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

Optimal. Leaf size=150 \[ \frac{B d i n \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac{d i \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}-\frac{i (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g^2 (a+b x)}-\frac{B i n (c+d x)}{b g^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.375878, antiderivative size = 233, normalized size of antiderivative = 1.55, number of steps used = 14, number of rules used = 11, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.268, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B d i n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac{d i \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}-\frac{i (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2 (a+b x)}-\frac{B i n (b c-a d)}{b^2 g^2 (a+b x)}+\frac{B d i n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{B d i n \log ^2(a+b x)}{2 b^2 g^2}-\frac{B d i n \log (a+b x)}{b^2 g^2}+\frac{B d i n \log (c+d x)}{b^2 g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 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{(113 c+113 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g^2 (a+b x)^2}+\frac{113 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g^2 (a+b x)}\right ) \, dx\\ &=\frac{(113 d) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g^2}+\frac{(113 (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b g^2}\\ &=-\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac{113 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{(113 B d n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 g^2}+\frac{(113 B (b c-a d) n) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac{113 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{(113 B d n) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 g^2}+\frac{\left (113 B (b c-a d)^2 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac{113 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{(113 B d n) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g^2}+\frac{\left (113 B d^2 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g^2}+\frac{\left (113 B (b c-a d)^2 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}\\ &=-\frac{113 B (b c-a d) n}{b^2 g^2 (a+b x)}-\frac{113 B d n \log (a+b x)}{b^2 g^2}-\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac{113 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac{113 B d n \log (c+d x)}{b^2 g^2}+\frac{113 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{(113 B d n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}-\frac{(113 B d n) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g^2}\\ &=-\frac{113 B (b c-a d) n}{b^2 g^2 (a+b x)}-\frac{113 B d n \log (a+b x)}{b^2 g^2}-\frac{113 B d n \log ^2(a+b x)}{2 b^2 g^2}-\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac{113 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac{113 B d n \log (c+d x)}{b^2 g^2}+\frac{113 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{(113 B d n) \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^2}\\ &=-\frac{113 B (b c-a d) n}{b^2 g^2 (a+b x)}-\frac{113 B d n \log (a+b x)}{b^2 g^2}-\frac{113 B d n \log ^2(a+b x)}{2 b^2 g^2}-\frac{113 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac{113 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac{113 B d n \log (c+d x)}{b^2 g^2}+\frac{113 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{113 B d n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}\\ \end{align*}

Mathematica [A]  time = 0.173961, size = 189, normalized size = 1.26 \[ \frac{i \left (-\frac{B d n \left (-2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )-2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )+\log ^2(a+b x)\right )}{2 b^2}+\frac{d \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2}-\frac{(b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 (a+b x)}-\frac{B n \left (\frac{b c-a d}{a+b x}+d \log (a+b x)-d \log (c+d x)\right )}{b^2}\right )}{g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.523, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{ \left ( bgx+ag \right ) ^{2}} \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((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2,x)

[Out]

int((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -B c i n{\left (\frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} + B d i{\left (\frac{{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b^{3} g^{2} x + a b^{2} g^{2}} + \int \frac{b^{2} d x^{2} \log \left (e\right ) + b^{2} c x \log \left (e\right ) - a b c n + a^{2} d n -{\left (a b c n - a^{2} d n +{\left (b^{2} c n - a b d n\right )} x\right )} \log \left (b x + a\right )}{b^{4} d g^{2} x^{3} + a^{2} b^{2} c g^{2} +{\left (b^{4} c g^{2} + 2 \, a b^{3} d g^{2}\right )} x^{2} +{\left (2 \, a b^{3} c g^{2} + a^{2} b^{2} d g^{2}\right )} x}\,{d x}\right )} + A d i{\left (\frac{a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac{\log \left (b x + a\right )}{b^{2} g^{2}}\right )} - \frac{B c i \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac{A c i}{b^{2} g^{2} x + a b g^{2}} \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))^n))/(b*g*x+a*g)^2,x, algorithm="maxima")

[Out]

-B*c*i*n*(1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*g^2
)) + B*d*i*((((b*x + a)*log(b*x + a) + a)*log((b*x + a)^n) - ((b*x + a)*log(b*x + a) + a)*log((d*x + c)^n))/(b
^3*g^2*x + a*b^2*g^2) + integrate((b^2*d*x^2*log(e) + b^2*c*x*log(e) - a*b*c*n + a^2*d*n - (a*b*c*n - a^2*d*n
+ (b^2*c*n - a*b*d*n)*x)*log(b*x + a))/(b^4*d*g^2*x^3 + a^2*b^2*c*g^2 + (b^4*c*g^2 + 2*a*b^3*d*g^2)*x^2 + (2*a
*b^3*c*g^2 + a^2*b^2*d*g^2)*x), x)) + A*d*i*(a/(b^3*g^2*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) - B*c*i*log(e
*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^2*g^2*x + a*b*g^2) - A*c*i/(b^2*g^2*x + a*b*g^2)

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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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, 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))^n))/(b*g*x+a*g)^2,x, algorithm="fricas")

[Out]

integral((A*d*i*x + A*c*i + (B*d*i*x + B*c*i)*log(e*((b*x + a)/(d*x + c))^n))/(b^2*g^2*x^2 + 2*a*b*g^2*x + a^2
*g^2), 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))**n))/(b*g*x+a*g)**2,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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{{\left (b g x + a g\right )}^{2}}\,{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))^n))/(b*g*x+a*g)^2,x, algorithm="giac")

[Out]

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