3.5.36 \(\int x (a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3 \, dx\) [436]

Optimal. Leaf size=569 \[ \frac {b^3 e^3 n^3 \sqrt {x}}{2 d^3}-\frac {b^3 e^4 n^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {5 b^2 e^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}-\frac {5 b^2 e^4 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}+\frac {3 b e^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {3 b e^4 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}+\frac {5 b^3 e^4 n^3 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{2 d^4}-\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{d^4}-\frac {3 b^3 e^4 n^3 \text {Li}_3\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4} \]

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

________________________________________________________________________________________

Rubi [A]
time = 0.87, antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \begin {gather*} -\frac {3 b^2 e^4 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^4}+\frac {5 b^3 e^4 n^3 \text {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{2 d^4}-\frac {3 b^3 e^4 n^3 \text {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right )}{d^4}-\frac {3 b^3 e^4 n^3 \text {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^4}-\frac {5 b^2 e^4 n^2 \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^4}-\frac {3 b^2 e^4 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^4}-\frac {5 b^2 e^3 n^2 \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^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {3 b e^4 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}{2 d^4}+\frac {3 b e^3 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}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {b^3 e^4 n^3 \log \left (d+\frac {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 \sqrt {x}}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e^3*n^3*Sqrt[x])/(2*d^3) - (b^3*e^4*n^3*Log[d + e/Sqrt[x]])/(2*d^4) - (5*b^2*e^3*n^2*(d + e/Sqrt[x])*Sqrt
[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^4) + (b^2*e^2*n^2*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^2) - (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) + (3*b*e^3*n*(d + e/Sqrt[x])*
Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d^4) - (3*b*e^2*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*d^2)
+ (b*e*n*x^(3/2)*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d) + (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) + (x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/2 - (3*b^2*e^4*n^2*(a + b*Log[c*(
d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))])/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/(d*Sqrt[x])])/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*} \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx &=-\left (2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^5} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\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,\frac {1}{\sqrt {x}}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\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+\frac {e}{\sqrt {x}}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\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+\frac {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+\frac {e}{\sqrt {x}}\right )}{2 d}\\ &=\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3+\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+\frac {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+\frac {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+\frac {e}{\sqrt {x}}\right )}{d}\\ &=-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\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+\frac {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+\frac {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+\frac {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+\frac {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+\frac {e}{\sqrt {x}}\right )}{2 d^2}\\ &=\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {3 b e^3 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}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^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}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {\left (3 b e^4 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^4}+\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+\frac {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+\frac {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+\frac {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+\frac {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+\frac {e}{\sqrt {x}}\right )}{2 d^3}-\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+\frac {e}{\sqrt {x}}\right )}{2 d^2}\\ &=-\frac {5 b^2 e^3 n^2 \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^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {3 b e^3 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}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {3 b^2 e^4 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^4}+\frac {3 b e^4 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 )}{2 d^4}-\frac {\left (3 e^4\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^4}-\frac {\left (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+\frac {e}{\sqrt {x}}\right )}{d^4}-\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+\frac {e}{\sqrt {x}}\right )}{2 d^4}+\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}+\frac {\left (3 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {\left (3 b^2 e^4 n^2\right ) \text {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^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+\frac {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+\frac {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+\frac {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+\frac {e}{\sqrt {x}}\right )}{d^4}\\ &=\frac {b^3 e^3 n^3 \sqrt {x}}{2 d^3}-\frac {b^3 e^4 n^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {5 b^2 e^3 n^2 \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^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {5 b e^4 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^4}+\frac {3 b e^3 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}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}-\frac {e^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {11 b^2 e^4 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 )}{2 d^4}+\frac {3 b e^4 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 )}{2 d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {3 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{d^4}+\frac {3 b^2 e^4 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^4}+\frac {\left (b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {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 {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^4}\\ &=\frac {b^3 e^3 n^3 \sqrt {x}}{2 d^3}-\frac {b^3 e^4 n^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}-\frac {5 b^2 e^3 n^2 \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^4}+\frac {b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 d^2}+\frac {5 b e^4 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^4}+\frac {3 b e^3 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}{2 d^4}-\frac {3 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 d^2}+\frac {b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 d}-\frac {e^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {11 b^2 e^4 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 )}{2 d^4}+\frac {3 b e^4 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 )}{2 d^4}-\frac {3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac {11 b^3 e^4 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{2 d^4}+\frac {3 b^2 e^4 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^4}-\frac {3 b^3 e^4 n^3 \text {Li}_3\left (1+\frac {e}{d \sqrt {x}}\right )}{d^4}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 777, normalized size = 1.37 \begin {gather*} \frac {6 b d e^3 n \sqrt {x} \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-3 b d^2 e^2 n x \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+2 b d^3 e n x^{3/2} \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+6 b d^4 n x^2 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+2 d^4 x^2 \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-6 b e^4 n \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \log \left (e+d \sqrt {x}\right )-2 b^2 n^2 \left (a-b n \log \left (d+\frac {e}{\sqrt {x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (3 \left (e^4-d^4 x^2\right ) \log ^2\left (d+\frac {e}{\sqrt {x}}\right )+e^2 \left (5 d e \sqrt {x}-d^2 x+11 e^2 \log \left (-\frac {e}{d \sqrt {x}}\right )\right )-e \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (11 e^3+6 d e^2 \sqrt {x}-3 d^2 e x+2 d^3 x^{3/2}+6 e^3 \log \left (-\frac {e}{d \sqrt {x}}\right )\right )-6 e^4 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )\right )+b^3 n^3 \left (d^2 e^2 x \left (2-3 \log \left (d+\frac {e}{\sqrt {x}}\right )\right ) \log \left (d+\frac {e}{\sqrt {x}}\right )+2 d^3 e x^{3/2} \log ^2\left (d+\frac {e}{\sqrt {x}}\right )+2 d^4 x^2 \log ^3\left (d+\frac {e}{\sqrt {x}}\right )+2 d e^3 \sqrt {x} \left (1-5 \log \left (d+\frac {e}{\sqrt {x}}\right )+3 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )\right )+12 e^4 \left (-\log \left (d+\frac {e}{\sqrt {x}}\right )+\log \left (-\frac {e}{d \sqrt {x}}\right )\right )+11 e^4 \left (\log \left (d+\frac {e}{\sqrt {x}}\right ) \left (\log \left (d+\frac {e}{\sqrt {x}}\right )-2 \log \left (-\frac {e}{d \sqrt {x}}\right )\right )-2 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )\right )-2 e^4 \left (\log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (\log \left (d+\frac {e}{\sqrt {x}}\right )-3 \log \left (-\frac {e}{d \sqrt {x}}\right )\right )-6 \log \left (d+\frac {e}{\sqrt {x}}\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )+6 \text {Li}_3\left (1+\frac {e}{d \sqrt {x}}\right )\right )\right )}{4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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