3.5.35 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^2}{x^4} \, dx\) [435]

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

[Out]

-1/3*b^2*d^6*n^2*ln(d+e/x^(1/2))^2/e^6+2/3*b*d^6*n*ln(d+e/x^(1/2))*(a+b*ln(c*(d+e/x^(1/2))^n))/e^6-1/3*(a+b*ln
(c*(d+e/x^(1/2))^n))^2/x^3-4*b*d^5*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))/e^6-5/2*b^2*d^4*n^2*(d+e/x^(1/2
))^2/e^6+5*b*d^4*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^2/e^6+40/27*b^2*d^3*n^2*(d+e/x^(1/2))^3/e^6-40/9*
b*d^3*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^3/e^6-5/8*b^2*d^2*n^2*(d+e/x^(1/2))^4/e^6+5/2*b*d^2*n*(a+b*l
n(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^4/e^6+4/25*b^2*d*n^2*(d+e/x^(1/2))^5/e^6-4/5*b*d*n*(a+b*ln(c*(d+e/x^(1/2))
^n))*(d+e/x^(1/2))^5/e^6-1/54*b^2*n^2*(d+e/x^(1/2))^6/e^6+1/9*b*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^6/
e^6+4*b^2*d^5*n^2/e^5/x^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 480, normalized size of antiderivative = 1.00, 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 = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \begin {gather*} \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 {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}+\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 {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} \end {gather*}

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) - (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)

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 45

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 2338

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]

Rule 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 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*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx &=-\left (2 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.23, size = 692, normalized size = 1.44 \begin {gather*} \frac {-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 \left (d+\frac {e}{\sqrt {x}}\right )-3600 a b e^6 \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 )-720 b^2 d e^5 n \sqrt {x} \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 )-1200 b^2 d^3 e^3 n x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+1800 b^2 d^4 e^2 n x^2 \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 )-3600 b^2 d^6 n x^3 \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 )+3600 a b d^6 n x^3 \log \left (e+d \sqrt {x}\right )+3600 b^2 d^6 n x^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )-1800 b^2 d^6 n^2 x^3 \log ^2\left (e+d \sqrt {x}\right )+3600 a b d^6 n x^3 \log \left (-\frac {e}{d \sqrt {x}}\right )+3600 b^2 d^6 n x^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )+3600 b^2 d^6 n^2 x^3 \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )+3600 b^2 d^6 n^2 x^3 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )+3600 b^2 d^6 n^2 x^3 \text {Li}_2\left (1+\frac {d \sqrt {x}}{e}\right )}{5400 e^6 x^3} \end {gather*}

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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )^{2}}{x^{4}}\, dx\]

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 = 0.33, size = 379, normalized size = 0.79 \begin {gather*} \frac {1}{90} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (d \sqrt {x} + e\right ) - 30 \, d^{6} e^{\left (-7\right )} \log \left (x\right ) - \frac {{\left (60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} x^{2} e + 20 \, d^{3} x^{\frac {3}{2}} e^{2} - 15 \, d^{2} x e^{3} + 12 \, d \sqrt {x} e^{4} - 10 \, e^{5}\right )} e^{\left (-6\right )}}{x^{3}}\right )} a b n e + \frac {1}{5400} \, {\left (60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (d \sqrt {x} + e\right ) - 30 \, d^{6} e^{\left (-7\right )} \log \left (x\right ) - \frac {{\left (60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} x^{2} e + 20 \, d^{3} x^{\frac {3}{2}} e^{2} - 15 \, d^{2} x e^{3} + 12 \, d \sqrt {x} e^{4} - 10 \, e^{5}\right )} e^{\left (-6\right )}}{x^{3}}\right )} n e \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} x^{\frac {5}{2}} e + 2610 \, d^{4} x^{2} e^{2} - 1140 \, d^{3} x^{\frac {3}{2}} e^{3} + 555 \, d^{2} x e^{4} - 264 \, d \sqrt {x} e^{5} - 180 \, {\left (10 \, d^{6} x^{3} \log \left (x\right ) - 49 \, d^{6} x^{3}\right )} \log \left (d \sqrt {x} + e\right ) + 100 \, e^{6}\right )} n^{2} e^{\left (-6\right )}}{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 {gather*}

