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

Optimal. Leaf size=595 \[ -\frac{9 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^4}-\frac{3 b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{16 e^4}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 e^4}+\frac{12 a b^2 d^3 n^2}{e^3 \sqrt{x}}-\frac{3 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}+\frac{9 b d^2 n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^4}+\frac{2 d^3 \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{6 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 e^4}+\frac{3 b n \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{8 e^4}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{2 b d n \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}+\frac{12 b^3 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^4}+\frac{9 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^4}-\frac{12 b^3 d^3 n^3}{e^3 \sqrt{x}}+\frac{3 b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{64 e^4}-\frac{4 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{9 e^4} \]

[Out]

(9*b^3*d^2*n^3*(d + e/Sqrt[x])^2)/(4*e^4) - (4*b^3*d*n^3*(d + e/Sqrt[x])^3)/(9*e^4) + (3*b^3*n^3*(d + e/Sqrt[x
])^4)/(64*e^4) + (12*a*b^2*d^3*n^2)/(e^3*Sqrt[x]) - (12*b^3*d^3*n^3)/(e^3*Sqrt[x]) + (12*b^3*d^3*n^2*(d + e/Sq
rt[x])*Log[c*(d + e/Sqrt[x])^n])/e^4 - (9*b^2*d^2*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e
^4) + (4*b^2*d*n^2*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*e^4) - (3*b^2*n^2*(d + e/Sqrt[x])^4*
(a + b*Log[c*(d + e/Sqrt[x])^n]))/(16*e^4) - (6*b*d^3*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^
4 + (9*b*d^2*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^4) - (2*b*d*n*(d + e/Sqrt[x])^3*(a +
b*Log[c*(d + e/Sqrt[x])^n])^2)/e^4 + (3*b*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(8*e^4) + (
2*d^3*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 - (3*d^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/S
qrt[x])^n])^3)/e^4 + (2*d*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 - ((d + e/Sqrt[x])^4*(a +
b*Log[c*(d + e/Sqrt[x])^n])^3)/(2*e^4)

________________________________________________________________________________________

Rubi [A]  time = 0.641699, antiderivative size = 595, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ -\frac{9 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^4}-\frac{3 b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{16 e^4}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 e^4}+\frac{12 a b^2 d^3 n^2}{e^3 \sqrt{x}}-\frac{3 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}+\frac{9 b d^2 n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^4}+\frac{2 d^3 \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{6 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 e^4}+\frac{3 b n \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{8 e^4}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{2 b d n \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}+\frac{12 b^3 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^4}+\frac{9 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^4}-\frac{12 b^3 d^3 n^3}{e^3 \sqrt{x}}+\frac{3 b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{64 e^4}-\frac{4 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{9 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*b^3*d^2*n^3*(d + e/Sqrt[x])^2)/(4*e^4) - (4*b^3*d*n^3*(d + e/Sqrt[x])^3)/(9*e^4) + (3*b^3*n^3*(d + e/Sqrt[x
])^4)/(64*e^4) + (12*a*b^2*d^3*n^2)/(e^3*Sqrt[x]) - (12*b^3*d^3*n^3)/(e^3*Sqrt[x]) + (12*b^3*d^3*n^2*(d + e/Sq
rt[x])*Log[c*(d + e/Sqrt[x])^n])/e^4 - (9*b^2*d^2*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e
^4) + (4*b^2*d*n^2*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*e^4) - (3*b^2*n^2*(d + e/Sqrt[x])^4*
(a + b*Log[c*(d + e/Sqrt[x])^n]))/(16*e^4) - (6*b*d^3*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^
4 + (9*b*d^2*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^4) - (2*b*d*n*(d + e/Sqrt[x])^3*(a +
b*Log[c*(d + e/Sqrt[x])^n])^2)/e^4 + (3*b*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(8*e^4) + (
2*d^3*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 - (3*d^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/S
qrt[x])^n])^3)/e^4 + (2*d*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 - ((d + e/Sqrt[x])^4*(a +
b*Log[c*(d + e/Sqrt[x])^n])^3)/(2*e^4)

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 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

