\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3}{x^3} \, dx\) [440]

Optimal result

Integrand size = 24, antiderivative size = 595 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\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} \]

[Out]

12*b^3*d^3*n^2*ln(c*(d+e/x^(1/2))^n)*(d+e/x^(1/2))/e^4-6*b*d^3*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))/e
^4+2*d^3*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))/e^4+9/4*b^3*d^2*n^3*(d+e/x^(1/2))^2/e^4-9/2*b^2*d^2*n^2*(
a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^2/e^4+9/2*b*d^2*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))^2/e^4-3
*d^2*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^2/e^4-4/9*b^3*d*n^3*(d+e/x^(1/2))^3/e^4+4/3*b^2*d*n^2*(a+b*ln
(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^3/e^4-2*b*d*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))^3/e^4+2*d*(a+b*ln
(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^3/e^4+3/64*b^3*n^3*(d+e/x^(1/2))^4/e^4-3/16*b^2*n^2*(a+b*ln(c*(d+e/x^(1/2
))^n))*(d+e/x^(1/2))^4/e^4+3/8*b*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))^4/e^4-1/2*(a+b*ln(c*(d+e/x^(1/2
))^n))^3*(d+e/x^(1/2))^4/e^4+12*a*b^2*d^3*n^2/e^3/x^(1/2)-12*b^3*d^3*n^3/e^3/x^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=-\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 {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 {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 {\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 {12 b^3 d^3 n^3}{e^3 \sqrt {x}}+\frac {9 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^4}+\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} \]

[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 2332

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

Rule 2333

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 2341

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
]

Rule 2342

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 2436

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 2437

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 2448

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 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}& = -\left (2 \text {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 \text {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 \text {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) \text {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 ) \text {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 ) \text {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 \text {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) \text {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 ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.65 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {-288 a^3 e^4+216 a^2 b e^4 n-108 a b^2 e^4 n^2+27 b^3 e^4 n^3-288 a^2 b d e^3 n \sqrt {x}+336 a b^2 d e^3 n^2 \sqrt {x}-148 b^3 d e^3 n^3 \sqrt {x}+432 a^2 b d^2 e^2 n x-936 a b^2 d^2 e^2 n^2 x+690 b^3 d^2 e^2 n^3 x-864 a^2 b d^3 e n x^{3/2}+3600 a b^2 d^3 e n^2 x^{3/2}-4980 b^3 d^3 e n^3 x^{3/2}-576 b^3 d^4 n^3 x^2 \log ^3\left (d+\frac {e}{\sqrt {x}}\right )-288 b^3 e^4 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+864 a^2 b d^4 n x^2 \log \left (e+d \sqrt {x}\right )-3600 a b^2 d^4 n^2 x^2 \log \left (e+d \sqrt {x}\right )+4980 b^3 d^4 n^3 x^2 \log \left (e+d \sqrt {x}\right )+72 b^2 d^4 n^2 x^2 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (-12 a+25 b n-12 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (2 \log \left (e+d \sqrt {x}\right )-\log (x)\right )-432 a^2 b d^4 n x^2 \log (x)+1800 a b^2 d^4 n^2 x^2 \log (x)-2490 b^3 d^4 n^3 x^2 \log (x)+72 b^2 d^4 n^2 x^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (12 a-25 b n+12 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+12 b n \log \left (e+d \sqrt {x}\right )-6 b n \log (x)\right )+72 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (-12 a e^3+3 b e^3 n-4 b d e^2 n \sqrt {x}+6 b d^2 e n x-12 b d^3 n x^{3/2}\right )+12 b d^4 n x^2 \log \left (e+d \sqrt {x}\right )-6 b d^4 n x^2 \log (x)\right )-12 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (72 a^2 e^4+b^2 e n^2 \left (9 e^3-28 d e^2 \sqrt {x}+78 d^2 e x-300 d^3 x^{3/2}\right )-12 a b e n \left (3 e^3-4 d e^2 \sqrt {x}+6 d^2 e x-12 d^3 x^{3/2}\right )+12 b d^4 n (-12 a+25 b n) x^2 \log \left (e+d \sqrt {x}\right )+6 b d^4 n (12 a-25 b n) x^2 \log (x)\right )}{576 e^4 x^2} \]

[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)

