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

Optimal. Leaf size=50 \[ \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 B g i n (b c-a d)} \]

[Out]

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

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Rubi [C]  time = 0.56, antiderivative size = 316, normalized size of antiderivative = 6.32, number of steps used = 18, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g i (b c-a d)}+\frac {B n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g i (b c-a d)}+\frac {\log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i (b c-a d)}-\frac {\log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i (b c-a d)}-\frac {B n \log ^2(a+b x)}{2 g i (b c-a d)}-\frac {B n \log ^2(c+d x)}{2 g i (b c-a d)}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g i (b c-a d)}+\frac {B n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g i (b c-a d)} \]

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)),x]

[Out]

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

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 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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(139 c+139 d x) (a g+b g x)} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac {b \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{139 (b c-a d) g}-\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {(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}{139 (b c-a d) g}+\frac {(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}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {(B n) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{139 (b c-a d) g}+\frac {(B n) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {(b B n) \int \frac {\log (a+b x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac {(b B n) \int \frac {\log (c+d x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac {(B d n) \int \frac {\log (a+b x)}{c+d x} \, dx}{139 (b c-a d) g}-\frac {(B d n) \int \frac {\log (c+d x)}{c+d x} \, dx}{139 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}-\frac {(B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{139 (b c-a d) g}-\frac {(B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{139 (b c-a d) g}-\frac {(b B n) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{139 (b c-a d) g}-\frac {(B d n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{139 (b c-a d) g}\\ &=-\frac {B n \log ^2(a+b x)}{278 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {B n \log ^2(c+d x)}{278 (b c-a d) g}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}-\frac {(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 )}{139 (b c-a d) g}-\frac {(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 )}{139 (b c-a d) g}\\ &=-\frac {B n \log ^2(a+b x)}{278 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac {B n \log ^2(c+d x)}{278 (b c-a d) g}+\frac {B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}+\frac {B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{139 (b c-a d) g}+\frac {B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 219, normalized size = 4.38 \[ \frac {2 A \log (a+b x)+2 B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B \log (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+2 B n \log (c+d x) \log \left (\frac {d (a+b x)}{a d-b c}\right )-B n \log ^2(a+b x)-2 A \log (c+d x)-B n \log ^2(c+d x)}{2 g i (b c-a d)} \]

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)),x]

[Out]

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

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fricas [A]  time = 0.93, size = 74, normalized size = 1.48 \[ \frac {B n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, B \log \relax (e) \log \left (\frac {b x + a}{d x + c}\right ) + 2 \, A \log \left (\frac {b x + a}{d x + c}\right )}{2 \, {\left (b c - a d\right )} g i} \]

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),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.22, size = 90, normalized size = 1.80 \[ -\frac {{\left (B i n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, A i \log \left (\frac {b x + a}{d x + c}\right ) + 2 \, B i \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, g} \]

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),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right ) \left (d i x +c i \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.22, size = 175, normalized size = 3.50 \[ B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} B n}{2 \, {\left (b c g i - a d g i\right )}} + A {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \]

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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.72, size = 76, normalized size = 1.52 \[ -\frac {B\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2-A\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{2\,g\,i\,n\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A}{a c + a d x + b c x + b d x^{2}}\, dx + \int \frac {B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a c + a d x + b c x + b d x^{2}}\, dx}{g i} \]

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),x)

[Out]

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

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