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

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{x^3} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (-\frac{d^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{3 d^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac{3 d (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{(d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^3}+\frac{(6 d) \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^3}-\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^3}+\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^3}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}+\frac{(6 d) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}-\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}+\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}\\ &=\frac{2 d^3 \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^4}-\frac{3 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 e^4}+\frac{(3 b n) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 e^4}-\frac{(6 b d n) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}+\frac{\left (9 b d^2 n\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}-\frac{\left (6 b d^3 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}\\ &=-\frac{6 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}+\frac{9 b d^2 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 )^2}{2 e^4}-\frac{2 b d 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 )^2}{e^4}+\frac{3 b 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}{8 e^4}+\frac{2 d^3 \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^4}-\frac{3 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 e^4}-\frac{\left (3 b^2 n^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{4 e^4}+\frac{\left (4 b^2 d n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}-\frac{\left (9 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}+\frac{\left (12 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}\\ &=\frac{9 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^4}-\frac{4 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{9 e^4}+\frac{3 b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{64 e^4}+\frac{12 a b^2 d^3 n^2}{e^3 \sqrt{x}}-\frac{9 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^4}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 e^4}-\frac{3 b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{16 e^4}-\frac{6 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}+\frac{9 b d^2 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 )^2}{2 e^4}-\frac{2 b d 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 )^2}{e^4}+\frac{3 b 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}{8 e^4}+\frac{2 d^3 \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^4}-\frac{3 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 e^4}+\frac{\left (12 b^3 d^3 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^4}\\ &=\frac{9 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^4}-\frac{4 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{9 e^4}+\frac{3 b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{64 e^4}+\frac{12 a b^2 d^3 n^2}{e^3 \sqrt{x}}-\frac{12 b^3 d^3 n^3}{e^3 \sqrt{x}}+\frac{12 b^3 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^4}-\frac{9 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^4}+\frac{4 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{3 e^4}-\frac{3 b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{16 e^4}-\frac{6 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^4}+\frac{9 b d^2 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 )^2}{2 e^4}-\frac{2 b d 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 )^2}{e^4}+\frac{3 b 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}{8 e^4}+\frac{2 d^3 \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^4}-\frac{3 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^4}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 e^4}\\ \end{align*}

Mathematica [A]  time = 1.00426, size = 766, normalized size = 1.29 \[ \frac{-12 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (72 a^2 e^4-12 a b e n \left (6 d^2 e x-12 d^3 x^{3/2}-4 d e^2 \sqrt{x}+3 e^3\right )+12 b d^4 n x^2 (25 b n-12 a) \log \left (d \sqrt{x}+e\right )+6 b d^4 n x^2 \log (x) (12 a-25 b n)+b^2 e n^2 \left (78 d^2 e x-300 d^3 x^{3/2}-28 d e^2 \sqrt{x}+9 e^3\right )\right )+432 a^2 b d^2 e^2 n x-864 a^2 b d^3 e n x^{3/2}+864 a^2 b d^4 n x^2 \log \left (d \sqrt{x}+e\right )-432 a^2 b d^4 n x^2 \log (x)-288 a^2 b d e^3 n \sqrt{x}+216 a^2 b e^4 n-288 a^3 e^4+72 b^2 \log ^2\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (e \left (-12 a e^3+6 b d^2 e n x-12 b d^3 n x^{3/2}-4 b d e^2 n \sqrt{x}+3 b e^3 n\right )+12 b d^4 n x^2 \log \left (d \sqrt{x}+e\right )-6 b d^4 n x^2 \log (x)\right )+72 b^2 d^4 n^2 x^2 \log ^2\left (d+\frac{e}{\sqrt{x}}\right ) \left (12 a+12 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+12 b n \log \left (d \sqrt{x}+e\right )-6 b n \log (x)-25 b n\right )+72 b^2 d^4 n^2 x^2 \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (2 \log \left (d \sqrt{x}+e\right )-\log (x)\right ) \left (-12 a-12 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+25 b n\right )-936 a b^2 d^2 e^2 n^2 x+3600 a b^2 d^3 e n^2 x^{3/2}-3600 a b^2 d^4 n^2 x^2 \log \left (d \sqrt{x}+e\right )+1800 a b^2 d^4 n^2 x^2 \log (x)+336 a b^2 d e^3 n^2 \sqrt{x}-108 a b^2 e^4 n^2-288 b^3 e^4 \log ^3\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+690 b^3 d^2 e^2 n^3 x-4980 b^3 d^3 e n^3 x^{3/2}-576 b^3 d^4 n^3 x^2 \log ^3\left (d+\frac{e}{\sqrt{x}}\right )+4980 b^3 d^4 n^3 x^2 \log \left (d \sqrt{x}+e\right )-2490 b^3 d^4 n^3 x^2 \log (x)-148 b^3 d e^3 n^3 \sqrt{x}+27 b^3 e^4 n^3}{576 e^4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-288*a^3*e^4 + 216*a^2*b*e^4*n - 108*a*b^2*e^4*n^2 + 27*b^3*e^4*n^3 - 288*a^2*b*d*e^3*n*Sqrt[x] + 336*a*b^2*d
*e^3*n^2*Sqrt[x] - 148*b^3*d*e^3*n^3*Sqrt[x] + 432*a^2*b*d^2*e^2*n*x - 936*a*b^2*d^2*e^2*n^2*x + 690*b^3*d^2*e
^2*n^3*x - 864*a^2*b*d^3*e*n*x^(3/2) + 3600*a*b^2*d^3*e*n^2*x^(3/2) - 4980*b^3*d^3*e*n^3*x^(3/2) - 576*b^3*d^4
*n^3*x^2*Log[d + e/Sqrt[x]]^3 - 288*b^3*e^4*Log[c*(d + e/Sqrt[x])^n]^3 + 864*a^2*b*d^4*n*x^2*Log[e + d*Sqrt[x]
] - 3600*a*b^2*d^4*n^2*x^2*Log[e + d*Sqrt[x]] + 4980*b^3*d^4*n^3*x^2*Log[e + d*Sqrt[x]] + 72*b^2*d^4*n^2*x^2*L
og[d + e/Sqrt[x]]*(-12*a + 25*b*n - 12*b*Log[c*(d + e/Sqrt[x])^n])*(2*Log[e + d*Sqrt[x]] - Log[x]) - 432*a^2*b
*d^4*n*x^2*Log[x] + 1800*a*b^2*d^4*n^2*x^2*Log[x] - 2490*b^3*d^4*n^3*x^2*Log[x] + 72*b^2*d^4*n^2*x^2*Log[d + e
/Sqrt[x]]^2*(12*a - 25*b*n + 12*b*Log[c*(d + e/Sqrt[x])^n] + 12*b*n*Log[e + d*Sqrt[x]] - 6*b*n*Log[x]) + 72*b^
2*Log[c*(d + e/Sqrt[x])^n]^2*(e*(-12*a*e^3 + 3*b*e^3*n - 4*b*d*e^2*n*Sqrt[x] + 6*b*d^2*e*n*x - 12*b*d^3*n*x^(3
/2)) + 12*b*d^4*n*x^2*Log[e + d*Sqrt[x]] - 6*b*d^4*n*x^2*Log[x]) - 12*b*Log[c*(d + e/Sqrt[x])^n]*(72*a^2*e^4 +
b^2*e*n^2*(9*e^3 - 28*d*e^2*Sqrt[x] + 78*d^2*e*x - 300*d^3*x^(3/2)) - 12*a*b*e*n*(3*e^3 - 4*d*e^2*Sqrt[x] + 6
*d^2*e*x - 12*d^3*x^(3/2)) + 12*b*d^4*n*(-12*a + 25*b*n)*x^2*Log[e + d*Sqrt[x]] + 6*b*d^4*n*(12*a - 25*b*n)*x^
2*Log[x]))/(576*e^4*x^2)

