3.637 \(\int (a+b \log (c (d+\frac{e}{f+g x})^p))^3 \, dx\)

Optimal. Leaf size=168 \[ -\frac{6 b^2 e p^2 \text{PolyLog}\left (2,\frac{e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )}{d g}+\frac{6 b^3 e p^3 \text{PolyLog}\left (3,\frac{e}{d (f+g x)}+1\right )}{d g}-\frac{3 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac{(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g} \]

[Out]

(-3*b*e*p*Log[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p])^2)/(d*g) + ((e + d*(f + g*x))*(a + b*Log[
c*(d + e/(f + g*x))^p])^3)/(d*g) - (6*b^2*e*p^2*(a + b*Log[c*(d + e/(f + g*x))^p])*PolyLog[2, 1 + e/(d*(f + g*
x))])/(d*g) + (6*b^3*e*p^3*PolyLog[3, 1 + e/(d*(f + g*x))])/(d*g)

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Rubi [A]  time = 0.184101, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2483, 2449, 2454, 2396, 2433, 2374, 6589} \[ -\frac{6 b^2 e p^2 \text{PolyLog}\left (2,\frac{e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )}{d g}+\frac{6 b^3 e p^3 \text{PolyLog}\left (3,\frac{e}{d (f+g x)}+1\right )}{d g}-\frac{3 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac{(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/(f + g*x))^p])^3,x]

[Out]

(-3*b*e*p*Log[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p])^2)/(d*g) + ((e + d*(f + g*x))*(a + b*Log[
c*(d + e/(f + g*x))^p])^3)/(d*g) - (6*b^2*e*p^2*(a + b*Log[c*(d + e/(f + g*x))^p])*PolyLog[2, 1 + e/(d*(f + g*
x))])/(d*g) + (6*b^3*e*p^3*PolyLog[3, 1 + e/(d*(f + g*x))])/(d*g)

Rule 2483

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

Rule 2449

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

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 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 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x}\right )^p\right )\right )^3 \, dx,x,f+g x\right )}{g}\\ &=\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{(3 b e p) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x}\right )^p\right )\right )^2}{x} \, dx,x,f+g x\right )}{d g}\\ &=\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac{(3 b e p) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^p\right )\right )^2}{x} \, dx,x,\frac{1}{f+g x}\right )}{d g}\\ &=-\frac{3 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{\left (6 b^2 e^2 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^p\right )\right )}{d+e x} \, dx,x,\frac{1}{f+g x}\right )}{d g}\\ &=-\frac{3 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac{\left (6 b^2 e p^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^p\right )\right ) \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac{e}{f+g x}\right )}{d g}\\ &=-\frac{3 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac{6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right ) \text{Li}_2\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}+\frac{\left (6 b^3 e p^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{f+g x}\right )}{d g}\\ &=-\frac{3 b e p \log \left (-\frac{e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac{(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac{6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right ) \text{Li}_2\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}+\frac{6 b^3 e p^3 \text{Li}_3\left (\frac{d+\frac{e}{f+g x}}{d}\right )}{d g}\\ \end{align*}

Mathematica [B]  time = 0.686147, size = 415, normalized size = 2.47 \[ \frac{3 b^2 p^2 \left (2 e \text{PolyLog}\left (2,\frac{d (f+g x)}{e}+1\right )+d (f+g x) \log ^2\left (d+\frac{e}{f+g x}\right )+e \left (2 \log \left (-\frac{d (f+g x)}{e}\right )-\log (d f+d g x+e)+2 \log \left (d+\frac{e}{f+g x}\right )\right ) \log (d (f+g x)+e)\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )+b^3 p^3 \left (6 e \text{PolyLog}\left (3,\frac{e}{d f+d g x}+1\right )-6 e \log \left (d+\frac{e}{f+g x}\right ) \text{PolyLog}\left (2,\frac{e}{d f+d g x}+1\right )+\left ((d f+d g x+e) \log \left (d+\frac{e}{f+g x}\right )-3 e \log \left (-\frac{e}{d f+d g x}\right )\right ) \log ^2\left (d+\frac{e}{f+g x}\right )\right )+3 b d p (f+g x) \log \left (d+\frac{e}{f+g x}\right ) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )^2+3 b e p \log (d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )^2+d (f+g x) \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b p \log \left (d+\frac{e}{f+g x}\right )\right )^3}{d g} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^3,x]

[Out]

