\(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^3}{x^3} \, dx\) [420]

Optimal result

Integrand size = 24, antiderivative size = 573 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^4} \]

[Out]

-3/2*b^3*e^4*n^3*ln(x)/d^4+1/2*b^3*e^4*n^3*ln(d+e*x^(1/2))/d^4-1/2*b^2*e^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))/d^2
/x+3*b^2*e^4*n^2*ln(-e*x^(1/2)/d)*(a+b*ln(c*(d+e*x^(1/2))^n))/d^4-1/2*b*e*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2/d/x^
(3/2)+3/4*b*e^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2/d^2/x-1/2*(a+b*ln(c*(d+e*x^(1/2))^n))^3/x^2+5/2*b^2*e^4*n^2*(a
+b*ln(c*(d+e*x^(1/2))^n))*ln(1-d/(d+e*x^(1/2)))/d^4-3/2*b*e^4*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*ln(1-d/(d+e*x^(1
/2)))/d^4-5/2*b^3*e^4*n^3*polylog(2,d/(d+e*x^(1/2)))/d^4+3*b^2*e^4*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*polylog(2,d
/(d+e*x^(1/2)))/d^4+3*b^3*e^4*n^3*polylog(2,1+e*x^(1/2)/d)/d^4+3*b^3*e^4*n^3*polylog(3,d/(d+e*x^(1/2)))/d^4-1/
2*b^3*e^3*n^3/d^3/x^(1/2)+5/2*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))/d^4/x^(1/2)-3/2*b*e^3*n*(a
+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))/d^4/x^(1/2)

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\frac {3 b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}-\frac {5 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} e}{d}+1\right )}{d^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}} \]

[In]

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

[Out]

-1/2*(b^3*e^3*n^3)/(d^3*Sqrt[x]) + (b^3*e^4*n^3*Log[d + e*Sqrt[x]])/(2*d^4) - (b^2*e^2*n^2*(a + b*Log[c*(d + e
*Sqrt[x])^n]))/(2*d^2*x) + (5*b^2*e^3*n^2*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^4*Sqrt[x]) +
(5*b^2*e^4*n^2*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^4) - (b*e*n*(a + b*Log[c*(d +
e*Sqrt[x])^n])^2)/(2*d*x^(3/2)) + (3*b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*d^2*x) - (3*b*e^3*n*(d +
e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*d^4*Sqrt[x]) - (3*b*e^4*n*Log[1 - d/(d + e*Sqrt[x])]*(a + b*
Log[c*(d + e*Sqrt[x])^n])^2)/(2*d^4) - (a + b*Log[c*(d + e*Sqrt[x])^n])^3/(2*x^2) + (3*b^2*e^4*n^2*(a + b*Log[
c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)])/d^4 - (3*b^3*e^4*n^3*Log[x])/(2*d^4) - (5*b^3*e^4*n^3*PolyLog[2,
d/(d + e*Sqrt[x])])/(2*d^4) + (3*b^2*e^4*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n])*PolyLog[2, d/(d + e*Sqrt[x])])/d
^4 + (3*b^3*e^4*n^3*PolyLog[2, 1 + (e*Sqrt[x])/d])/d^4 + (3*b^3*e^4*n^3*PolyLog[3, d/(d + e*Sqrt[x])])/d^4

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2354

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 2355

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 2356

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

Rule 2379

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

Rule 2389

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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]

