\(\int x^2 (a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^2 \, dx\) [429]

Optimal result

Integrand size = 24, antiderivative size = 404 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=-\frac {77 b^2 e^5 n^2 \sqrt {x}}{90 d^5}+\frac {47 b^2 e^4 n^2 x}{180 d^4}-\frac {b^2 e^3 n^2 x^{3/2}}{10 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{90 d^6}+\frac {2 b e^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^6}-\frac {b e^4 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^4}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {2 b e^6 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {2 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{3 d^6} \]

[Out]

47/180*b^2*e^4*n^2*x/d^4-1/10*b^2*e^3*n^2*x^(3/2)/d^3+1/30*b^2*e^2*n^2*x^2/d^2+137/180*b^2*e^6*n^2*ln(x)/d^6+7
7/90*b^2*e^6*n^2*ln(d+e/x^(1/2))/d^6-1/3*b*e^4*n*x*(a+b*ln(c*(d+e/x^(1/2))^n))/d^4+2/9*b*e^3*n*x^(3/2)*(a+b*ln
(c*(d+e/x^(1/2))^n))/d^3-1/6*b*e^2*n*x^2*(a+b*ln(c*(d+e/x^(1/2))^n))/d^2+2/15*b*e*n*x^(5/2)*(a+b*ln(c*(d+e/x^(
1/2))^n))/d+2/3*b*e^6*n*ln(1-d/(d+e/x^(1/2)))*(a+b*ln(c*(d+e/x^(1/2))^n))/d^6+1/3*x^3*(a+b*ln(c*(d+e/x^(1/2))^
n))^2-2/3*b^2*e^6*n^2*polylog(2,d/(d+e/x^(1/2)))/d^6-77/90*b^2*e^5*n^2*x^(1/2)/d^5+2/3*b*e^5*n*(a+b*ln(c*(d+e/
x^(1/2))^n))*(d+e/x^(1/2))*x^(1/2)/d^6

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {2 b e^6 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^6}+\frac {2 b e^5 n \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^6}-\frac {b e^4 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^4}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {2 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{3 d^6}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{90 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {77 b^2 e^5 n^2 \sqrt {x}}{90 d^5}+\frac {47 b^2 e^4 n^2 x}{180 d^4}-\frac {b^2 e^3 n^2 x^{3/2}}{10 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2} \]

[In]

Int[x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2,x]

[Out]

(-77*b^2*e^5*n^2*Sqrt[x])/(90*d^5) + (47*b^2*e^4*n^2*x)/(180*d^4) - (b^2*e^3*n^2*x^(3/2))/(10*d^3) + (b^2*e^2*
n^2*x^2)/(30*d^2) + (77*b^2*e^6*n^2*Log[d + e/Sqrt[x]])/(90*d^6) + (2*b*e^5*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*L
og[c*(d + e/Sqrt[x])^n]))/(3*d^6) - (b*e^4*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*d^4) + (2*b*e^3*n*x^(3/2)*
(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*d^3) - (b*e^2*n*x^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(6*d^2) + (2*b*e*n*
x^(5/2)*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(15*d) + (2*b*e^6*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/S
qrt[x])^n]))/(3*d^6) + (x^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/3 + (137*b^2*e^6*n^2*Log[x])/(180*d^6) - (2*b^
2*e^6*n^2*PolyLog[2, d/(d + e/Sqrt[x])])/(3*d^6)

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 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 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])

