3.235 \(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{f+g x} \, dx\)

Optimal. Leaf size=140 \[ \frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}-\frac {B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {B \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{g} \]

[Out]

-B*ln(-g*(b*x+a)/(-a*g+b*f))*ln(g*x+f)/g+(A+B*ln(e*(b*x+a)/(d*x+c)))*ln(g*x+f)/g+B*ln(-g*(d*x+c)/(-c*g+d*f))*l
n(g*x+f)/g-B*polylog(2,b*(g*x+f)/(-a*g+b*f))/g+B*polylog(2,d*(g*x+f)/(-c*g+d*f))/g

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Rubi [A]  time = 0.25, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2524, 12, 2418, 2394, 2393, 2391} \[ -\frac {B \text {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B \text {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}-\frac {B \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {B \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*Log[-((g*(a + b*x))/(b*f - a*g))]*Log[f + g*x])/g) + ((A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[f + g*x])/
g + (B*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g - (B*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (B*Po
lyLog[2, (d*(f + g*x))/(d*f - c*g)])/g

Rule 12

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 115, normalized size = 0.82 \[ \frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )-B \log \left (\frac {g (a+b x)}{a g-b f}\right )+A+B \log \left (\frac {g (c+d x)}{c g-d f}\right )\right )-B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )+B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \log \left (\frac {b e x + a e}{d x + c}\right ) + A}{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.20, size = 1400, normalized size = 10.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*A/g/(a*d-b*c)*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a+A/g/(a*d-b*c)*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e
)*d)*b*c+d*A/g/(a*d-b*c)*ln((b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c*g-d*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*f-a*e*g+b*e*f)*a
-A/g/(a*d-b*c)*ln((b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c*g-d*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*f-a*e*g+b*e*f)*b*c-d*B/g/(
a*d-b*c)*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*a+B/g/(a*d-b*c)*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x
+c)/d*e)*d)/b/e)*b*c-d*B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d
)/b/e)*a+B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*b*c+d*B
/(a*d-b*c)*dilog(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*c*a-B/(a*d-b*
c)*dilog(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*c^2*b-d^2*B/g/(a*d-b*
c)*dilog(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*f*a+d*B/g/(a*d-b*c)*d
ilog(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*f*b*c+d*B/(a*d-b*c)*ln(b/
d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*
c*a-B/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e
*g+b*e*f))/(c*g-d*f)*c^2*b-d^2*B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(((c*g-d*f)*(b/d*e+(a*d-b*c)/(d
*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*f*a+d*B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(((c*g
-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e)-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f)*f*b*c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -B \int -\frac {\log \left (b x + a\right ) - \log \left (d x + c\right ) + \log \relax (e)}{g x + f}\,{d x} + \frac {A \log \left (g x + f\right )}{g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*integrate(-(log(b*x + a) - log(d*x + c) + log(e))/(g*x + f), x) + A*log(g*x + f)/g

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{f+g\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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