3.435 \(\int \frac{(a+b \log (c (d+\frac{e}{\sqrt{x}})^n))^2}{x^4} \, dx\)

Optimal. Leaf size=480 \[ \frac{2 b d^6 n \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 e^6}-\frac{4 b d^5 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^6}+\frac{5 b d^4 n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^6}-\frac{40 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{9 e^6}+\frac{5 b d^2 n \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^6}-\frac{4 b d n \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{5 e^6}+\frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{9 e^6}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}+\frac{4 b^2 d^5 n^2}{e^5 \sqrt{x}}-\frac{5 b^2 d^4 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^6}+\frac{40 b^2 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}-\frac{5 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{8 e^6}-\frac{b^2 d^6 n^2 \log ^2\left (d+\frac{e}{\sqrt{x}}\right )}{3 e^6}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^5}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^6}{54 e^6} \]

[Out]

(-5*b^2*d^4*n^2*(d + e/Sqrt[x])^2)/(2*e^6) + (40*b^2*d^3*n^2*(d + e/Sqrt[x])^3)/(27*e^6) - (5*b^2*d^2*n^2*(d +
e/Sqrt[x])^4)/(8*e^6) + (4*b^2*d*n^2*(d + e/Sqrt[x])^5)/(25*e^6) - (b^2*n^2*(d + e/Sqrt[x])^6)/(54*e^6) + (4*
b^2*d^5*n^2)/(e^5*Sqrt[x]) - (b^2*d^6*n^2*Log[d + e/Sqrt[x]]^2)/(3*e^6) - (4*b*d^5*n*(d + e/Sqrt[x])*(a + b*Lo
g[c*(d + e/Sqrt[x])^n]))/e^6 + (5*b*d^4*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^6 - (40*b*d^3*
n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) + (5*b*d^2*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d +
e/Sqrt[x])^n]))/(2*e^6) - (4*b*d*n*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(5*e^6) + (b*n*(d + e/
Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) + (2*b*d^6*n*Log[d + e/Sqrt[x]]*(a + b*Log[c*(d + e/Sqrt[
x])^n]))/(3*e^6) - (a + b*Log[c*(d + e/Sqrt[x])^n])^2/(3*x^3)

________________________________________________________________________________________

Rubi [A]  time = 0.470705, antiderivative size = 355, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{1}{90} b n \left (\frac{360 d^5 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{72 d \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{\sqrt{x}}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}+\frac{4 b^2 d^5 n^2}{e^5 \sqrt{x}}-\frac{5 b^2 d^4 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^6}+\frac{40 b^2 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}-\frac{5 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{8 e^6}-\frac{b^2 d^6 n^2 \log ^2\left (d+\frac{e}{\sqrt{x}}\right )}{3 e^6}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^5}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^6}{54 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x^4,x]

[Out]

(-5*b^2*d^4*n^2*(d + e/Sqrt[x])^2)/(2*e^6) + (40*b^2*d^3*n^2*(d + e/Sqrt[x])^3)/(27*e^6) - (5*b^2*d^2*n^2*(d +
e/Sqrt[x])^4)/(8*e^6) + (4*b^2*d*n^2*(d + e/Sqrt[x])^5)/(25*e^6) - (b^2*n^2*(d + e/Sqrt[x])^6)/(54*e^6) + (4*
b^2*d^5*n^2)/(e^5*Sqrt[x]) - (b^2*d^6*n^2*Log[d + e/Sqrt[x]]^2)/(3*e^6) - (b*n*((360*d^5*(d + e/Sqrt[x]))/e^6
- (450*d^4*(d + e/Sqrt[x])^2)/e^6 + (400*d^3*(d + e/Sqrt[x])^3)/e^6 - (225*d^2*(d + e/Sqrt[x])^4)/e^6 + (72*d*
(d + e/Sqrt[x])^5)/e^6 - (10*(d + e/Sqrt[x])^6)/e^6 - (60*d^6*Log[d + e/Sqrt[x]])/e^6)*(a + b*Log[c*(d + e/Sqr
t[x])^n]))/90 - (a + b*Log[c*(d + e/Sqrt[x])^n])^2/(3*x^3)

