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

Optimal. Leaf size=134 \[ \frac{B g n (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 i}+\frac{g (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A+B n\right )}{d^2 i}+\frac{g (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d i} \]

[Out]

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

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Rubi [A]  time = 0.388567, antiderivative size = 223, normalized size of antiderivative = 1.66, number of steps used = 13, number of rules used = 10, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.244, Rules used = {2528, 2486, 31, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B g n (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac{g (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}-\frac{B g n (b c-a d) \log ^2(c+d x)}{2 d^2 i}-\frac{B g n (b c-a d) \log (c+d x)}{d^2 i}+\frac{B g n (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac{B g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d i}+\frac{A b g x}{d i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{137 c+137 d x} \, dx &=\int \left (\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{137 d}+\frac{(-b c+a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{137 d (c+d x)}\right ) \, dx\\ &=\frac{(b g) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{137 d}-\frac{((b c-a d) g) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{137 d}\\ &=\frac{A b g x}{137 d}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac{(b B g) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{137 d}+\frac{(B (b c-a d) g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{137 d^2}\\ &=\frac{A b g x}{137 d}+\frac{B g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{137 d}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac{(B (b c-a d) g n) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{137 d^2}-\frac{(B (b c-a d) g n) \int \frac{1}{c+d x} \, dx}{137 d}\\ &=\frac{A b g x}{137 d}+\frac{B g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{137 d}-\frac{B (b c-a d) g n \log (c+d x)}{137 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac{(b B (b c-a d) g n) \int \frac{\log (c+d x)}{a+b x} \, dx}{137 d^2}-\frac{(B (b c-a d) g n) \int \frac{\log (c+d x)}{c+d x} \, dx}{137 d}\\ &=\frac{A b g x}{137 d}+\frac{B g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{137 d}-\frac{B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac{B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac{(B (b c-a d) g n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{137 d^2}-\frac{(B (b c-a d) g n) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{137 d}\\ &=\frac{A b g x}{137 d}+\frac{B g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{137 d}-\frac{B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac{B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac{B (b c-a d) g n \log ^2(c+d x)}{274 d^2}-\frac{(B (b c-a d) g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{137 d^2}\\ &=\frac{A b g x}{137 d}+\frac{B g (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{137 d}-\frac{B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac{B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac{B (b c-a d) g n \log ^2(c+d x)}{274 d^2}+\frac{B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{137 d^2}\\ \end{align*}

Mathematica [A]  time = 0.118595, size = 170, normalized size = 1.27 \[ \frac{g \left (B n (b c-a d) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n (b c-a d) \log (c+d x)+2 A b d x\right )}{2 d^2 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.645, size = 0, normalized size = 0. \begin{align*} \int{\frac{bgx+ag}{dix+ci} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.68274, size = 413, normalized size = 3.08 \begin{align*} A b g{\left (\frac{x}{d i} - \frac{c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac{A a g \log \left (d i x + c i\right )}{d i} - \frac{{\left (b c g n - a d g n\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}{d^{2} i} + \frac{{\left (a d g \log \left (e\right ) -{\left (g n + g \log \left (e\right )\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} + \frac{2 \, B a d g n \log \left (b x + a\right ) + 2 \, B b d g x \log \left (e\right ) + 2 \,{\left (b c g n - a d g n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) -{\left (b c g n - a d g n\right )} B \log \left (d x + c\right )^{2} + 2 \,{\left (B b d g x -{\left (b c g - a d g\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B b d g x -{\left (b c g - a d g\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, d^{2} i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((A*b*g*x + A*a*g + (B*b*g*x + B*a*g)*log(e*((b*x + a)/(d*x + c))^n))/(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((b*g*x+a*g)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(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{{\left (b g x + a g\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{d i x + c i}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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