3.62 \(\int \frac{(a+b \log (c (d+e x)^n))^4}{f+g x} \, dx\)

Optimal. Leaf size=205 \[ \frac{24 b^3 n^3 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{12 b^2 n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac{4 b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac{24 b^4 n^4 \text{PolyLog}\left (5,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g} \]

[Out]

((a + b*Log[c*(d + e*x)^n])^4*Log[(e*(f + g*x))/(e*f - d*g)])/g + (4*b*n*(a + b*Log[c*(d + e*x)^n])^3*PolyLog[
2, -((g*(d + e*x))/(e*f - d*g))])/g - (12*b^2*n^2*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[3, -((g*(d + e*x))/(e*f
- d*g))])/g + (24*b^3*n^3*(a + b*Log[c*(d + e*x)^n])*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g - (24*b^4*n^
4*PolyLog[5, -((g*(d + e*x))/(e*f - d*g))])/g

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Rubi [A]  time = 0.231313, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2396, 2433, 2374, 2383, 6589} \[ \frac{24 b^3 n^3 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{12 b^2 n^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac{4 b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac{24 b^4 n^4 \text{PolyLog}\left (5,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^4/(f + g*x),x]

[Out]

((a + b*Log[c*(d + e*x)^n])^4*Log[(e*(f + g*x))/(e*f - d*g)])/g + (4*b*n*(a + b*Log[c*(d + e*x)^n])^3*PolyLog[
2, -((g*(d + e*x))/(e*f - d*g))])/g - (12*b^2*n^2*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[3, -((g*(d + e*x))/(e*f
- d*g))])/g + (24*b^3*n^3*(a + b*Log[c*(d + e*x)^n])*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g - (24*b^4*n^
4*PolyLog[5, -((g*(d + e*x))/(e*f - d*g))])/g

Rule 2396

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(4 b e n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac{e \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{\left (12 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\left (24 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{24 b^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{\left (24 b^4 n^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_3\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{24 b^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}-\frac{24 b^4 n^4 \text{Li}_5\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}\\ \end{align*}

Mathematica [B]  time = 0.223017, size = 503, normalized size = 2.45 \[ \frac{-4 b^3 n^3 \left (6 \text{PolyLog}\left (4,\frac{g (d+e x)}{d g-e f}\right )+3 \log ^2(d+e x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )-6 \log (d+e x) \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )+\log ^3(d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) \left (-a-b \log \left (c (d+e x)^n\right )+b n \log (d+e x)\right )+6 b^2 n^2 \left (-2 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+\log ^2(d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+4 b n \left (\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+\log (d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^3+b^4 n^4 \left (-24 \text{PolyLog}\left (5,\frac{g (d+e x)}{d g-e f}\right )+4 \log ^3(d+e x) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )-12 \log ^2(d+e x) \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )+24 \log (d+e x) \text{PolyLog}\left (4,\frac{g (d+e x)}{d g-e f}\right )+\log ^4(d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )\right )+\log (f+g x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^4}{g} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^4/(f + g*x),x]

[Out]

((a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^4*Log[f + g*x] + 4*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x
)^n])^3*(Log[d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + 6*b^2*n^2*(
a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)] + 2*Log[d + e*x]
*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]) - 4*b^3*n^3*(-a + b*n*
Log[d + e*x] - b*Log[c*(d + e*x)^n])*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*Log[d + e*x]^2*PolyLog
[2, (g*(d + e*x))/(-(e*f) + d*g)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 6*PolyLog[4, (g*
(d + e*x))/(-(e*f) + d*g)]) + b^4*n^4*(Log[d + e*x]^4*Log[(e*(f + g*x))/(e*f - d*g)] + 4*Log[d + e*x]^3*PolyLo
g[2, (g*(d + e*x))/(-(e*f) + d*g)] - 12*Log[d + e*x]^2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 24*Log[d + e
*x]*PolyLog[4, (g*(d + e*x))/(-(e*f) + d*g)] - 24*PolyLog[5, (g*(d + e*x))/(-(e*f) + d*g)]))/g

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Maple [C]  time = 1.559, size = 33189, normalized size = 161.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^4/(g*x+f),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{4} \log \left (g x + f\right )}{g} + \int \frac{b^{4} \log \left ({\left (e x + d\right )}^{n}\right )^{4} + b^{4} \log \left (c\right )^{4} + 4 \, a b^{3} \log \left (c\right )^{3} + 6 \, a^{2} b^{2} \log \left (c\right )^{2} + 4 \, a^{3} b \log \left (c\right ) + 4 \,{\left (b^{4} \log \left (c\right ) + a b^{3}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + 6 \,{\left (b^{4} \log \left (c\right )^{2} + 2 \, a b^{3} \log \left (c\right ) + a^{2} b^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 4 \,{\left (b^{4} \log \left (c\right )^{3} + 3 \, a b^{3} \log \left (c\right )^{2} + 3 \, a^{2} b^{2} \log \left (c\right ) + a^{3} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^4/(g*x+f),x, algorithm="maxima")

[Out]

a^4*log(g*x + f)/g + integrate((b^4*log((e*x + d)^n)^4 + b^4*log(c)^4 + 4*a*b^3*log(c)^3 + 6*a^2*b^2*log(c)^2
+ 4*a^3*b*log(c) + 4*(b^4*log(c) + a*b^3)*log((e*x + d)^n)^3 + 6*(b^4*log(c)^2 + 2*a*b^3*log(c) + a^2*b^2)*log
((e*x + d)^n)^2 + 4*(b^4*log(c)^3 + 3*a*b^3*log(c)^2 + 3*a^2*b^2*log(c) + a^3*b)*log((e*x + d)^n))/(g*x + f),
x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{4} \log \left ({\left (e x + d\right )}^{n} c\right )^{4} + 4 \, a b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 6 \, a^{2} b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 4 \, a^{3} b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{4}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^4/(g*x+f),x, algorithm="fricas")

[Out]

integral((b^4*log((e*x + d)^n*c)^4 + 4*a*b^3*log((e*x + d)^n*c)^3 + 6*a^2*b^2*log((e*x + d)^n*c)^2 + 4*a^3*b*l
og((e*x + d)^n*c) + a^4)/(g*x + f), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{4}}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**4/(g*x+f),x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))**4/(f + g*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^4/(g*x+f),x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^4/(g*x + f), x)