Maple [F]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {27 \, b^{3} e^{4} n^{3} - 288 \, b^{3} e^{4} \log \left (c\right )^{3} - 108 \, a b^{2} e^{4} n^{2} + 216 \, a^{2} b e^{4} n - 288 \, a^{3} e^{4} + 288 \, {\left (b^{3} d^{4} n^{3} x^{2} - b^{3} e^{4} n^{3}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{3} + 216 \, {\left (2 \, b^{3} d^{2} e^{2} n x + b^{3} e^{4} n - 4 \, a b^{2} e^{4}\right )} \log \left (c\right )^{2} + 72 \, {\left (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} - {\left (25 \, b^{3} d^{4} n^{3} - 12 \, a b^{2} d^{4} n^{2}\right )} x^{2} + 12 \, {\left (b^{3} d^{4} n^{2} x^{2} - b^{3} e^{4} n^{2}\right )} \log \left (c\right ) - 4 \, {\left (3 \, b^{3} d^{3} e n^{3} x + b^{3} d e^{3} n^{3}\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{2} + 6 \, {\left (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\right )} x - 36 \, {\left (3 \, b^{3} e^{4} n^{2} - 12 \, a b^{2} e^{4} n + 24 \, a^{2} b e^{4} + 2 \, {\left (13 \, b^{3} d^{2} e^{2} n^{2} - 12 \, a b^{2} d^{2} e^{2} n\right )} x\right )} \log \left (c\right ) - 12 \, {\left (9 \, b^{3} e^{4} n^{3} - 36 \, a b^{2} e^{4} n^{2} + 72 \, a^{2} b e^{4} n - {\left (415 \, b^{3} d^{4} n^{3} - 300 \, a b^{2} d^{4} n^{2} + 72 \, a^{2} b d^{4} n\right )} x^{2} - 72 \, {\left (b^{3} d^{4} n x^{2} - b^{3} e^{4} n\right )} \log \left (c\right )^{2} + 6 \, {\left (13 \, b^{3} d^{2} e^{2} n^{3} - 12 \, a b^{2} d^{2} e^{2} n^{2}\right )} x - 12 \, {\left (6 \, b^{3} d^{2} e^{2} n^{2} x + 3 \, b^{3} e^{4} n^{2} - 12 \, a b^{2} e^{4} n - {\left (25 \, b^{3} d^{4} n^{2} - 12 \, a b^{2} d^{4} n\right )} x^{2}\right )} \log \left (c\right ) - 4 \, {\left (7 \, b^{3} d e^{3} n^{3} - 12 \, a b^{2} d e^{3} n^{2} + 3 \, {\left (25 \, b^{3} d^{3} e n^{3} - 12 \, a b^{2} d^{3} e n^{2}\right )} x - 12 \, {\left (3 \, b^{3} d^{3} e n^{2} x + b^{3} d e^{3} n^{2}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 4 \, {\left (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 \, {\left (3 \, b^{3} d^{3} e n x + b^{3} d e^{3} n\right )} \log \left (c\right )^{2} + 3 \, {\left (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\right )} x - 12 \, {\left (7 \, b^{3} d e^{3} n^{2} - 12 \, a b^{2} d e^{3} n + 3 \, {\left (25 \, b^{3} d^{3} e n^{2} - 12 \, a b^{2} d^{3} e n\right )} x\right )} \log \left (c\right )\right )} \sqrt {x}}{576 \, e^{4} x^{2}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {1}{8} \, a^{2} b e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} + \frac {1}{48} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (72 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{2} + 18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) - 300 \, d^{3} e x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x - 28 \, d e^{3} \sqrt {x} + 9 \, e^{4} - 12 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{4} x^{2}}\right )} a b^{2} + \frac {1}{576} \, {\left (72 \, e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2} + e n {\left (\frac {{\left (288 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{3} - 36 \, d^{4} x^{2} \log \left (x\right )^{3} + 450 \, d^{4} x^{2} \log \left (x\right )^{2} - 2490 \, d^{4} x^{2} \log \left (x\right ) - 4980 \, d^{3} e x^{\frac {3}{2}} + 690 \, d^{2} e^{2} x - 148 \, d e^{3} \sqrt {x} + 27 \, e^{4} - 72 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )^{2} + 12 \, {\left (18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) + 415 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{5} x^{2}} - \frac {12 \, {\left (72 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{2} + 18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) - 300 \, d^{3} e x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x - 28 \, d e^{3} \sqrt {x} + 9 \, e^{4} - 12 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{e^{5} x^{2}}\right )}\right )} b^{3} - \frac {b^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1147 vs. \(2 (519) = 1038\).

