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

Optimal. Leaf size=76 \[ -\frac{B \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d i}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i} \]

[Out]

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

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Rubi [A]  time = 0.215483, antiderivative size = 122, normalized size of antiderivative = 1.61, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2524, 12, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{\log (c i+d i x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i}-\frac{B \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}+\frac{B \log ^2(i (c+d x))}{2 d i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{34 c+34 d x} \, dx &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\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 (34 c+34 d x)}{e (a+b x)} \, dx}{34 d}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\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 (34 c+34 d x)}{a+b x} \, dx}{34 d e}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\frac{B \int \left (\frac{b e \log (34 c+34 d x)}{a+b x}-\frac{d e \log (34 c+34 d x)}{c+d x}\right ) \, dx}{34 d e}\\ &=\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}+\frac{1}{34} B \int \frac{\log (34 c+34 d x)}{c+d x} \, dx-\frac{(b B) \int \frac{\log (34 c+34 d x)}{a+b x} \, dx}{34 d}\\ &=-\frac{B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (34 c+34 d x)}{34 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}+B \int \frac{\log \left (\frac{34 d (a+b x)}{-34 b c+34 a d}\right )}{34 c+34 d x} \, dx+\frac{B \operatorname{Subst}\left (\int \frac{34 \log (x)}{x} \, dx,x,34 c+34 d x\right )}{1156 d}\\ &=-\frac{B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (34 c+34 d x)}{34 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}+\frac{B \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,34 c+34 d x\right )}{34 d}+\frac{B \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-34 b c+34 a d}\right )}{x} \, dx,x,34 c+34 d x\right )}{34 d}\\ &=\frac{B \log ^2(34 (c+d x))}{68 d}-\frac{B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (34 c+34 d x)}{34 d}+\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (34 c+34 d x)}{34 d}-\frac{B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{34 d}\\ \end{align*}

Mathematica [A]  time = 0.0317992, size = 95, normalized size = 1.25 \[ \frac{\log (i (c+d x)) \left (2 B \log \left (\frac{e (a+b x)}{c+d x}\right )-2 B \log \left (\frac{d (a+b x)}{a d-b c}\right )+2 A+B \log (i (c+d x))\right )-2 B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 d i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.056, size = 411, normalized size = 5.4 \begin{align*} -{\frac{Aa}{i \left ( ad-bc \right ) }\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }+{\frac{Abc}{di \left ( ad-bc \right ) }\ln \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) }-{\frac{Ba}{i \left ( ad-bc \right ) }{\it dilog} \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) }+{\frac{Bbc}{di \left ( ad-bc \right ) }{\it dilog} \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) }-{\frac{Ba}{i \left ( ad-bc \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \ln \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) }+{\frac{Bbc}{di \left ( ad-bc \right ) }\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \ln \left ( -{\frac{1}{be} \left ( d \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) -be \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, B{\left (\frac{\log \left (d x + c\right )^{2}}{d i} - 2 \, \int \frac{\log \left (b x + a\right ) + \log \left (e\right )}{d i x + c i}\,{d x}\right )} + \frac{A \log \left (d i x + c i\right )}{d i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B \log \left (\frac{b e x + a e}{d x + c}\right ) + A}{d i x + c i}, x\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)))/(d*i*x+c*i),x, algorithm="fricas")

[Out]

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

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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)))/(d*i*x+c*i),x)

[Out]

Timed out

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

[Out]

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