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*(60*d^6*e^(-7)*log(d*sqrt(x) + e) - 30*d^6*e^(-7)*log(x) - (60*d^5*x^(5/2) - 30*d^4*x^2*e + 20*d^3*x^(3/2
)*e^2 - 15*d^2*x*e^3 + 12*d*sqrt(x)*e^4 - 10*e^5)*e^(-6)/x^3)*a*b*n*e + 1/5400*(60*(60*d^6*e^(-7)*log(d*sqrt(x
) + e) - 30*d^6*e^(-7)*log(x) - (60*d^5*x^(5/2) - 30*d^4*x^2*e + 20*d^3*x^(3/2)*e^2 - 15*d^2*x*e^3 + 12*d*sqrt
(x)*e^4 - 10*e^5)*e^(-6)/x^3)*n*e*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 - 4410*d^6*x^3*log(x) - 8820*d^5*x^(5/2)*e + 2610*d^4*x^2*e^2 - 1140*d^3*x^(3/2)*e^3 + 555*d^2*x*e^4
- 264*d*sqrt(x)*e^5 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e) + 100*e^6)*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 = 0.38, size = 460, normalized size = 0.96 \begin {gather*} -\frac {{\left (1800 \, b^{2} e^{6} \log \left (c\right )^{2} + 90 \, {\left (29 \, b^{2} d^{4} n^{2} - 20 \, a b d^{4} n\right )} x^{2} e^{2} + 15 \, {\left (37 \, b^{2} d^{2} n^{2} - 60 \, a b d^{2} n\right )} x e^{4} - 1800 \, {\left (b^{2} d^{6} n^{2} x^{3} - b^{2} n^{2} e^{6}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right )^{2} + 100 \, {\left (b^{2} n^{2} - 6 \, a b n + 18 \, a^{2}\right )} e^{6} - 300 \, {\left (6 \, b^{2} d^{4} n x^{2} e^{2} + 3 \, b^{2} d^{2} n x e^{4} + 2 \, {\left (b^{2} n - 6 \, a b\right )} e^{6}\right )} \log \left (c\right ) - 60 \, {\left (30 \, b^{2} d^{4} n^{2} x^{2} e^{2} + 15 \, b^{2} d^{2} n^{2} x e^{4} - 3 \, {\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{3} + 10 \, {\left (b^{2} n^{2} - 6 \, a b n\right )} e^{6} + 60 \, {\left (b^{2} d^{6} n x^{3} - b^{2} n e^{6}\right )} \log \left (c\right ) - 4 \, {\left (15 \, b^{2} d^{5} n^{2} x^{2} e + 5 \, b^{2} d^{3} n^{2} x e^{3} + 3 \, b^{2} d n^{2} e^{5}\right )} \sqrt {x}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right ) - 12 \, {\left (15 \, {\left (49 \, b^{2} d^{5} n^{2} - 20 \, a b d^{5} n\right )} x^{2} e + 5 \, {\left (19 \, b^{2} d^{3} n^{2} - 20 \, a b d^{3} n\right )} x e^{3} + 2 \, {\left (11 \, b^{2} d n^{2} - 30 \, a b d n\right )} e^{5} - 20 \, {\left (15 \, b^{2} d^{5} n x^{2} e + 5 \, b^{2} d^{3} n x e^{3} + 3 \, b^{2} d n e^{5}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} e^{\left (-6\right )}}{5400 \, x^{3}} \end {gather*}

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*(1800*b^2*e^6*log(c)^2 + 90*(29*b^2*d^4*n^2 - 20*a*b*d^4*n)*x^2*e^2 + 15*(37*b^2*d^2*n^2 - 60*a*b*d^2*
n)*x*e^4 - 1800*(b^2*d^6*n^2*x^3 - b^2*n^2*e^6)*log((d*x + sqrt(x)*e)/x)^2 + 100*(b^2*n^2 - 6*a*b*n + 18*a^2)*
e^6 - 300*(6*b^2*d^4*n*x^2*e^2 + 3*b^2*d^2*n*x*e^4 + 2*(b^2*n - 6*a*b)*e^6)*log(c) - 60*(30*b^2*d^4*n^2*x^2*e^
2 + 15*b^2*d^2*n^2*x*e^4 - 3*(49*b^2*d^6*n^2 - 20*a*b*d^6*n)*x^3 + 10*(b^2*n^2 - 6*a*b*n)*e^6 + 60*(b^2*d^6*n*
x^3 - b^2*n*e^6)*log(c) - 4*(15*b^2*d^5*n^2*x^2*e + 5*b^2*d^3*n^2*x*e^3 + 3*b^2*d*n^2*e^5)*sqrt(x))*log((d*x +
sqrt(x)*e)/x) - 12*(15*(49*b^2*d^5*n^2 - 20*a*b*d^5*n)*x^2*e + 5*(19*b^2*d^3*n^2 - 20*a*b*d^3*n)*x*e^3 + 2*(1
1*b^2*d*n^2 - 30*a*b*d*n)*e^5 - 20*(15*b^2*d^5*n*x^2*e + 5*b^2*d^3*n*x*e^3 + 3*b^2*d*n*e^5)*log(c))*sqrt(x))*e
^(-6)/x^3