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^7} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^6 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {1}{3} (2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt {x}}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d}+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d} \\ & = \frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d^2}-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d^2}-\frac {\left (2 b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{15 d} \\ & = -\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d^3}+\frac {\left (2 b e^3 n\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 )}{3 d^3}-\frac {\left (2 b^2 e n^2\right ) \text {Subst}\left (\int \left (-\frac {e^5}{d (d-x)^5}-\frac {e^5}{d^2 (d-x)^4}-\frac {e^5}{d^3 (d-x)^3}-\frac {e^5}{d^4 (d-x)^2}-\frac {e^5}{d^5 (d-x)}-\frac {e^5}{d^5 x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{15 d}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{6 d^2} \\ & = -\frac {2 b^2 e^5 n^2 \sqrt {x}}{15 d^5}+\frac {b^2 e^4 n^2 x}{15 d^4}-\frac {2 b^2 e^3 n^2 x^{3/2}}{45 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2}+\frac {2 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{15 d^6}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b^2 e^6 n^2 \log (x)}{15 d^6}+\frac {\left (2 b e^3 n\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 )}{3 d^4}-\frac {\left (2 b e^4 n\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 )}{3 d^4}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (\frac {e^4}{d (d-x)^4}+\frac {e^4}{d^2 (d-x)^3}+\frac {e^4}{d^3 (d-x)^2}+\frac {e^4}{d^4 (d-x)}+\frac {e^4}{d^4 x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{6 d^2}-\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{9 d^3} \\ & = -\frac {3 b^2 e^5 n^2 \sqrt {x}}{10 d^5}+\frac {3 b^2 e^4 n^2 x}{20 d^4}-\frac {b^2 e^3 n^2 x^{3/2}}{10 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2}+\frac {3 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{10 d^6}-\frac {b e^4 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^4}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {3 b^2 e^6 n^2 \log (x)}{20 d^6}-\frac {\left (2 b e^4 n\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 )}{3 d^5}+\frac {\left (2 b e^5 n\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 )}{3 d^5}-\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^3}{d (d-x)^3}-\frac {e^3}{d^2 (d-x)^2}-\frac {e^3}{d^3 (d-x)}-\frac {e^3}{d^3 x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{9 d^3}+\frac {\left (b^2 e^4 n^2\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 )}{3 d^4} \\ & = -\frac {47 b^2 e^5 n^2 \sqrt {x}}{90 d^5}+\frac {47 b^2 e^4 n^2 x}{180 d^4}-\frac {b^2 e^3 n^2 x^{3/2}}{10 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2}+\frac {47 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{90 d^6}+\frac {2 b e^5 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 )}{3 d^6}-\frac {b e^4 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^4}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {2 b e^6 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 )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {47 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {\left (b^2 e^4 n^2\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 )}{3 d^4}-\frac {\left (2 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d^6}-\frac {\left (2 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 d^6} \\ & = -\frac {77 b^2 e^5 n^2 \sqrt {x}}{90 d^5}+\frac {47 b^2 e^4 n^2 x}{180 d^4}-\frac {b^2 e^3 n^2 x^{3/2}}{10 d^3}+\frac {b^2 e^2 n^2 x^2}{30 d^2}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt {x}}\right )}{90 d^6}+\frac {2 b e^5 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 )}{3 d^6}-\frac {b e^4 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 d^4}+\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 d^3}-\frac {b e^2 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{6 d^2}+\frac {2 b e n x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{15 d}+\frac {2 b e^6 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 )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {2 b^2 e^6 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{3 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \left (x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e n \left (120 a d e^4 \sqrt {x}-154 b d e^4 n \sqrt {x}-60 a d^2 e^3 x+47 b d^2 e^3 n x+40 a d^3 e^2 x^{3/2}-18 b d^3 e^2 n x^{3/2}-30 a d^4 e x^2+6 b d^4 e n x^2+24 a d^5 x^{5/2}+214 b e^5 n \log \left (d+\frac {e}{\sqrt {x}}\right )+120 b d e^4 \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-60 b d^2 e^3 x \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+40 b d^3 e^2 x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-30 b d^4 e x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+24 b d^5 x^{5/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-120 a e^5 \log \left (e+d \sqrt {x}\right )+60 b e^5 n \log \left (e+d \sqrt {x}\right )-120 b e^5 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )+60 b e^5 n \log ^2\left (e+d \sqrt {x}\right )-120 b e^5 n \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )+107 b e^5 n \log (x)-120 b e^5 n \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )}{60 d^6}\right ) \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2,x]

[Out]

(x^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b*e*n*(120*a*d*e^4*Sqrt[x] - 154*b*d*e^4*n*Sqrt[x] - 60*a*d^2*e^3*x
+ 47*b*d^2*e^3*n*x + 40*a*d^3*e^2*x^(3/2) - 18*b*d^3*e^2*n*x^(3/2) - 30*a*d^4*e*x^2 + 6*b*d^4*e*n*x^2 + 24*a*
d^5*x^(5/2) + 214*b*e^5*n*Log[d + e/Sqrt[x]] + 120*b*d*e^4*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] - 60*b*d^2*e^3*x*L
og[c*(d + e/Sqrt[x])^n] + 40*b*d^3*e^2*x^(3/2)*Log[c*(d + e/Sqrt[x])^n] - 30*b*d^4*e*x^2*Log[c*(d + e/Sqrt[x])
^n] + 24*b*d^5*x^(5/2)*Log[c*(d + e/Sqrt[x])^n] - 120*a*e^5*Log[e + d*Sqrt[x]] + 60*b*e^5*n*Log[e + d*Sqrt[x]]
- 120*b*e^5*Log[c*(d + e/Sqrt[x])^n]*Log[e + d*Sqrt[x]] + 60*b*e^5*n*Log[e + d*Sqrt[x]]^2 - 120*b*e^5*n*Log[e
+ d*Sqrt[x]]*Log[-((d*Sqrt[x])/e)] + 107*b*e^5*n*Log[x] - 120*b*e^5*n*PolyLog[2, 1 + (d*Sqrt[x])/e]))/(60*d^6
))/3

Maple [F]

\[\int x^{2} {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int x^{2} \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

1/3*b^2*x^3*log((d*sqrt(x) + e)^n)^2 - integrate(-1/3*(3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^3 + 3*(b^
2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(5/2) + 3*(b^2*d*x^3 + b^2*e*x^(5/2))*log(x^(1/2*n))^2 - (b^2*d*n*x^3
- 6*(b^2*d*log(c) + a*b*d)*x^3 - 6*(b^2*e*log(c) + a*b*e)*x^(5/2) + 6*(b^2*d*x^3 + b^2*e*x^(5/2))*log(x^(1/2*
n)))*log((d*sqrt(x) + e)^n) - 6*((b^2*d*log(c) + a*b*d)*x^3 + (b^2*e*log(c) + a*b*e)*x^(5/2))*log(x^(1/2*n)))/
(d*x + e*sqrt(x)), x)

Giac [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2 \,d x \]

[In]

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

[Out]

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