\(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^2}{x^3} \, dx\) [413]

Optimal result

Integrand size = 24, antiderivative size = 293 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b^2 e^2 n^2}{6 d^2 x}+\frac {5 b^2 e^3 n^2}{6 d^3 \sqrt {x}}-\frac {5 b^2 e^4 n^2 \log \left (d+e \sqrt {x}\right )}{6 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4 \sqrt {x}}-\frac {b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}+\frac {b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^4} \]

[Out]

-1/6*b^2*e^2*n^2/d^2/x+11/12*b^2*e^4*n^2*ln(x)/d^4-5/6*b^2*e^4*n^2*ln(d+e*x^(1/2))/d^4-1/3*b*e*n*(a+b*ln(c*(d+
e*x^(1/2))^n))/d/x^(3/2)+1/2*b*e^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))/d^2/x-1/2*(a+b*ln(c*(d+e*x^(1/2))^n))^2/x^2-b
*e^4*n*(a+b*ln(c*(d+e*x^(1/2))^n))*ln(1-d/(d+e*x^(1/2)))/d^4+b^2*e^4*n^2*polylog(2,d/(d+e*x^(1/2)))/d^4+5/6*b^
2*e^3*n^2/d^3/x^(1/2)-b*e^3*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))/d^4/x^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 16, 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} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}-\frac {b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4 \sqrt {x}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 e^4 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^4}-\frac {5 b^2 e^4 n^2 \log \left (d+e \sqrt {x}\right )}{6 d^4}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}+\frac {5 b^2 e^3 n^2}{6 d^3 \sqrt {x}}-\frac {b^2 e^2 n^2}{6 d^2 x} \]

[In]

Int[(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^3,x]

[Out]

-1/6*(b^2*e^2*n^2)/(d^2*x) + (5*b^2*e^3*n^2)/(6*d^3*Sqrt[x]) - (5*b^2*e^4*n^2*Log[d + e*Sqrt[x]])/(6*d^4) - (b
*e*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d*x^(3/2)) + (b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^2*x) -
(b*e^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(d^4*Sqrt[x]) - (b*e^4*n*Log[1 - d/(d + e*Sqrt[x])]
*(a + b*Log[c*(d + e*Sqrt[x])^n]))/d^4 - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/(2*x^2) + (11*b^2*e^4*n^2*Log[x])/
(12*d^4) + (b^2*e^4*n^2*PolyLog[2, 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 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}& = 2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^5} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4 (d+e x)} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^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 )^4} \, dx,x,d+e \sqrt {x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {(b n) \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+e \sqrt {x}\right )}{d}-\frac {(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 )^3} \, dx,x,d+e \sqrt {x}\right )}{d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}-\frac {(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (b 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 )^2} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt {x}\right )}{3 d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {\left (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 )^2} \, dx,x,d+e \sqrt {x}\right )}{d^3}-\frac {\left (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 )} \, dx,x,d+e \sqrt {x}\right )}{d^3}+\frac {\left (b^2 e 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+e \sqrt {x}\right )}{3 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 )^2} \, dx,x,d+e \sqrt {x}\right )}{2 d^2} \\ & = -\frac {b^2 e^2 n^2}{6 d^2 x}+\frac {b^2 e^3 n^2}{3 d^3 \sqrt {x}}-\frac {b^2 e^4 n^2 \log \left (d+e \sqrt {x}\right )}{3 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4 \sqrt {x}}-\frac {b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 e^4 n^2 \log (x)}{6 d^4}-\frac {\left (b^2 e^2 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+e \sqrt {x}\right )}{2 d^2}+\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^4}+\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^4} \\ & = -\frac {b^2 e^2 n^2}{6 d^2 x}+\frac {5 b^2 e^3 n^2}{6 d^3 \sqrt {x}}-\frac {5 b^2 e^4 n^2 \log \left (d+e \sqrt {x}\right )}{6 d^4}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 d x^{3/2}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 d^2 x}-\frac {b e^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4 \sqrt {x}}-\frac {b e^4 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^4}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 x^2}+\frac {11 b^2 e^4 n^2 \log (x)}{12 d^4}+\frac {b^2 e^4 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {e \sqrt {x} \left (4 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-6 b d^2 e n \sqrt {x} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+12 b d e^2 n x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-6 e^3 x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+12 b e^3 n x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b^2 e^3 n^2 x^{3/2} \left (2 \log \left (d+e \sqrt {x}\right )-\log (x)\right )-3 b^2 e^2 n^2 x \left (2 d-2 e \sqrt {x} \log \left (d+e \sqrt {x}\right )+e \sqrt {x} \log (x)\right )+2 b^2 e n^2 \sqrt {x} \left (d \left (d-2 e \sqrt {x}\right )+2 e^2 x \log \left (d+e \sqrt {x}\right )-e^2 x \log (x)\right )+12 b^2 e^3 n^2 x^{3/2} \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )}{d^4}}{12 x^2} \]

[In]

Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^3,x]

[Out]

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

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^2}{x^3} \,d x \]

[In]

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

[Out]

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