3.437 \(\int (a+b \log (c (d+\frac{e}{\sqrt{x}})^n))^3 \, dx\)

Optimal. Leaf size=260 \[ -\frac{6 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (3,\frac{d}{d+\frac{e}{\sqrt{x}}}\right )}{d^2}-\frac{6 b^2 e^2 n^2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}+\frac{3 b e^2 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}+\frac{3 b e n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.616107, antiderivative size = 281, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {2451, 2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ \frac{6 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right )}{d^2}-\frac{6 b^2 e^2 n^2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}+\frac{3 b e^2 n \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}-\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{d^2}+\frac{3 b e n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/Sqrt[x])^n])^3,x]

[Out]

(3*b*e*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/d^2 - (e^2*(a + b*Log[c*(d + e/Sqrt[x])^n
])^3)/d^2 + x*(a + b*Log[c*(d + e/Sqrt[x])^n])^3 - (6*b^2*e^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[-(e/(d*
Sqrt[x]))])/d^2 + (3*b*e^2*n*(a + b*Log[c*(d + e/Sqrt[x])^n])^2*Log[-(e/(d*Sqrt[x]))])/d^2 - (6*b^3*e^2*n^3*Po
lyLog[2, 1 + e/(d*Sqrt[x])])/d^2 + (6*b^2*e^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[2, 1 + e/(d*Sqrt[x]
)])/d^2 - (6*b^3*e^2*n^3*PolyLog[3, 1 + e/(d*Sqrt[x])])/d^2

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

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 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

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

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 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 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \, dx &=2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c \left (d+\frac{e}{x}\right )^n\right )\right )^3 \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^3} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 (d+e x)} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d}+\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d}\\ &=\frac{3 b e n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3+\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}-\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}-\frac{\left (6 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}\\ &=\frac{3 b e n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^2}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^2}-\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}-\frac{\left (6 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}+\frac{\left (6 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}\\ &=\frac{3 b e n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}-\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^2}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^2}+\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^2}-\frac{\left (6 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}\\ &=\frac{3 b e n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}-\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^2}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^2}+\frac{6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^2}-\frac{6 b^3 e^2 n^3 \text{Li}_3\left (1+\frac{e}{d \sqrt{x}}\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.665092, size = 476, normalized size = 1.83 \[ \frac{3 b^2 n^2 \left (2 e^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+\left (d^2 x-e^2\right ) \log ^2\left (d+\frac{e}{\sqrt{x}}\right )-2 e^2 \log \left (-\frac{e}{d \sqrt{x}}\right )+2 e \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (e \log \left (-\frac{e}{d \sqrt{x}}\right )+d \sqrt{x}+e\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )+b^3 n^3 \left (-6 e^2 \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right )+6 e^2 \left (\log \left (d+\frac{e}{\sqrt{x}}\right )-1\right ) \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+\log \left (d+\frac{e}{\sqrt{x}}\right ) \left (\left (d^2 x-e^2\right ) \log ^2\left (d+\frac{e}{\sqrt{x}}\right )-6 e^2 \log \left (-\frac{e}{d \sqrt{x}}\right )+3 e \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (e \log \left (-\frac{e}{d \sqrt{x}}\right )+d \sqrt{x}+e\right )\right )\right )+3 b d^2 n x \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2+d^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^3-3 b e^2 n \log \left (d \sqrt{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2+3 b d e n \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2}{d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^n])^3,x]

[Out]

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

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Maple [F]  time = 0.427, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b^{3} x \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{3} - 3 \,{\left (e n{\left (\frac{e \log \left (d \sqrt{x} + e\right )}{d^{2}} - \frac{\sqrt{x}}{d}\right )} - x \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )\right )} a^{2} b + a^{3} x - \int \frac{2 \,{\left (b^{3} d x + b^{3} e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{3} + 3 \,{\left (b^{3} d n x - 2 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x + 2 \,{\left (b^{3} d x + b^{3} e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right ) - 2 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} \sqrt{x}\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{2} - 6 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} - 2 \,{\left (b^{3} d \log \left (c\right )^{3} + 3 \, a b^{2} d \log \left (c\right )^{2}\right )} x - 6 \,{\left ({\left (b^{3} d x + b^{3} e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} +{\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right )\right )} x - 2 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right ) +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right )\right )} \sqrt{x}\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right ) + 6 \,{\left ({\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right )\right )} x +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right )\right )} \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right ) - 2 \,{\left (b^{3} e \log \left (c\right )^{3} + 3 \, a b^{2} e \log \left (c\right )^{2}\right )} \sqrt{x}}{2 \,{\left (d x + e \sqrt{x}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3,x, algorithm="maxima")

[Out]

b^3*x*log((d*sqrt(x) + e)^n)^3 - 3*(e*n*(e*log(d*sqrt(x) + e)/d^2 - sqrt(x)/d) - x*log(c*(d + e/sqrt(x))^n))*a
^2*b + a^3*x - integrate(1/2*(2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n))^3 + 3*(b^3*d*n*x - 2*(b^3*d*log(c) +
a*b^2*d)*x + 2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c) + a*b^2*e)*sqrt(x))*log((d*sqrt(x) +
e)^n)^2 - 6*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n))^2 - 2*(b^3*d*log(c
)^3 + 3*a*b^2*d*log(c)^2)*x - 6*((b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n))^2 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(
c))*x - 2*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n)) + (b^3*e*log(c)^2 + 2
*a*b^2*e*log(c))*sqrt(x))*log((d*sqrt(x) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c))*x + (b^3*e*log(c)^2
+ 2*a*b^2*e*log(c))*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2)*sqrt(x))/(d*x + e*sqrt(x
)), 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 x + e \sqrt{x}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\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/x^(1/2))^n))^3,x, algorithm="fricas")

[Out]

integral(b^3*log(c*((d*x + e*sqrt(x))/x)^n)^3 + 3*a*b^2*log(c*((d*x + e*sqrt(x))/x)^n)^2 + 3*a^2*b*log(c*((d*x
+ e*sqrt(x))/x)^n) + a^3, 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((a+b*ln(c*(d+e/x**(1/2))**n))**3,x)

[Out]

Timed out

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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}{\sqrt{x}}\right )}^{n}\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/x^(1/2))^n))^3,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/sqrt(x))^n) + a)^3, x)