(3*b*d*p*(f + g*x)*Log[d + e/(f + g*x)]*(a - b*p*Log[d + e/(f + g*x)] + b*Log[c*(d + e/(f + g*x))^p])^2 + d*(f
+ g*x)*(a - b*p*Log[d + e/(f + g*x)] + b*Log[c*(d + e/(f + g*x))^p])^3 + 3*b*e*p*(a - b*p*Log[d + e/(f + g*x)
] + b*Log[c*(d + e/(f + g*x))^p])^2*Log[e + d*(f + g*x)] + 3*b^2*p^2*(a - b*p*Log[d + e/(f + g*x)] + b*Log[c*(
d + e/(f + g*x))^p])*(d*(f + g*x)*Log[d + e/(f + g*x)]^2 + e*(2*Log[-((d*(f + g*x))/e)] - Log[e + d*f + d*g*x]
+ 2*Log[d + e/(f + g*x)])*Log[e + d*(f + g*x)] + 2*e*PolyLog[2, 1 + (d*(f + g*x))/e]) + b^3*p^3*(Log[d + e/(f
+ g*x)]^2*(-3*e*Log[-(e/(d*f + d*g*x))] + (e + d*f + d*g*x)*Log[d + e/(f + g*x)]) - 6*e*Log[d + e/(f + g*x)]*
PolyLog[2, 1 + e/(d*f + d*g*x)] + 6*e*PolyLog[3, 1 + e/(d*f + d*g*x)]))/(d*g)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{\frac{e}{gx+f}} \right ) ^{p} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/(g*x+f))^p))^3,x)

[Out]

int((a+b*ln(c*(d+e/(g*x+f))^p))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/(g*x+f))^p))^3,x, algorithm="maxima")

[Out]

-3*a^2*b*e*g*p*(f*log(g*x + f)/(e*g^2) - (d*f + e)*log(d*g*x + d*f + e)/(d*e*g^2)) + 3*a^2*b*x*log(c*(d + e/(g
*x + f))^p) + a^3*x + (b^3*d*g*x*log((d*g*x + d*f + e)^p)^3 - 3*(b^3*d*f*p*log(g*x + f) + b^3*d*g*x*log((g*x +
f)^p) - (d*f*p + e*p)*b^3*log(d*g*x + d*f + e) - (b^3*d*g*log(c) + a*b^2*d*g)*x)*log((d*g*x + d*f + e)^p)^2)/
(d*g) + integrate(((d*f + e)*b^3*log(c)^3 + 3*(d*f + e)*a*b^2*log(c)^2 - (b^3*d*g*x + (d*f + e)*b^3)*log((g*x
+ f)^p)^3 + 3*((d*f + e)*b^3*log(c) + (d*f + e)*a*b^2 + (b^3*d*g*log(c) + a*b^2*d*g)*x)*log((g*x + f)^p)^2 + (
b^3*d*g*log(c)^3 + 3*a*b^2*d*g*log(c)^2)*x + 3*(2*b^3*d*f*p^2*log(g*x + f) + (d*f + e)*b^3*log(c)^2 - 2*(d*f*p
^2 + e*p^2)*b^3*log(d*g*x + d*f + e) + 2*(d*f + e)*a*b^2*log(c) + (b^3*d*g*x + (d*f + e)*b^3)*log((g*x + f)^p)
^2 - (2*(d*g*p - d*g*log(c))*a*b^2 + (2*d*g*p*log(c) - d*g*log(c)^2)*b^3)*x - 2*((d*f + e)*b^3*log(c) + (d*f +
e)*a*b^2 + (a*b^2*d*g - (d*g*p - d*g*log(c))*b^3)*x)*log((g*x + f)^p))*log((d*g*x + d*f + e)^p) - 3*((d*f + e
)*b^3*log(c)^2 + 2*(d*f + e)*a*b^2*log(c) + (b^3*d*g*log(c)^2 + 2*a*b^2*d*g*log(c))*x)*log((g*x + f)^p))/(d*g*
x + d*f + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/(g*x+f))^p))^3,x, algorithm="fricas")

[Out]

integral(b^3*log(c*((d*g*x + d*f + e)/(g*x + f))^p)^3 + 3*a*b^2*log(c*((d*g*x + d*f + e)/(g*x + f))^p)^2 + 3*a
^2*b*log(c*((d*g*x + d*f + e)/(g*x + f))^p) + a^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e/(g*x+f))**p))**3,x)

[Out]

Integral((a + b*log(c*(d + e/(f + g*x))**p))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/(g*x+f))^p))^3,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/(g*x + f))^p) + a)^3, x)