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{x^4} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}+\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \frac{x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}+\frac{1}{3} (2 b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=-\frac{1}{90} b n \left (\frac{360 d^5 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{\sqrt{x}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}-\frac{1}{3} \left (2 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=-\frac{1}{90} b n \left (\frac{360 d^5 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{\sqrt{x}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}-\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{90 e^6}\\ &=-\frac{1}{90} b n \left (\frac{360 d^5 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{\sqrt{x}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}-\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac{60 d^6 \log (x)}{x}\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{90 e^6}\\ &=-\frac{5 b^2 d^4 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^6}+\frac{40 b^2 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}-\frac{5 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{8 e^6}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^5}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^6}{54 e^6}+\frac{4 b^2 d^5 n^2}{e^5 \sqrt{x}}-\frac{1}{90} b n \left (\frac{360 d^5 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{\sqrt{x}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}-\frac{\left (2 b^2 d^6 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{3 e^6}\\ &=-\frac{5 b^2 d^4 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^6}+\frac{40 b^2 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}-\frac{5 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{8 e^6}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^5}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^6}{54 e^6}+\frac{4 b^2 d^5 n^2}{e^5 \sqrt{x}}-\frac{b^2 d^6 n^2 \log ^2\left (d+\frac{e}{\sqrt{x}}\right )}{3 e^6}-\frac{1}{90} b n \left (\frac{360 d^5 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{\sqrt{x}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 x^3}\\ \end{align*}

Mathematica [C]  time = 0.34083, size = 692, normalized size = 1.44 \[ \frac{3600 b^2 d^6 n^2 x^3 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+3600 b^2 d^6 n^2 x^3 \text{PolyLog}\left (2,\frac{d \sqrt{x}}{e}+1\right )-1800 a^2 e^6-3600 a b e^6 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+1800 a b d^4 e^2 n x^2-1200 a b d^3 e^3 n x^{3/2}+900 a b d^2 e^4 n x-3600 a b d^5 e n x^{5/2}+3600 a b d^6 n x^3 \log \left (d \sqrt{x}+e\right )+3600 a b d^6 n x^3 \log \left (-\frac{e}{d \sqrt{x}}\right )-720 a b d e^5 n \sqrt{x}+600 a b e^6 n+1800 b^2 d^4 e^2 n x^2 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-1200 b^2 d^3 e^3 n x^{3/2} \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+900 b^2 d^2 e^4 n x \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-3600 b^2 d^6 n x^3 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+3600 b^2 d^6 n x^3 \log \left (d \sqrt{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+3600 b^2 d^6 n x^3 \log \left (-\frac{e}{d \sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-3600 b^2 d^5 e n x^{5/2} \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-1800 b^2 e^6 \log ^2\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-720 b^2 d e^5 n \sqrt{x} \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+600 b^2 e^6 n \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-2610 b^2 d^4 e^2 n^2 x^2+1140 b^2 d^3 e^3 n^2 x^{3/2}-555 b^2 d^2 e^4 n^2 x+8820 b^2 d^5 e n^2 x^{5/2}-1800 b^2 d^6 n^2 x^3 \log ^2\left (d \sqrt{x}+e\right )-5220 b^2 d^6 n^2 x^3 \log \left (d+\frac{e}{\sqrt{x}}\right )+3600 b^2 d^6 n^2 x^3 \log \left (d \sqrt{x}+e\right ) \log \left (-\frac{d \sqrt{x}}{e}\right )+264 b^2 d e^5 n^2 \sqrt{x}-100 b^2 e^6 n^2}{5400 e^6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x^4,x]

[Out]

