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

Optimal. Leaf size=166 \[ \frac{b \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 B g i^2 n (b c-a d)^2}-\frac{A d (a+b x)}{g i^2 (c+d x) (b c-a d)^2}-\frac{B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g i^2 (c+d x) (b c-a d)^2}+\frac{B d n (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.678837, antiderivative size = 450, normalized size of antiderivative = 2.71, number of steps used = 22, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 12, 44} \[ \frac{b B n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac{b B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac{b \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g i^2 (b c-a d)^2}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{g i^2 (c+d x) (b c-a d)}-\frac{b \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g i^2 (b c-a d)^2}-\frac{B n}{g i^2 (c+d x) (b c-a d)}-\frac{b B n \log ^2(a+b x)}{2 g i^2 (b c-a d)^2}-\frac{b B n \log ^2(c+d x)}{2 g i^2 (b c-a d)^2}-\frac{b B n \log (a+b x)}{g i^2 (b c-a d)^2}+\frac{b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g i^2 (b c-a d)^2}+\frac{b B n \log (c+d x)}{g i^2 (b c-a d)^2}+\frac{b B n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g i^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*n)/((b*c - a*d)*g*i^2*(c + d*x))) - (b*B*n*Log[a + b*x])/((b*c - a*d)^2*g*i^2) - (b*B*n*Log[a + b*x]^2)/(
2*(b*c - a*d)^2*g*i^2) + (A + B*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)*g*i^2*(c + d*x)) + (b*Log[a + b*x
]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^2*g*i^2) + (b*B*n*Log[c + d*x])/((b*c - a*d)^2*g*i^2) +
(b*B*n*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))^n])*Log[c + d*x])/((b*c - a*d)^2*g*i^2) - (b*B*n*Log[c + d*x]^2)/(2*(b*c - a*d)^2*g*i^2) + (b*B*n*Log[
a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*g*i^2) + (b*B*n*PolyLog[2, -((d*(a + b*x))/(b*c - a*d)
)])/((b*c - a*d)^2*g*i^2) + (b*B*n*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 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 12

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

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 (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(147 c+147 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d) g (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac{b^2 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{21609 (b c-a d)^2 g}-\frac{(b d) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{21609 (b c-a d)^2 g}-\frac{d \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{21609 (b c-a d) g}\\ &=\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{21609 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{(b B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{21609 (b c-a d)^2 g}+\frac{(b B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{21609 (b c-a d)^2 g}-\frac{(B n) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{21609 (b c-a d) g}\\ &=\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{21609 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{(B n) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{21609 g}-\frac{(b B n) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{21609 (b c-a d)^2 g}+\frac{(b B n) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{21609 (b c-a d)^2 g}\\ &=\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{21609 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{(B n) \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}{21609 g}-\frac{\left (b^2 B n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{21609 (b c-a d)^2 g}+\frac{\left (b^2 B n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{21609 (b c-a d)^2 g}+\frac{(b B d n) \int \frac{\log (a+b x)}{c+d x} \, dx}{21609 (b c-a d)^2 g}-\frac{(b B d n) \int \frac{\log (c+d x)}{c+d x} \, dx}{21609 (b c-a d)^2 g}\\ &=-\frac{B n}{21609 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x)}{21609 (b c-a d)^2 g}+\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{21609 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g}+\frac{b B n \log (c+d x)}{21609 (b c-a d)^2 g}+\frac{b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{21609 (b c-a d)^2 g}+\frac{b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{21609 (b c-a d)^2 g}-\frac{(b B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{21609 (b c-a d)^2 g}-\frac{(b B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{21609 (b c-a d)^2 g}-\frac{\left (b^2 B n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{21609 (b c-a d)^2 g}-\frac{(b B d n) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{21609 (b c-a d)^2 g}\\ &=-\frac{B n}{21609 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x)}{21609 (b c-a d)^2 g}-\frac{b B n \log ^2(a+b x)}{43218 (b c-a d)^2 g}+\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{21609 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g}+\frac{b B n \log (c+d x)}{21609 (b c-a d)^2 g}+\frac{b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{b B n \log ^2(c+d x)}{43218 (b c-a d)^2 g}+\frac{b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{21609 (b c-a d)^2 g}-\frac{(b B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{21609 (b c-a d)^2 g}-\frac{(b B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{21609 (b c-a d)^2 g}\\ &=-\frac{B n}{21609 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x)}{21609 (b c-a d)^2 g}-\frac{b B n \log ^2(a+b x)}{43218 (b c-a d)^2 g}+\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{21609 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{21609 (b c-a d)^2 g}+\frac{b B n \log (c+d x)}{21609 (b c-a d)^2 g}+\frac{b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{21609 (b c-a d)^2 g}-\frac{b B n \log ^2(c+d x)}{43218 (b c-a d)^2 g}+\frac{b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{21609 (b c-a d)^2 g}+\frac{b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{21609 (b c-a d)^2 g}+\frac{b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{21609 (b c-a d)^2 g}\\ \end{align*}

Mathematica [C]  time = 0.313787, size = 304, normalized size = 1.83 \[ \frac{-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B n (c+d x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 B n (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{2 g i^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 0.762, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) \left ( dix+ci \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.29679, size = 572, normalized size = 3.45 \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 (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\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 n}{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 )}} + 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(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(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 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*n/(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) + 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))

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Fricas [A]  time = 0.52912, size = 444, normalized size = 2.67 \begin{align*} \frac{2 \, A b c - 2 \, A a d +{\left (B b d n x + B b c n\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} - 2 \,{\left (B b c - B a d\right )} n + 2 \,{\left (B b c - B a d +{\left (B b d x + B b c\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) - 2 \,{\left (B a d n - A b c +{\left (B b d n - A b d\right )} x\right )} \log \left (\frac{b x + a}{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))^n))/(b*g*x+a*g)/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

1/2*(2*A*b*c - 2*A*a*d + (B*b*d*n*x + B*b*c*n)*log((b*x + a)/(d*x + c))^2 - 2*(B*b*c - B*a*d)*n + 2*(B*b*c - B
*a*d + (B*b*d*x + B*b*c)*log((b*x + a)/(d*x + c)))*log(e) - 2*(B*a*d*n - A*b*c + (B*b*d*n - A*b*d)*x)*log((b*x
+ a)/(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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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