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Maple [F]  time = 0.344, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{3}}\, 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))^3/x^3,x)

[Out]

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

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Maxima [A]  time = 1.1549, size = 988, normalized size = 1.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^2*b*e*n*(12*d^4*log(d*sqrt(x) + e)/e^5 - 6*d^4*log(x)/e^5 - (12*d^3*x^(3/2) - 6*d^2*e*x + 4*d*e^2*sqrt(x
) - 3*e^3)/(e^4*x^2)) + 1/48*(12*e*n*(12*d^4*log(d*sqrt(x) + e)/e^5 - 6*d^4*log(x)/e^5 - (12*d^3*x^(3/2) - 6*d
^2*e*x + 4*d*e^2*sqrt(x) - 3*e^3)/(e^4*x^2))*log(c*(d + e/sqrt(x))^n) - (72*d^4*x^2*log(d*sqrt(x) + e)^2 + 18*
d^4*x^2*log(x)^2 - 150*d^4*x^2*log(x) - 300*d^3*e*x^(3/2) + 78*d^2*e^2*x - 28*d*e^3*sqrt(x) + 9*e^4 - 12*(6*d^
4*x^2*log(x) - 25*d^4*x^2)*log(d*sqrt(x) + e))*n^2/(e^4*x^2))*a*b^2 + 1/576*(72*e*n*(12*d^4*log(d*sqrt(x) + e)
/e^5 - 6*d^4*log(x)/e^5 - (12*d^3*x^(3/2) - 6*d^2*e*x + 4*d*e^2*sqrt(x) - 3*e^3)/(e^4*x^2))*log(c*(d + e/sqrt(
x))^n)^2 + e*n*((288*d^4*x^2*log(d*sqrt(x) + e)^3 - 36*d^4*x^2*log(x)^3 + 450*d^4*x^2*log(x)^2 - 2490*d^4*x^2*
log(x) - 4980*d^3*e*x^(3/2) + 690*d^2*e^2*x - 148*d*e^3*sqrt(x) + 27*e^4 - 72*(6*d^4*x^2*log(x) - 25*d^4*x^2)*
log(d*sqrt(x) + e)^2 + 12*(18*d^4*x^2*log(x)^2 - 150*d^4*x^2*log(x) + 415*d^4*x^2)*log(d*sqrt(x) + e))*n^2/(e^
5*x^2) - 12*(72*d^4*x^2*log(d*sqrt(x) + e)^2 + 18*d^4*x^2*log(x)^2 - 150*d^4*x^2*log(x) - 300*d^3*e*x^(3/2) +
78*d^2*e^2*x - 28*d*e^3*sqrt(x) + 9*e^4 - 12*(6*d^4*x^2*log(x) - 25*d^4*x^2)*log(d*sqrt(x) + e))*n*log(c*(d +
e/sqrt(x))^n)/(e^5*x^2)))*b^3 - 1/2*b^3*log(c*(d + e/sqrt(x))^n)^3/x^2 - 3/2*a*b^2*log(c*(d + e/sqrt(x))^n)^2/
x^2 - 3/2*a^2*b*log(c*(d + e/sqrt(x))^n)/x^2 - 1/2*a^3/x^2