Rule 2445

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 2458

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 2504

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 6724

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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^5} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 (d+e x)} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {1}{2} (3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt {x}\right )}{2 d}-\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{2 d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}-\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{2 d^2}+\frac {\left (3 b e^2 n\right ) \text {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+e \sqrt {x}\right )}{2 d^2}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {\left (3 b e^2 n\right ) \text {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+e \sqrt {x}\right )}{2 d^3}-\frac {\left (3 b e^3 n\right ) \text {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+e \sqrt {x}\right )}{2 d^3}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{d^2}-\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{d^2}-\frac {\left (3 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{2 d^2} \\ & = -\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}-\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{d^3}-\frac {\left (3 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{2 d^3}+\frac {\left (3 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^4}+\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt {x}\right )}{d^3}+\frac {\left (3 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt {x}\right )}{2 d^3}+\frac {\left (3 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^4}+\frac {\left (b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt {x}\right )}{2 d^2} \\ & = -\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {\left (b^3 e^2 n^3\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+e \sqrt {x}\right )}{2 d^2}-\frac {\left (b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^4}-\frac {\left (3 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{2 d^4}-\frac {\left (b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^4}-\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 d^4}-\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^4}-\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^4} \\ & = -\frac {b^3 e^3 n^3}{2 d^3 \sqrt {x}}+\frac {b^3 e^4 n^3 \log \left (d+e \sqrt {x}\right )}{2 d^4}-\frac {b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}+\frac {5 b^2 e^3 n^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4 \sqrt {x}}+\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d x^{3/2}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 d^2 x}-\frac {3 b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4 \sqrt {x}}-\frac {3 b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {5 b^3 e^4 n^3 \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{2 d^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^4}+\frac {3 b^3 e^4 n^3 \text {Li}_3\left (\frac {d}{d+e \sqrt {x}}\right )}{d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {2 b d^3 e n \sqrt {x} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-3 b d^2 e^2 n x \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+6 b d e^3 n x^{3/2} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+6 b d^4 n \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-6 b e^4 n x^2 \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+2 d^4 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+3 b e^4 n x^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log (x)-2 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (-3 \left (d^4-e^4 x^2\right ) \log ^2\left (d+e \sqrt {x}\right )+e^2 x \left (-d^2+5 d e \sqrt {x}+11 e^2 x \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-\log \left (d+e \sqrt {x}\right ) \left (2 d^3 e \sqrt {x}-3 d^2 e^2 x+6 d e^3 x^{3/2}+11 e^4 x^2+6 e^4 x^2 \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-6 e^4 x^2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )+b^3 n^3 \left (d^2 e^2 x \left (2-3 \log \left (d+e \sqrt {x}\right )\right ) \log \left (d+e \sqrt {x}\right )+2 d^3 e \sqrt {x} \log ^2\left (d+e \sqrt {x}\right )+2 d^4 \log ^3\left (d+e \sqrt {x}\right )+2 d e^3 x^{3/2} \left (1-5 \log \left (d+e \sqrt {x}\right )+3 \log ^2\left (d+e \sqrt {x}\right )\right )+12 e^4 x^2 \left (-\log \left (d+e \sqrt {x}\right )+\log \left (-\frac {e \sqrt {x}}{d}\right )\right )+11 e^4 x^2 \left (\log \left (d+e \sqrt {x}\right ) \left (\log \left (d+e \sqrt {x}\right )-2 \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )-2 e^4 x^2 \left (\log ^2\left (d+e \sqrt {x}\right ) \left (\log \left (d+e \sqrt {x}\right )-3 \log \left (-\frac {e \sqrt {x}}{d}\right )\right )-6 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )+6 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right )\right )}{4 d^4 x^2} \]

[In]

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

[Out]

-1/4*(2*b*d^3*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*e^2*n*x*(a - b
*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2 + 6*b*d*e^3*n*x^(3/2)*(a - b*n*Log[d + e*Sqrt[x]] + b*Lo
g[c*(d + e*Sqrt[x])^n])^2 + 6*b*d^4*n*Log[d + e*Sqrt[x]]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])
^n])^2 - 6*b*e^4*n*x^2*Log[d + e*Sqrt[x]]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2 + 2*d^4*
(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^3 + 3*b*e^4*n*x^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Lo
g[c*(d + e*Sqrt[x])^n])^2*Log[x] - 2*b^2*n^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])*(-3*(d^
4 - e^4*x^2)*Log[d + e*Sqrt[x]]^2 + e^2*x*(-d^2 + 5*d*e*Sqrt[x] + 11*e^2*x*Log[-((e*Sqrt[x])/d)]) - Log[d + e*
Sqrt[x]]*(2*d^3*e*Sqrt[x] - 3*d^2*e^2*x + 6*d*e^3*x^(3/2) + 11*e^4*x^2 + 6*e^4*x^2*Log[-((e*Sqrt[x])/d)]) - 6*
e^4*x^2*PolyLog[2, 1 + (e*Sqrt[x])/d]) + b^3*n^3*(d^2*e^2*x*(2 - 3*Log[d + e*Sqrt[x]])*Log[d + e*Sqrt[x]] + 2*
d^3*e*Sqrt[x]*Log[d + e*Sqrt[x]]^2 + 2*d^4*Log[d + e*Sqrt[x]]^3 + 2*d*e^3*x^(3/2)*(1 - 5*Log[d + e*Sqrt[x]] +
3*Log[d + e*Sqrt[x]]^2) + 12*e^4*x^2*(-Log[d + e*Sqrt[x]] + Log[-((e*Sqrt[x])/d)]) + 11*e^4*x^2*(Log[d + e*Sqr
t[x]]*(Log[d + e*Sqrt[x]] - 2*Log[-((e*Sqrt[x])/d)]) - 2*PolyLog[2, 1 + (e*Sqrt[x])/d]) - 2*e^4*x^2*(Log[d + e
*Sqrt[x]]^2*(Log[d + e*Sqrt[x]] - 3*Log[-((e*Sqrt[x])/d)]) - 6*Log[d + e*Sqrt[x]]*PolyLog[2, 1 + (e*Sqrt[x])/d
] + 6*PolyLog[3, 1 + (e*Sqrt[x])/d])))/(d^4*x^2)

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

-1/2*b^3*log((e*sqrt(x) + d)^n)^3/x^2 + integrate(1/4*(3*(b^3*e*n*x + 4*(b^3*e*log(c) + a*b^2*e)*x + 4*(b^3*d*
log(c) + a*b^2*d)*sqrt(x))*log((e*sqrt(x) + d)^n)^2 + 4*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c
) + a^3*e)*x + 12*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*
b*d)*sqrt(x))*log((e*sqrt(x) + d)^n) + 4*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*sqrt
(x))/(e*x^4 + d*x^(7/2)), x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x^3} \,d x \]

[In]

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

[Out]

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