+b g x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{(c i+d i x)^2} \, dx\) [41]

Optimal result

Integrand size = 38, antiderivative size = 160 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=-\frac {A g (a+b x)}{d i^2 (c+d x)}+\frac {B g (a+b x)}{d i^2 (c+d x)}-\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d i^2 (c+d x)}-\frac {b g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i^2}-\frac {b B g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2562, 45, 2393, 2332, 2354, 2438} \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=-\frac {b g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i^2}-\frac {A g (a+b x)}{d i^2 (c+d x)}-\frac {b B g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2}-\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d i^2 (c+d x)}+\frac {B g (a+b x)}{d i^2 (c+d x)} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\frac {g \left (\frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+2 b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B \left (\frac {b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )-b B \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^2 i^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.84

method	result	size
parts	\(\frac {g A \left (\frac {b \ln \left (d x +c \right )}{d^{2}}-\frac {a d -c b}{d^{2} \left (d x +c \right )}\right )}{i^{2}}-\frac {g B \left (\frac {\left (a d -c b \right ) \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{d}+\frac {b e \left (a d -c b \right ) \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{d}\right )}{i^{2} \left (a d -c b \right ) e}\)	\(295\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {g \,d^{2} A \left (-\frac {\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}{d}-\frac {b e \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{2}}\right )}{\left (a d -c b \right ) e^{2} i^{2}}-\frac {g \,d^{2} B \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}}{d}-\frac {b e \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{d}\right )}{\left (a d -c b \right ) e^{2} i^{2}}\right )}{d^{2}}\)	\(358\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {g \,d^{2} A \left (-\frac {\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}{d}-\frac {b e \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{2}}\right )}{\left (a d -c b \right ) e^{2} i^{2}}-\frac {g \,d^{2} B \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}}{d}-\frac {b e \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{d}\right )}{\left (a d -c b \right ) e^{2} i^{2}}\right )}{d^{2}}\)	\(358\)
risch	\(\frac {g A b \ln \left (d x +c \right )}{i^{2} d^{2}}-\frac {g A a}{i^{2} d \left (d x +c \right )}+\frac {g A c b}{i^{2} d^{2} \left (d x +c \right )}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b a}{i^{2} \left (a d -c b \right ) 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 ) b^{2} c}{i^{2} \left (a d -c b \right ) 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 ) a^{2}}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}+\frac {2 g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2} b^{2}}{i^{2} \left (a d -c b \right ) d^{2} \left (d x +c \right )}+\frac {g B \,a^{2}}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}-\frac {2 g B a c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}+\frac {g B \,c^{2} b^{2}}{i^{2} \left (a d -c b \right ) d^{2} \left (d x +c \right )}+\frac {g B a b}{i^{2} \left (a d -c b \right ) d}-\frac {g B \,b^{2} c}{i^{2} \left (a d -c b \right ) d^{2}}-\frac {g B 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^{2} \left (a d -c b \right ) d}+\frac {g B \,b^{2} \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}{i^{2} \left (a d -c b \right ) d^{2}}-\frac {g B 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^{2} \left (a d -c b \right ) d}+\frac {g B \,b^{2} \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}{i^{2} \left (a d -c b \right ) d^{2}}\)	\(780\)

[In]

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

[Out]

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

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

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

-1/2*B*b*g*(((d*x + c)*log(d*x + c)^2 + 2*c*log(d*x + c))/(d^3*i^2*x + c*d^2*i^2) - 2*integrate((d*x*log(b*x +
a) + d*x*log(e) + c)/(d^3*i^2*x^2 + 2*c*d^2*i^2*x + c^2*d*i^2), x)) + A*b*g*(c/(d^3*i^2*x + c*d^2*i^2) + log(
d*x + c)/(d^2*i^2)) - B*a*g*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^2*i^2*x + c*d*i^2) - 1/(d^2*i^2*x + c*d*i
^2) - b*log(b*x + a)/((b*c*d - a*d^2)*i^2) + b*log(d*x + c)/((b*c*d - a*d^2)*i^2)) - A*a*g/(d^2*i^2*x + c*d*i^
2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (159) = 318\).

Time = 48.78 (sec) , antiderivative size = 895, normalized size of antiderivative = 5.59 \[ \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)^2} \, dx=-\frac {1}{2} \, {\left (\frac {{\left (B b^{4} c^{3} e^{3} g - 3 \, B a b^{3} c^{2} d e^{3} g + 3 \, B a^{2} b^{2} c d^{2} e^{3} g - B a^{3} b d^{3} e^{3} g - \frac {2 \, {\left (b e x + a e\right )} B b^{3} c^{3} d e^{2} g}{d x + c} + \frac {6 \, {\left (b e x + a e\right )} B a b^{2} c^{2} d^{2} e^{2} g}{d x + c} - \frac {6 \, {\left (b e x + a e\right )} B a^{2} b c d^{3} e^{2} g}{d x + c} + \frac {2 \, {\left (b e x + a e\right )} B a^{3} d^{4} e^{2} g}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d^{2} e^{2} i^{2} - \frac {2 \, {\left (b e x + a e\right )} b d^{3} e i^{2}}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} d^{4} i^{2}}{{\left (d x + c\right )}^{2}}} + \frac {A b^{4} c^{3} e^{3} g + B b^{4} c^{3} e^{3} g - 3 \, A a b^{3} c^{2} d e^{3} g - 3 \, B a b^{3} c^{2} d e^{3} g + 3 \, A a^{2} b^{2} c d^{2} e^{3} g + 3 \, B a^{2} b^{2} c d^{2} e^{3} g - A a^{3} b d^{3} e^{3} g - B a^{3} b d^{3} e^{3} g - \frac {2 \, {\left (b e x + a e\right )} A b^{3} c^{3} d e^{2} g}{d x + c} - \frac {{\left (b e x + a e\right )} B b^{3} c^{3} d e^{2} g}{d x + c} + \frac {6 \, {\left (b e x + a e\right )} A a b^{2} c^{2} d^{2} e^{2} g}{d x + c} + \frac {3 \, {\left (b e x + a e\right )} B a b^{2} c^{2} d^{2} e^{2} g}{d x + c} - \frac {6 \, {\left (b e x + a e\right )} A a^{2} b c d^{3} e^{2} g}{d x + c} - \frac {3 \, {\left (b e x + a e\right )} B a^{2} b c d^{3} e^{2} g}{d x + c} + \frac {2 \, {\left (b e x + a e\right )} A a^{3} d^{4} e^{2} g}{d x + c} + \frac {{\left (b e x + a e\right )} B a^{3} d^{4} e^{2} g}{d x + c}}{b^{2} d^{2} e^{2} i^{2} - \frac {2 \, {\left (b e x + a e\right )} b d^{3} e i^{2}}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} d^{4} i^{2}}{{\left (d x + c\right )}^{2}}} + \frac {{\left (B b^{3} c^{3} e g - 3 \, B a b^{2} c^{2} d e g + 3 \, B a^{2} b c d^{2} e g - B a^{3} d^{3} e g\right )} \log \left (-b e + \frac {{\left (b e x + a e\right )} d}{d x + c}\right )}{b d^{2} i^{2}} - \frac {{\left (B b^{3} c^{3} e g - 3 \, B a b^{2} c^{2} d e g + 3 \, B a^{2} b c d^{2} e g - B a^{3} d^{3} e g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b d^{2} i^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \]

[In]

integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

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

[In]

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

[Out]

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