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Fricas [A]  time = 1.88304, size = 1894, normalized size = 3.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(27*b^3*e^4*n^3 - 288*b^3*e^4*log(c)^3 - 108*a*b^2*e^4*n^2 + 216*a^2*b*e^4*n - 288*a^3*e^4 + 288*(b^3*d^
4*n^3*x^2 - b^3*e^4*n^3)*log((d*x + e*sqrt(x))/x)^3 + 216*(2*b^3*d^2*e^2*n*x + b^3*e^4*n - 4*a*b^2*e^4)*log(c)
^2 + 72*(6*b^3*d^2*e^2*n^3*x + 3*b^3*e^4*n^3 - 12*a*b^2*e^4*n^2 - (25*b^3*d^4*n^3 - 12*a*b^2*d^4*n^2)*x^2 + 12
*(b^3*d^4*n^2*x^2 - b^3*e^4*n^2)*log(c) - 4*(3*b^3*d^3*e*n^3*x + b^3*d*e^3*n^3)*sqrt(x))*log((d*x + e*sqrt(x))
/x)^2 + 6*(115*b^3*d^2*e^2*n^3 - 156*a*b^2*d^2*e^2*n^2 + 72*a^2*b*d^2*e^2*n)*x - 36*(3*b^3*e^4*n^2 - 12*a*b^2*
e^4*n + 24*a^2*b*e^4 + 2*(13*b^3*d^2*e^2*n^2 - 12*a*b^2*d^2*e^2*n)*x)*log(c) - 12*(9*b^3*e^4*n^3 - 36*a*b^2*e^
4*n^2 + 72*a^2*b*e^4*n - (415*b^3*d^4*n^3 - 300*a*b^2*d^4*n^2 + 72*a^2*b*d^4*n)*x^2 - 72*(b^3*d^4*n*x^2 - b^3*
e^4*n)*log(c)^2 + 6*(13*b^3*d^2*e^2*n^3 - 12*a*b^2*d^2*e^2*n^2)*x - 12*(6*b^3*d^2*e^2*n^2*x + 3*b^3*e^4*n^2 -
12*a*b^2*e^4*n - (25*b^3*d^4*n^2 - 12*a*b^2*d^4*n)*x^2)*log(c) - 4*(7*b^3*d*e^3*n^3 - 12*a*b^2*d*e^3*n^2 + 3*(
25*b^3*d^3*e*n^3 - 12*a*b^2*d^3*e*n^2)*x - 12*(3*b^3*d^3*e*n^2*x + b^3*d*e^3*n^2)*log(c))*sqrt(x))*log((d*x +
e*sqrt(x))/x) - 4*(37*b^3*d*e^3*n^3 - 84*a*b^2*d*e^3*n^2 + 72*a^2*b*d*e^3*n + 72*(3*b^3*d^3*e*n*x + b^3*d*e^3*
n)*log(c)^2 + 3*(415*b^3*d^3*e*n^3 - 300*a*b^2*d^3*e*n^2 + 72*a^2*b*d^3*e*n)*x - 12*(7*b^3*d*e^3*n^2 - 12*a*b^
2*d*e^3*n + 3*(25*b^3*d^3*e*n^2 - 12*a*b^2*d^3*e*n)*x)*log(c))*sqrt(x))/(e^4*x^2)

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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))**3/x**3,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 )}^{3}}{x^{3}}\,{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))^3/x^3,x, algorithm="giac")

[Out]

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