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

Optimal. Leaf size=408 \[ \frac{2 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt{x}}\right )}{3 d^6}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{6 d^2 x^2}-\frac{2 b e^6 n \log \left (1-\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d^6}-\frac{2 b e^5 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d^6 \sqrt{x}}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d^4 x}-\frac{2 b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{15 d x^{5/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{3 x^3}+\frac{b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac{b^2 e^2 n^2}{30 d^2 x^2}+\frac{77 b^2 e^5 n^2}{90 d^5 \sqrt{x}}-\frac{47 b^2 e^4 n^2}{180 d^4 x}-\frac{77 b^2 e^6 n^2 \log \left (d+e \sqrt{x}\right )}{90 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]

[Out]

-(b^2*e^2*n^2)/(30*d^2*x^2) + (b^2*e^3*n^2)/(10*d^3*x^(3/2)) - (47*b^2*e^4*n^2)/(180*d^4*x) + (77*b^2*e^5*n^2)
/(90*d^5*Sqrt[x]) - (77*b^2*e^6*n^2*Log[d + e*Sqrt[x]])/(90*d^6) - (2*b*e*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/
(15*d*x^(5/2)) + (b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(6*d^2*x^2) - (2*b*e^3*n*(a + b*Log[c*(d + e*Sqrt[
x])^n]))/(9*d^3*x^(3/2)) + (b*e^4*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d^4*x) - (2*b*e^5*n*(d + e*Sqrt[x])*(
a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d^6*Sqrt[x]) - (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) - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/(3*x^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)

________________________________________________________________________________________

Rubi [A]  time = 1.03361, antiderivative size = 432, normalized size of antiderivative = 1.06, number of steps used = 26, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{2 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{3 d^6}-\frac{2 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{9 d^3 x^{3/2}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{6 d^2 x^2}+\frac{e^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{3 d^6}-\frac{2 b e^6 n \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d^6}-\frac{2 b e^5 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d^6 \sqrt{x}}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 d^4 x}-\frac{2 b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{15 d x^{5/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{3 x^3}+\frac{b^2 e^3 n^2}{10 d^3 x^{3/2}}-\frac{b^2 e^2 n^2}{30 d^2 x^2}+\frac{77 b^2 e^5 n^2}{90 d^5 \sqrt{x}}-\frac{47 b^2 e^4 n^2}{180 d^4 x}-\frac{77 b^2 e^6 n^2 \log \left (d+e \sqrt{x}\right )}{90 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2347

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 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

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]

Rule 2317

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 2391

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 2314

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2319

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 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.279603, size = 538, normalized size = 1.32 \[ -\frac{120 b^2 e^6 n^2 x^3 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )+60 a^2 d^6-60 a^2 e^6 x^3+120 a b d^6 \log \left (c \left (d+e \sqrt{x}\right )^n\right )-120 a b e^6 x^3 \log \left (c \left (d+e \sqrt{x}\right )^n\right )+40 a b d^3 e^3 n x^{3/2}-60 a b d^2 e^4 n x^2-30 a b d^4 e^2 n x+24 a b d^5 e n \sqrt{x}+120 a b d e^5 n x^{5/2}+120 a b e^6 n x^3 \log \left (-\frac{e \sqrt{x}}{d}\right )+40 b^2 d^3 e^3 n x^{3/2} \log \left (c \left (d+e \sqrt{x}\right )^n\right )-60 b^2 d^2 e^4 n x^2 \log \left (c \left (d+e \sqrt{x}\right )^n\right )-30 b^2 d^4 e^2 n x \log \left (c \left (d+e \sqrt{x}\right )^n\right )+60 b^2 d^6 \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )+24 b^2 d^5 e n \sqrt{x} \log \left (c \left (d+e \sqrt{x}\right )^n\right )-60 b^2 e^6 x^3 \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )+120 b^2 d e^5 n x^{5/2} \log \left (c \left (d+e \sqrt{x}\right )^n\right )+120 b^2 e^6 n x^3 \log \left (-\frac{e \sqrt{x}}{d}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )-18 b^2 d^3 e^3 n^2 x^{3/2}+47 b^2 d^2 e^4 n^2 x^2+6 b^2 d^4 e^2 n^2 x-154 b^2 d e^5 n^2 x^{5/2}+274 b^2 e^6 n^2 x^3 \log \left (d+e \sqrt{x}\right )-137 b^2 e^6 n^2 x^3 \log (x)}{180 d^6 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(60*a^2*d^6 + 24*a*b*d^5*e*n*Sqrt[x] - 30*a*b*d^4*e^2*n*x + 6*b^2*d^4*e^2*n^2*x + 40*a*b*d^3*e^3*n*x^(3/2) -
18*b^2*d^3*e^3*n^2*x^(3/2) - 60*a*b*d^2*e^4*n*x^2 + 47*b^2*d^2*e^4*n^2*x^2 + 120*a*b*d*e^5*n*x^(5/2) - 154*b^2
*d*e^5*n^2*x^(5/2) - 60*a^2*e^6*x^3 + 274*b^2*e^6*n^2*x^3*Log[d + e*Sqrt[x]] + 120*a*b*d^6*Log[c*(d + e*Sqrt[x
])^n] + 24*b^2*d^5*e*n*Sqrt[x]*Log[c*(d + e*Sqrt[x])^n] - 30*b^2*d^4*e^2*n*x*Log[c*(d + e*Sqrt[x])^n] + 40*b^2
*d^3*e^3*n*x^(3/2)*Log[c*(d + e*Sqrt[x])^n] - 60*b^2*d^2*e^4*n*x^2*Log[c*(d + e*Sqrt[x])^n] + 120*b^2*d*e^5*n*
x^(5/2)*Log[c*(d + e*Sqrt[x])^n] - 120*a*b*e^6*x^3*Log[c*(d + e*Sqrt[x])^n] + 60*b^2*d^6*Log[c*(d + e*Sqrt[x])
^n]^2 - 60*b^2*e^6*x^3*Log[c*(d + e*Sqrt[x])^n]^2 + 120*a*b*e^6*n*x^3*Log[-((e*Sqrt[x])/d)] + 120*b^2*e^6*n*x^
3*Log[c*(d + e*Sqrt[x])^n]*Log[-((e*Sqrt[x])/d)] - 137*b^2*e^6*n^2*x^3*Log[x] + 120*b^2*e^6*n^2*x^3*PolyLog[2,
1 + (e*Sqrt[x])/d])/(180*d^6*x^3)

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+e\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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{2}}{3 \, x^{3}} + \int \frac{3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x +{\left (b^{2} e n x + 6 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x + 6 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right ) + 3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} \sqrt{x}}{3 \,{\left (e x^{5} + d x^{\frac{9}{2}}\right )}}\,{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="maxima")

[Out]

-1/3*b^2*log((e*sqrt(x) + d)^n)^2/x^3 + integrate(1/3*(3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x + (b^2*e*
n*x + 6*(b^2*e*log(c) + a*b*e)*x + 6*(b^2*d*log(c) + a*b*d)*sqrt(x))*log((e*sqrt(x) + d)^n) + 3*(b^2*d*log(c)^
2 + 2*a*b*d*log(c) + a^2*d)*sqrt(x))/(e*x^5 + d*x^(9/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + a^{2}}{x^{4}}, x\right ) \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]

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

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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 ({\left (e \sqrt{x} + d\right )}^{n} c\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((e*sqrt(x) + d)^n*c) + a)^2/x^4, x)