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1639 vs. \(2 (419) = 838\).
time = 3.76, size = 1639, normalized size = 3.41 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/5400*(10800*(d*sqrt(x) + e)*b^2*d^5*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/sqrt(x) - 27000*(d*sqrt(x) + e)^2*b^2
*d^4*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/x - 21600*(d*sqrt(x) + e)*b^2*d^5*n^2*log((d*sqrt(x) + e)/sqrt(x))/sqr
t(x) + 21600*(d*sqrt(x) + e)*b^2*d^5*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) + 36000*(d*sqrt(x) + e)^3*b
^2*d^3*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/x^(3/2) + 27000*(d*sqrt(x) + e)^2*b^2*d^4*n^2*log((d*sqrt(x) + e)/sq
rt(x))/x - 54000*(d*sqrt(x) + e)^2*b^2*d^4*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x - 27000*(d*sqrt(x) + e)^4*b
^2*d^2*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/x^2 + 21600*(d*sqrt(x) + e)*b^2*d^5*n^2/sqrt(x) - 21600*(d*sqrt(x) +
e)*b^2*d^5*n*log(c)/sqrt(x) + 10800*(d*sqrt(x) + e)*b^2*d^5*log(c)^2/sqrt(x) - 24000*(d*sqrt(x) + e)^3*b^2*d^
3*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 21600*(d*sqrt(x) + e)*a*b*d^5*n*log((d*sqrt(x) + e)/sqrt(x))/sqrt
(x) + 72000*(d*sqrt(x) + e)^3*b^2*d^3*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 10800*(d*sqrt(x) + e)^5*
b^2*d*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/x^(5/2) - 13500*(d*sqrt(x) + e)^2*b^2*d^4*n^2/x + 27000*(d*sqrt(x) +
e)^2*b^2*d^4*n*log(c)/x - 27000*(d*sqrt(x) + e)^2*b^2*d^4*log(c)^2/x + 13500*(d*sqrt(x) + e)^4*b^2*d^2*n^2*log
((d*sqrt(x) + e)/sqrt(x))/x^2 - 54000*(d*sqrt(x) + e)^2*a*b*d^4*n*log((d*sqrt(x) + e)/sqrt(x))/x - 54000*(d*sq
rt(x) + e)^4*b^2*d^2*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 1800*(d*sqrt(x) + e)^6*b^2*n^2*log((d*sqrt(x)
+ e)/sqrt(x))^2/x^3 + 8000*(d*sqrt(x) + e)^3*b^2*d^3*n^2/x^(3/2) - 21600*(d*sqrt(x) + e)*a*b*d^5*n/sqrt(x) -
24000*(d*sqrt(x) + e)^3*b^2*d^3*n*log(c)/x^(3/2) + 21600*(d*sqrt(x) + e)*a*b*d^5*log(c)/sqrt(x) + 36000*(d*sqr
t(x) + e)^3*b^2*d^3*log(c)^2/x^(3/2) - 4320*(d*sqrt(x) + e)^5*b^2*d*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) +
72000*(d*sqrt(x) + e)^3*a*b*d^3*n*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 21600*(d*sqrt(x) + e)^5*b^2*d*n*log(
c)*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) - 3375*(d*sqrt(x) + e)^4*b^2*d^2*n^2/x^2 + 27000*(d*sqrt(x) + e)^2*a*b
*d^4*n/x + 13500*(d*sqrt(x) + e)^4*b^2*d^2*n*log(c)/x^2 - 54000*(d*sqrt(x) + e)^2*a*b*d^4*log(c)/x - 27000*(d*
sqrt(x) + e)^4*b^2*d^2*log(c)^2/x^2 + 600*(d*sqrt(x) + e)^6*b^2*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^3 - 54000*(
d*sqrt(x) + e)^4*a*b*d^2*n*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 3600*(d*sqrt(x) + e)^6*b^2*n*log(c)*log((d*sqrt(
x) + e)/sqrt(x))/x^3 + 864*(d*sqrt(x) + e)^5*b^2*d*n^2/x^(5/2) - 24000*(d*sqrt(x) + e)^3*a*b*d^3*n/x^(3/2) + 1
0800*(d*sqrt(x) + e)*a^2*d^5/sqrt(x) - 4320*(d*sqrt(x) + e)^5*b^2*d*n*log(c)/x^(5/2) + 72000*(d*sqrt(x) + e)^3
*a*b*d^3*log(c)/x^(3/2) + 10800*(d*sqrt(x) + e)^5*b^2*d*log(c)^2/x^(5/2) + 21600*(d*sqrt(x) + e)^5*a*b*d*n*log
((d*sqrt(x) + e)/sqrt(x))/x^(5/2) - 100*(d*sqrt(x) + e)^6*b^2*n^2/x^3 + 13500*(d*sqrt(x) + e)^4*a*b*d^2*n/x^2
- 27000*(d*sqrt(x) + e)^2*a^2*d^4/x + 600*(d*sqrt(x) + e)^6*b^2*n*log(c)/x^3 - 54000*(d*sqrt(x) + e)^4*a*b*d^2
*log(c)/x^2 - 1800*(d*sqrt(x) + e)^6*b^2*log(c)^2/x^3 - 3600*(d*sqrt(x) + e)^6*a*b*n*log((d*sqrt(x) + e)/sqrt(
x))/x^3 - 4320*(d*sqrt(x) + e)^5*a*b*d*n/x^(5/2) + 36000*(d*sqrt(x) + e)^3*a^2*d^3/x^(3/2) + 21600*(d*sqrt(x)
+ e)^5*a*b*d*log(c)/x^(5/2) + 600*(d*sqrt(x) + e)^6*a*b*n/x^3 - 27000*(d*sqrt(x) + e)^4*a^2*d^2/x^2 - 3600*(d*
sqrt(x) + e)^6*a*b*log(c)/x^3 + 10800*(d*sqrt(x) + e)^5*a^2*d/x^(5/2) - 1800*(d*sqrt(x) + e)^6*a^2/x^3)*e^(-6)

