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

Optimal. Leaf size=221 \[ \frac {24 b^3 e p^3 \text {Li}_3\left (\frac {e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}-\frac {12 b^2 e p^2 \text {Li}_2\left (\frac {e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac {4 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^4}{d g}-\frac {24 b^4 e p^4 \text {Li}_4\left (\frac {e}{d (f+g x)}+1\right )}{d g} \]

[Out]

-4*b*e*p*ln(-e/d/(g*x+f))*(a+b*ln(c*(d+e/(g*x+f))^p))^3/d/g+(e+d*(g*x+f))*(a+b*ln(c*(d+e/(g*x+f))^p))^4/d/g-12
*b^2*e*p^2*(a+b*ln(c*(d+e/(g*x+f))^p))^2*polylog(2,1+e/d/(g*x+f))/d/g+24*b^3*e*p^3*(a+b*ln(c*(d+e/(g*x+f))^p))
*polylog(3,1+e/d/(g*x+f))/d/g-24*b^4*e*p^4*polylog(4,1+e/d/(g*x+f))/d/g

________________________________________________________________________________________

Rubi [A]  time = 0.28, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2483, 2449, 2454, 2396, 2433, 2374, 2383, 6589} \[ \frac {24 b^3 e p^3 \text {PolyLog}\left (3,\frac {e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}-\frac {12 b^2 e p^2 \text {PolyLog}\left (2,\frac {e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac {24 b^4 e p^4 \text {PolyLog}\left (4,\frac {e}{d (f+g x)}+1\right )}{d g}-\frac {4 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^4}{d g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*e*p*Log[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p])^3)/(d*g) + ((e + d*(f + g*x))*(a + b*Log[
c*(d + e/(f + g*x))^p])^4)/(d*g) - (12*b^2*e*p^2*(a + b*Log[c*(d + e/(f + g*x))^p])^2*PolyLog[2, 1 + e/(d*(f +
g*x))])/(d*g) + (24*b^3*e*p^3*(a + b*Log[c*(d + e/(f + g*x))^p])*PolyLog[3, 1 + e/(d*(f + g*x))])/(d*g) - (24
*b^4*e*p^4*PolyLog[4, 1 + e/(d*(f + g*x))])/(d*g)

