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

Optimal. Leaf size=133 \[ \frac{B i (b c-a d) \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right )}{b^2 g}-\frac{i (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A-B\right )}{b^2 g}+\frac{i (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b g} \]

[Out]

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

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Rubi [A]  time = 0.3527, antiderivative size = 213, normalized size of antiderivative = 1.6, number of steps used = 14, number of rules used = 11, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2528, 2486, 31, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B i (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{i (b c-a d) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g}+\frac{B d i (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}-\frac{B i (b c-a d) \log ^2(a+b x)}{2 b^2 g}+\frac{B i (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{B i (b c-a d) \log (c+d x)}{b^2 g}+\frac{A d i x}{b g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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]

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{(5 c+5 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx &=\int \left (\frac{5 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g}+\frac{5 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g (a+b x)}\right ) \, dx\\ &=\frac{(5 d) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b g}+\frac{(5 (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g}\\ &=\frac{5 A d x}{b g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{(5 B d) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b g}-\frac{(5 B (b c-a d)) \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+b x)}{e (a+b x)} \, dx}{b^2 g}\\ &=\frac{5 A d x}{b g}+\frac{5 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}-\frac{(5 B d (b c-a d)) \int \frac{1}{c+d x} \, dx}{b^2 g}-\frac{(5 B (b c-a d)) \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+b x)}{a+b x} \, dx}{b^2 e g}\\ &=\frac{5 A d x}{b g}+\frac{5 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}-\frac{5 B (b c-a d) \log (c+d x)}{b^2 g}-\frac{(5 B (b c-a d)) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 e g}\\ &=\frac{5 A d x}{b g}+\frac{5 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}-\frac{5 B (b c-a d) \log (c+d x)}{b^2 g}-\frac{(5 B (b c-a d)) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g}+\frac{(5 B d (b c-a d)) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g}\\ &=\frac{5 A d x}{b g}+\frac{5 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}-\frac{5 B (b c-a d) \log (c+d x)}{b^2 g}+\frac{5 B (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{(5 B (b c-a d)) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac{(5 B (b c-a d)) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}\\ &=\frac{5 A d x}{b g}-\frac{5 B (b c-a d) \log ^2(a+b x)}{2 b^2 g}+\frac{5 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}-\frac{5 B (b c-a d) \log (c+d x)}{b^2 g}+\frac{5 B (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{(5 B (b c-a d)) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}\\ &=\frac{5 A d x}{b g}-\frac{5 B (b c-a d) \log ^2(a+b x)}{2 b^2 g}+\frac{5 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{5 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}-\frac{5 B (b c-a d) \log (c+d x)}{b^2 g}+\frac{5 B (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{5 B (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}\\ \end{align*}

Mathematica [A]  time = 0.113839, size = 164, normalized size = 1.23 \[ \frac{i \left (2 B (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+2 (b c-a d) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+B \log \left (\frac{b (c+d x)}{b c-a d}\right )+A\right )+2 \left (B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )+\log (c+d x) (a B d-b B c)+A b d x\right )+\log ^2(a+b x) (a B d-b B c)\right )}{2 b^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.191, size = 1044, 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((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x)

[Out]

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

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Maxima [A]  time = 1.5801, size = 325, normalized size = 2.44 \begin{align*} A d i{\left (\frac{x}{b g} - \frac{a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac{A c i \log \left (b g x + a g\right )}{b g} - \frac{B c i \log \left (d x + c\right )}{b g} + \frac{{\left (b c i - a d i\right )}{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )} B}{b^{2} g} + \frac{2 \, B b d i x \log \left (e\right ) +{\left (b c i - a d i\right )} B \log \left (b x + a\right )^{2} + 2 \,{\left (B b d i x +{\left (b c i \log \left (e\right ) -{\left (i \log \left (e\right ) - i\right )} a d\right )} B\right )} \log \left (b x + a\right ) - 2 \,{\left (B b d i x +{\left (b c i - a d i\right )} B \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{2 \, b^{2} g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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