(-1800*a^2*e^6 + 600*a*b*e^6*n - 100*b^2*e^6*n^2 - 720*a*b*d*e^5*n*Sqrt[x] + 264*b^2*d*e^5*n^2*Sqrt[x] + 900*a
*b*d^2*e^4*n*x - 555*b^2*d^2*e^4*n^2*x - 1200*a*b*d^3*e^3*n*x^(3/2) + 1140*b^2*d^3*e^3*n^2*x^(3/2) + 1800*a*b*
d^4*e^2*n*x^2 - 2610*b^2*d^4*e^2*n^2*x^2 - 3600*a*b*d^5*e*n*x^(5/2) + 8820*b^2*d^5*e*n^2*x^(5/2) - 5220*b^2*d^
6*n^2*x^3*Log[d + e/Sqrt[x]] - 3600*a*b*e^6*Log[c*(d + e/Sqrt[x])^n] + 600*b^2*e^6*n*Log[c*(d + e/Sqrt[x])^n]
- 720*b^2*d*e^5*n*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] + 900*b^2*d^2*e^4*n*x*Log[c*(d + e/Sqrt[x])^n] - 1200*b^2*d
^3*e^3*n*x^(3/2)*Log[c*(d + e/Sqrt[x])^n] + 1800*b^2*d^4*e^2*n*x^2*Log[c*(d + e/Sqrt[x])^n] - 3600*b^2*d^5*e*n
*x^(5/2)*Log[c*(d + e/Sqrt[x])^n] - 3600*b^2*d^6*n*x^3*Log[c*(d + e/Sqrt[x])^n] - 1800*b^2*e^6*Log[c*(d + e/Sq
rt[x])^n]^2 + 3600*a*b*d^6*n*x^3*Log[e + d*Sqrt[x]] + 3600*b^2*d^6*n*x^3*Log[c*(d + e/Sqrt[x])^n]*Log[e + d*Sq
rt[x]] - 1800*b^2*d^6*n^2*x^3*Log[e + d*Sqrt[x]]^2 + 3600*a*b*d^6*n*x^3*Log[-(e/(d*Sqrt[x]))] + 3600*b^2*d^6*n
*x^3*Log[c*(d + e/Sqrt[x])^n]*Log[-(e/(d*Sqrt[x]))] + 3600*b^2*d^6*n^2*x^3*Log[e + d*Sqrt[x]]*Log[-((d*Sqrt[x]
)/e)] + 3600*b^2*d^6*n^2*x^3*PolyLog[2, 1 + e/(d*Sqrt[x])] + 3600*b^2*d^6*n^2*x^3*PolyLog[2, 1 + (d*Sqrt[x])/e
])/(5400*e^6*x^3)

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Maple [F]  time = 0.339, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{2}}\, 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))^2/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/2))^n))^2/x^4,x)

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Maxima [A]  time = 1.07684, size = 522, normalized size = 1.09 \begin{align*} \frac{1}{90} \, a b e n{\left (\frac{60 \, d^{6} \log \left (d \sqrt{x} + e\right )}{e^{7}} - \frac{30 \, d^{6} \log \left (x\right )}{e^{7}} - \frac{60 \, d^{5} x^{\frac{5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac{3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt{x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} + \frac{1}{5400} \,{\left (60 \, e n{\left (\frac{60 \, d^{6} \log \left (d \sqrt{x} + e\right )}{e^{7}} - \frac{30 \, d^{6} \log \left (x\right )}{e^{7}} - \frac{60 \, d^{5} x^{\frac{5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac{3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt{x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) - \frac{{\left (1800 \, d^{6} x^{3} \log \left (d \sqrt{x} + e\right )^{2} + 450 \, d^{6} x^{3} \log \left (x\right )^{2} - 4410 \, d^{6} x^{3} \log \left (x\right ) - 8820 \, d^{5} e x^{\frac{5}{2}} + 2610 \, d^{4} e^{2} x^{2} - 1140 \, d^{3} e^{3} x^{\frac{3}{2}} + 555 \, d^{2} e^{4} x - 264 \, d e^{5} \sqrt{x} + 100 \, e^{6} - 180 \,{\left (10 \, d^{6} x^{3} \log \left (x\right ) - 49 \, d^{6} x^{3}\right )} \log \left (d \sqrt{x} + e\right )\right )} n^{2}}{e^{6} x^{3}}\right )} b^{2} - \frac{b^{2} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )^{2}}{3 \, x^{3}} - \frac{2 \, a b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )}{3 \, x^{3}} - \frac{a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x^4,x, algorithm="maxima")

[Out]

