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

Optimal. Leaf size=156 \[ \frac{b \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 B g i^2 (b c-a d)^2}-\frac{A d (a+b x)}{g i^2 (c+d x) (b c-a d)^2}-\frac{B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{g i^2 (c+d x) (b c-a d)^2}+\frac{B d (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.713646, antiderivative size = 432, normalized size of antiderivative = 2.77, number of steps used = 24, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44} \[ \frac{b B \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac{b B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac{b \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^2 (b c-a d)^2}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{g i^2 (c+d x) (b c-a d)}-\frac{b \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^2 (b c-a d)^2}-\frac{B}{g i^2 (c+d x) (b c-a d)}-\frac{b B \log ^2(a+b x)}{2 g i^2 (b c-a d)^2}-\frac{b B \log ^2(c+d x)}{2 g i^2 (b c-a d)^2}+\frac{b B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}-\frac{b B \log (a+b x)}{g i^2 (b c-a d)^2}+\frac{b B \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac{b B \log (c+d x)}{g i^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(43 c+43 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d) g (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac{b^2 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1849 (b c-a d)^2 g}-\frac{(b d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1849 (b c-a d)^2 g}-\frac{d \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1849 (b c-a d) g}\\ &=\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{(b B) \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}{1849 (b c-a d)^2 g}+\frac{(b B) \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}{1849 (b c-a d)^2 g}-\frac{B \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1849 (b c-a d) g}\\ &=\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{B \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1849 g}-\frac{(b B) \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}{1849 (b c-a d)^2 e g}+\frac{(b B) \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}{1849 (b c-a d)^2 e g}\\ &=\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{B \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}{1849 g}-\frac{(b B) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{1849 (b c-a d)^2 e g}+\frac{(b B) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{1849 (b c-a d)^2 e g}\\ &=-\frac{B}{1849 (b c-a d) g (c+d x)}-\frac{b B \log (a+b x)}{1849 (b c-a d)^2 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac{b B \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{\left (b^2 B\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{1849 (b c-a d)^2 g}+\frac{\left (b^2 B\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{1849 (b c-a d)^2 g}+\frac{(b B d) \int \frac{\log (a+b x)}{c+d x} \, dx}{1849 (b c-a d)^2 g}-\frac{(b B d) \int \frac{\log (c+d x)}{c+d x} \, dx}{1849 (b c-a d)^2 g}\\ &=-\frac{B}{1849 (b c-a d) g (c+d x)}-\frac{b B \log (a+b x)}{1849 (b c-a d)^2 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac{b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac{b B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}+\frac{b B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}-\frac{(b B) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{1849 (b c-a d)^2 g}-\frac{(b B) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{1849 (b c-a d)^2 g}-\frac{\left (b^2 B\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1849 (b c-a d)^2 g}-\frac{(b B d) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1849 (b c-a d)^2 g}\\ &=-\frac{B}{1849 (b c-a d) g (c+d x)}-\frac{b B \log (a+b x)}{1849 (b c-a d)^2 g}-\frac{b B \log ^2(a+b x)}{3698 (b c-a d)^2 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac{b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac{b B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{b B \log ^2(c+d x)}{3698 (b c-a d)^2 g}+\frac{b B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}-\frac{(b B) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1849 (b c-a d)^2 g}-\frac{(b B) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1849 (b c-a d)^2 g}\\ &=-\frac{B}{1849 (b c-a d) g (c+d x)}-\frac{b B \log (a+b x)}{1849 (b c-a d)^2 g}-\frac{b B \log ^2(a+b x)}{3698 (b c-a d)^2 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{1849 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1849 (b c-a d)^2 g}+\frac{b B \log (c+d x)}{1849 (b c-a d)^2 g}+\frac{b B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1849 (b c-a d)^2 g}-\frac{b B \log ^2(c+d x)}{3698 (b c-a d)^2 g}+\frac{b B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}+\frac{b B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}+\frac{b B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{1849 (b c-a d)^2 g}\\ \end{align*}

Mathematica [C]  time = 0.288676, size = 292, normalized size = 1.87 \[ \frac{-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{2 g i^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 759, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.2849, size = 568, normalized size = 3.64 \begin{align*} B{\left (\frac{1}{{\left (b c d - a d^{2}\right )} g i^{2} x +{\left (b c^{2} - a c d\right )} g i^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) + A{\left (\frac{1}{{\left (b c d - a d^{2}\right )} g i^{2} x +{\left (b c^{2} - a c d\right )} g i^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} - \frac{{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} +{\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \,{\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \,{\left (b d x + b c +{\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{2 \,{\left (b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} +{\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.511147, size = 328, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (A - B\right )} b c - 2 \,{\left (A - B\right )} a d +{\left (B b d x + B b c\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left ({\left (A - B\right )} b d x + A b c - B a d\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{2 \,{\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x +{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.11933, size = 386, normalized size = 2.47 \begin{align*} \frac{B b \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g i^{2} - 4 a b c d g i^{2} + 2 b^{2} c^{2} g i^{2}} - \frac{B \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A - B\right ) \left (- \frac{b \log{\left (x + \frac{- \frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac{1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}{\left (d i x + c i\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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