3.1.92 \(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{a g+b g x} \, dx\) [92]

Optimal. Leaf size=80 \[ -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}+\frac {B \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b g} \]

[Out]

-ln((a*d-b*c)/d/(b*x+a))*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/g+B*polylog(2,1+(-a*d+b*c)/d/(b*x+a))/b/g

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Rubi [A]
time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2542, 2458, 2378, 2370, 2352} \begin {gather*} \frac {B \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*g)) + (B*PolyLog[2, 1 + (b*c - a
*d)/(d*(a + b*x))])/(b*g)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2370

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2378

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Dist[1/n, Subst[Int[(a
+ b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2542

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 95, normalized size = 1.19 \begin {gather*} \frac {\log (g (a+b x)) \left (-B \log (g (a+b x))+2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+2 B \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )}{2 b g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(79)=158\).
time = 0.60, size = 345, normalized size = 4.31

method	result	size
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{g \left (a d -c b \right ) b e}-\frac {d^{2} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g \left (a d -c b \right ) b e}+\frac {d^{2} B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{g \left (a d -c b \right ) b e}+\frac {d^{2} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{g \left (a d -c b \right ) b e}-\frac {d^{2} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g \left (a d -c b \right ) b e}\right )}{d^{2}}\)	\(345\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{g \left (a d -c b \right ) b e}-\frac {d^{2} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g \left (a d -c b \right ) b e}+\frac {d^{2} B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{g \left (a d -c b \right ) b e}+\frac {d^{2} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{g \left (a d -c b \right ) b e}-\frac {d^{2} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g \left (a d -c b \right ) b e}\right )}{d^{2}}\)	\(345\)
risch	\(\frac {A \ln \left (b x +a \right )}{g b}+\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} a}{2 g \left (a d -c b \right ) b}-\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} c}{2 g \left (a d -c b \right )}-\frac {B d \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) a}{g \left (a d -c b \right ) b}+\frac {B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) c}{g \left (a d -c b \right )}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) a}{g \left (a d -c b \right ) b}+\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) c}{g \left (a d -c b \right )}\)	\(416\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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),x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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),x, algorithm="fricas")

[Out]

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

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

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),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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