1/90*a*b*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*
x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3)) + 1/5400*(60*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7
- 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) -
10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt(x))^n) - (1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 441
0*d^6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x - 264*d*e^5*sq
rt(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e))*n^2/(e^6*x^3))*b^2 - 1/3*b^2*log(c*
(d + e/sqrt(x))^n)^2/x^3 - 2/3*a*b*log(c*(d + e/sqrt(x))^n)/x^3 - 1/3*a^2/x^3

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Fricas [A]  time = 1.84532, size = 1094, normalized size = 2.28 \begin{align*} -\frac{100 \, b^{2} e^{6} n^{2} + 1800 \, b^{2} e^{6} \log \left (c\right )^{2} - 600 \, a b e^{6} n + 1800 \, a^{2} e^{6} + 90 \,{\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x^{2} - 1800 \,{\left (b^{2} d^{6} n^{2} x^{3} - b^{2} e^{6} n^{2}\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right )^{2} + 15 \,{\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x - 300 \,{\left (6 \, b^{2} d^{4} e^{2} n x^{2} + 3 \, b^{2} d^{2} e^{4} n x + 2 \, b^{2} e^{6} n - 12 \, a b e^{6}\right )} \log \left (c\right ) - 60 \,{\left (30 \, b^{2} d^{4} e^{2} n^{2} x^{2} + 15 \, b^{2} d^{2} e^{4} n^{2} x + 10 \, b^{2} e^{6} n^{2} - 60 \, a b e^{6} n - 3 \,{\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{3} + 60 \,{\left (b^{2} d^{6} n x^{3} - b^{2} e^{6} n\right )} \log \left (c\right ) - 4 \,{\left (15 \, b^{2} d^{5} e n^{2} x^{2} + 5 \, b^{2} d^{3} e^{3} n^{2} x + 3 \, b^{2} d e^{5} n^{2}\right )} \sqrt{x}\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) - 12 \,{\left (22 \, b^{2} d e^{5} n^{2} - 60 \, a b d e^{5} n + 15 \,{\left (49 \, b^{2} d^{5} e n^{2} - 20 \, a b d^{5} e n\right )} x^{2} + 5 \,{\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x - 20 \,{\left (15 \, b^{2} d^{5} e n x^{2} + 5 \, b^{2} d^{3} e^{3} n x + 3 \, b^{2} d e^{5} n\right )} \log \left (c\right )\right )} \sqrt{x}}{5400 \, e^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x^4,x, algorithm="fricas")

[Out]

-1/5400*(100*b^2*e^6*n^2 + 1800*b^2*e^6*log(c)^2 - 600*a*b*e^6*n + 1800*a^2*e^6 + 90*(29*b^2*d^4*e^2*n^2 - 20*
a*b*d^4*e^2*n)*x^2 - 1800*(b^2*d^6*n^2*x^3 - b^2*e^6*n^2)*log((d*x + e*sqrt(x))/x)^2 + 15*(37*b^2*d^2*e^4*n^2
- 60*a*b*d^2*e^4*n)*x - 300*(6*b^2*d^4*e^2*n*x^2 + 3*b^2*d^2*e^4*n*x + 2*b^2*e^6*n - 12*a*b*e^6)*log(c) - 60*(
30*b^2*d^4*e^2*n^2*x^2 + 15*b^2*d^2*e^4*n^2*x + 10*b^2*e^6*n^2 - 60*a*b*e^6*n - 3*(49*b^2*d^6*n^2 - 20*a*b*d^6
*n)*x^3 + 60*(b^2*d^6*n*x^3 - b^2*e^6*n)*log(c) - 4*(15*b^2*d^5*e*n^2*x^2 + 5*b^2*d^3*e^3*n^2*x + 3*b^2*d*e^5*
n^2)*sqrt(x))*log((d*x + e*sqrt(x))/x) - 12*(22*b^2*d*e^5*n^2 - 60*a*b*d*e^5*n + 15*(49*b^2*d^5*e*n^2 - 20*a*b
*d^5*e*n)*x^2 + 5*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x - 20*(15*b^2*d^5*e*n*x^2 + 5*b^2*d^3*e^3*n*x + 3*b
^2*d*e^5*n)*log(c))*sqrt(x))/(e^6*x^3)

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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))**2/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) + a\right )}^{2}}{x^{4}}\,{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))^2/x^4,x, algorithm="giac")

[Out]

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