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Mupad [B]
time = 1.77, size = 440, normalized size = 0.92 \begin {gather*} \frac {b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2}{3\,e^6}-\frac {b^2\,{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2}{3\,x^3}-\frac {b^2\,n^2}{54\,x^3}-\frac {2\,a\,b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,x^3}-\frac {a^2}{3\,x^3}+\frac {a\,b\,n}{9\,x^3}+\frac {b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{9\,x^3}-\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{30\,e^6}-\frac {37\,b^2\,d^2\,n^2}{360\,e^2\,x^2}-\frac {29\,b^2\,d^4\,n^2}{60\,e^4\,x}+\frac {19\,b^2\,d^3\,n^2}{90\,e^3\,x^{3/2}}+\frac {49\,b^2\,d^5\,n^2}{30\,e^5\,\sqrt {x}}+\frac {11\,b^2\,d\,n^2}{225\,e\,x^{5/2}}+\frac {b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{6\,e^2\,x^2}+\frac {b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,e^4\,x}-\frac {2\,b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{9\,e^3\,x^{3/2}}-\frac {2\,b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,e^5\,\sqrt {x}}-\frac {2\,a\,b\,d\,n}{15\,e\,x^{5/2}}+\frac {2\,a\,b\,d^6\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{3\,e^6}-\frac {2\,b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{15\,e\,x^{5/2}}+\frac {a\,b\,d^2\,n}{6\,e^2\,x^2}+\frac {a\,b\,d^4\,n}{3\,e^4\,x}-\frac {2\,a\,b\,d^3\,n}{9\,e^3\,x^{3/2}}-\frac {2\,a\,b\,d^5\,n}{3\,e^5\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*d^6*log(c*(d + e/x^(1/2))^n)^2)/(3*e^6) - (b^2*log(c*(d + e/x^(1/2))^n)^2)/(3*x^3) - (b^2*n^2)/(54*x^3) -
(2*a*b*log(c*(d + e/x^(1/2))^n))/(3*x^3) - a^2/(3*x^3) + (a*b*n)/(9*x^3) + (b^2*n*log(c*(d + e/x^(1/2))^n))/(
9*x^3) - (49*b^2*d^6*n^2*log(d + e/x^(1/2)))/(30*e^6) - (37*b^2*d^2*n^2)/(360*e^2*x^2) - (29*b^2*d^4*n^2)/(60*
e^4*x) + (19*b^2*d^3*n^2)/(90*e^3*x^(3/2)) + (49*b^2*d^5*n^2)/(30*e^5*x^(1/2)) + (11*b^2*d*n^2)/(225*e*x^(5/2)
) + (b^2*d^2*n*log(c*(d + e/x^(1/2))^n))/(6*e^2*x^2) + (b^2*d^4*n*log(c*(d + e/x^(1/2))^n))/(3*e^4*x) - (2*b^2
*d^3*n*log(c*(d + e/x^(1/2))^n))/(9*e^3*x^(3/2)) - (2*b^2*d^5*n*log(c*(d + e/x^(1/2))^n))/(3*e^5*x^(1/2)) - (2
*a*b*d*n)/(15*e*x^(5/2)) + (2*a*b*d^6*n*log(d + e/x^(1/2)))/(3*e^6) - (2*b^2*d*n*log(c*(d + e/x^(1/2))^n))/(15
*e*x^(5/2)) + (a*b*d^2*n)/(6*e^2*x^2) + (a*b*d^4*n)/(3*e^4*x) - (2*a*b*d^3*n)/(9*e^3*x^(3/2)) - (2*a*b*d^5*n)/
(3*e^5*x^(1/2))

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