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 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 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 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 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 )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^4 \, 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 )^4}{d g}+\frac {(4 b e p) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^3}{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 )^4}{d g}-\frac {(4 b e p) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^p\right )\right )^3}{x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {4 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 )^3}{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 )^4}{d g}+\frac {\left (12 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 )^2}{d+e x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {4 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 )^3}{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 )^4}{d g}+\frac {\left (12 b^2 e p^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^p\right )\right )^2 \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 {4 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 )^3}{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 )^4}{d g}-\frac {12 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {\left (24 b^3 e p^3\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^p\right )\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{f+g x}\right )}{d g}\\ &=-\frac {4 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 )^3}{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 )^4}{d g}-\frac {12 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {24 b^3 e p^3 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_3\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}-\frac {\left (24 b^4 e p^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{f+g x}\right )}{d g}\\ &=-\frac {4 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 )^3}{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 )^4}{d g}-\frac {12 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {24 b^3 e p^3 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_3\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}-\frac {24 b^4 e p^4 \text {Li}_4\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 1.42, size = 739, normalized size = 3.34 \[ \frac {4 b^3 p^3 \left (6 e \text {Li}_3\left (\frac {e}{d f+d g x}+1\right )-6 e \text {Li}_2\left (\frac {e}{d f+d g x}+1\right ) \log \left (d+\frac {e}{f+g x}\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 ) \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 )+6 b^2 p^2 \left (-d f \left (\log \left (-\frac {e}{d f+d g x}\right ) \left (\log \left (-\frac {e}{d f+d g x}\right )+2 \log \left (\frac {d f+d g x+e}{e}\right )\right )-2 \text {Li}_2\left (-\frac {d (f+g x)}{e}\right )\right )+(d f+e) \left (2 \text {Li}_2\left (\frac {e+d f+d g x}{e}\right )+\left (2 \log \left (-\frac {d (f+g x)}{e}\right )-\log (d f+d g x+e)\right ) \log (d f+d g x+e)\right )+d g x \log ^2\left (\frac {d f+d g x+e}{f+g x}\right )+2 d f \log \left (-\frac {e}{d f+d g x}\right ) \log \left (\frac {d f+d g x+e}{f+g x}\right )+2 (d f+e) \log (d f+d g x+e) \log \left (\frac {d f+d g x+e}{f+g x}\right )\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 )^2-4 b p \left (-(d f+e) \log (d f+d g x+e)-d g x \log \left (\frac {d f+d g x+e}{f+g x}\right )+d f \log (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 )^3+d 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 )^4-b^4 p^4 \left (24 e \text {Li}_4\left (\frac {e}{d f+d g x}+1\right )+12 e \text {Li}_2\left (\frac {e}{d f+d g x}+1\right ) \log ^2\left (d+\frac {e}{f+g x}\right )-24 e \text {Li}_3\left (\frac {e}{d f+d g x}+1\right ) \log \left (d+\frac {e}{f+g x}\right )-e \log ^4\left (d+\frac {e}{f+g x}\right )-d f \log ^4\left (d+\frac {e}{f+g x}\right )-d g x \log ^4\left (d+\frac {e}{f+g x}\right )+4 e \log \left (-\frac {e}{d f+d g x}\right ) \log ^3\left (d+\frac {e}{f+g x}\right )\right )}{d g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*p*(d*f*Log[f + g*x] - (e + d*f)*Log[e + d*f + d*g*x] - d*g*x*Log[(e + d*f + d*g*x)/(f + g*x)])*(a - b*p*
Log[d + e/(f + g*x)] + b*Log[c*(d + e/(f + g*x))^p])^3 + d*g*x*(a - b*p*Log[d + e/(f + g*x)] + b*Log[c*(d + e/
(f + g*x))^p])^4 + 6*b^2*p^2*(a - b*p*Log[d + e/(f + g*x)] + b*Log[c*(d + e/(f + g*x))^p])^2*(2*d*f*Log[-(e/(d
*f + d*g*x))]*Log[(e + d*f + d*g*x)/(f + g*x)] + 2*(e + d*f)*Log[e + d*f + d*g*x]*Log[(e + d*f + d*g*x)/(f + g
*x)] + d*g*x*Log[(e + d*f + d*g*x)/(f + g*x)]^2 - d*f*(Log[-(e/(d*f + d*g*x))]*(Log[-(e/(d*f + d*g*x))] + 2*Lo
g[(e + d*f + d*g*x)/e]) - 2*PolyLog[2, -((d*(f + g*x))/e)]) + (e + d*f)*((2*Log[-((d*(f + g*x))/e)] - Log[e +
d*f + d*g*x])*Log[e + d*f + d*g*x] + 2*PolyLog[2, (e + d*f + d*g*x)/e])) + 4*b^3*p^3*(a - b*p*Log[d + e/(f + g
*x)] + b*Log[c*(d + e/(f + g*x))^p])*(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)]) - b^4*p^4*(4*e*Log[-(e/(d*f + d*g*x))]*Log[d + e/(f + g*x)]^3 - e*Log[d + e/(f + g*x)]^4 - d*f*Log[
d + e/(f + g*x)]^4 - d*g*x*Log[d + e/(f + g*x)]^4 + 12*e*Log[d + e/(f + g*x)]^2*PolyLog[2, 1 + e/(d*f + d*g*x)
] - 24*e*Log[d + e/(f + g*x)]*PolyLog[3, 1 + e/(d*f + d*g*x)] + 24*e*PolyLog[4, 1 + e/(d*f + d*g*x)]))/(d*g)

________________________________________________________________________________________

fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{4} \log \left (c \left (\frac {d g x + d f + e}{g x + f}\right )^{p}\right )^{4} + 4 \, a b^{3} \log \left (c \left (\frac {d g x + d f + e}{g x + f}\right )^{p}\right )^{3} + 6 \, a^{2} b^{2} \log \left (c \left (\frac {d g x + d f + e}{g x + f}\right )^{p}\right )^{2} + 4 \, a^{3} b \log \left (c \left (\frac {d g x + d f + e}{g x + f}\right )^{p}\right ) + a^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )+a \right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(d+1/(g*x+f)*e)^p)+a)^4,x)

[Out]

int((b*ln(c*(d+1/(g*x+f)*e)^p)+a)^4,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________