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

Optimal. Leaf size=144 \[ -\frac{2 B \text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )}{g}+\frac{2 B \text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )}{g}+\frac{\log (f+g x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g}-\frac{2 B \log (f+g x) \log \left (-\frac{g (a+b x)}{b f-a g}\right )}{g}+\frac{2 B \log (f+g x) \log \left (-\frac{g (c+d x)}{d f-c g}\right )}{g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.308878, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2524, 12, 2418, 2394, 2393, 2391} \[ -\frac{2 B \text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )}{g}+\frac{2 B \text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )}{g}+\frac{\log (f+g x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g}-\frac{2 B \log (f+g x) \log \left (-\frac{g (a+b x)}{b f-a g}\right )}{g}+\frac{2 B \log (f+g x) \log \left (-\frac{g (c+d x)}{d f-c g}\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx &=\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}-\frac{B \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (f+g x)}{e (a+b x)^2} \, dx}{g}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}-\frac{B \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (f+g x)}{(a+b x)^2} \, dx}{e g}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}-\frac{B \int \left (\frac{2 b e \log (f+g x)}{a+b x}-\frac{2 d e \log (f+g x)}{c+d x}\right ) \, dx}{e g}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}-\frac{(2 b B) \int \frac{\log (f+g x)}{a+b x} \, dx}{g}+\frac{(2 B d) \int \frac{\log (f+g x)}{c+d x} \, dx}{g}\\ &=-\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac{2 B \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+(2 B) \int \frac{\log \left (\frac{g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-(2 B) \int \frac{\log \left (\frac{g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx\\ &=-\frac{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac{2 B \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac{(2 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{(2 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{2 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)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac{2 B \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac{2 B \text{Li}_2\left (\frac{b (f+g x)}{b f-a g}\right )}{g}+\frac{2 B \text{Li}_2\left (\frac{d (f+g x)}{d f-c g}\right )}{g}\\ \end{align*}

Mathematica [A]  time = 0.0591387, size = 119, normalized size = 0.83 \[ \frac{-2 B \text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )+2 B \text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )+\log (f+g x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )-2 B \log \left (\frac{g (a+b x)}{a g-b f}\right )+A+2 B \log \left (\frac{g (c+d x)}{c g-d f}\right )\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.445, size = 1143, normalized size = 7.9 \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)^2/(d*x+c)^2))/(g*x+f),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)**2/(d*x+c)**2))/(g*x+f),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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