Time = 0.40 (sec) , antiderivative size = 1147, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/576*(288*(4*(d*sqrt(x) + e)*b^3*d^3*n^3/(e^3*sqrt(x)) - 6*(d*sqrt(x) + e)^2*b^3*d^2*n^3/(e^3*x) + 4*(d*sqrt(
x) + e)^3*b^3*d*n^3/(e^3*x^(3/2)) - (d*sqrt(x) + e)^4*b^3*n^3/(e^3*x^2))*log((d*sqrt(x) + e)/sqrt(x))^3 + 72*(
3*(b^3*n^3 - 4*b^3*n^2*log(c) - 4*a*b^2*n^2)*(d*sqrt(x) + e)^4/(e^3*x^2) - 16*(b^3*d*n^3 - 3*b^3*d*n^2*log(c)
- 3*a*b^2*d*n^2)*(d*sqrt(x) + e)^3/(e^3*x^(3/2)) + 36*(b^3*d^2*n^3 - 2*b^3*d^2*n^2*log(c) - 2*a*b^2*d^2*n^2)*(
d*sqrt(x) + e)^2/(e^3*x) - 48*(b^3*d^3*n^3 - b^3*d^3*n^2*log(c) - a*b^2*d^3*n^2)*(d*sqrt(x) + e)/(e^3*sqrt(x))
)*log((d*sqrt(x) + e)/sqrt(x))^2 - 12*(9*(b^3*n^3 - 4*b^3*n^2*log(c) + 8*b^3*n*log(c)^2 - 4*a*b^2*n^2 + 16*a*b
^2*n*log(c) + 8*a^2*b*n)*(d*sqrt(x) + e)^4/(e^3*x^2) - 32*(2*b^3*d*n^3 - 6*b^3*d*n^2*log(c) + 9*b^3*d*n*log(c)
^2 - 6*a*b^2*d*n^2 + 18*a*b^2*d*n*log(c) + 9*a^2*b*d*n)*(d*sqrt(x) + e)^3/(e^3*x^(3/2)) + 216*(b^3*d^2*n^3 - 2
*b^3*d^2*n^2*log(c) + 2*b^3*d^2*n*log(c)^2 - 2*a*b^2*d^2*n^2 + 4*a*b^2*d^2*n*log(c) + 2*a^2*b*d^2*n)*(d*sqrt(x
) + e)^2/(e^3*x) - 288*(2*b^3*d^3*n^3 - 2*b^3*d^3*n^2*log(c) + b^3*d^3*n*log(c)^2 - 2*a*b^2*d^3*n^2 + 2*a*b^2*
d^3*n*log(c) + a^2*b*d^3*n)*(d*sqrt(x) + e)/(e^3*sqrt(x)))*log((d*sqrt(x) + e)/sqrt(x)) + 9*(3*b^3*n^3 - 12*b^
3*n^2*log(c) + 24*b^3*n*log(c)^2 - 32*b^3*log(c)^3 - 12*a*b^2*n^2 + 48*a*b^2*n*log(c) - 96*a*b^2*log(c)^2 + 24
*a^2*b*n - 96*a^2*b*log(c) - 32*a^3)*(d*sqrt(x) + e)^4/(e^3*x^2) - 128*(2*b^3*d*n^3 - 6*b^3*d*n^2*log(c) + 9*b
^3*d*n*log(c)^2 - 9*b^3*d*log(c)^3 - 6*a*b^2*d*n^2 + 18*a*b^2*d*n*log(c) - 27*a*b^2*d*log(c)^2 + 9*a^2*b*d*n -
27*a^2*b*d*log(c) - 9*a^3*d)*(d*sqrt(x) + e)^3/(e^3*x^(3/2)) + 432*(3*b^3*d^2*n^3 - 6*b^3*d^2*n^2*log(c) + 6*
b^3*d^2*n*log(c)^2 - 4*b^3*d^2*log(c)^3 - 6*a*b^2*d^2*n^2 + 12*a*b^2*d^2*n*log(c) - 12*a*b^2*d^2*log(c)^2 + 6*
a^2*b*d^2*n - 12*a^2*b*d^2*log(c) - 4*a^3*d^2)*(d*sqrt(x) + e)^2/(e^3*x) - 1152*(6*b^3*d^3*n^3 - 6*b^3*d^3*n^2
*log(c) + 3*b^3*d^3*n*log(c)^2 - b^3*d^3*log(c)^3 - 6*a*b^2*d^3*n^2 + 6*a*b^2*d^3*n*log(c) - 3*a*b^2*d^3*log(c
)^2 + 3*a^2*b*d^3*n - 3*a^2*b*d^3*log(c) - a^3*d^3)*(d*sqrt(x) + e)/(e^3*sqrt(x)))/e

Mupad [B] (verification not implemented)

Time = 2.03 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{3\,e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{36\,e}}{x^{3/2}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^3\,\left (\frac {b^3}{2\,x^2}-\frac {b^3\,d^4}{2\,e^4}\right )+\frac {\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{8\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2\,\left (12\,a-25\,b\,n\right )}{4\,e^3}}{\sqrt {x}}+{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {\frac {b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b^2\,d}{e}}{2\,x^{3/2}}-\frac {3\,b^2\,\left (4\,a-b\,n\right )}{8\,x^2}+\frac {d\,\left (12\,a\,b^2\,d^3-25\,b^3\,d^3\,n\right )}{8\,e^4}-\frac {d\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{8\,e\,x}+\frac {d^2\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{4\,e^2\,\sqrt {x}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{16\,e^2}}{x}-\frac {\frac {a^3}{2}-\frac {3\,a^2\,b\,n}{8}+\frac {3\,a\,b^2\,n^2}{16}-\frac {3\,b^3\,n^3}{64}}{x^2}-\frac {\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{12\,e^2\,x^{3/2}}+\frac {\frac {d\,\left (\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2\right )}{e}-48\,b^3\,d^3\,e\,n^2}{4\,e^2\,\sqrt {x}}-\frac {\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2}{8\,e^2\,x}+\frac {3\,b\,e^2\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{4\,x^2}\right )}{4\,e^2}+\frac {\ln \left (d+\frac {e}{\sqrt {x}}\right )\,\left (72\,a^2\,b\,d^4\,n-300\,a\,b^2\,d^4\,n^2+415\,b^3\,d^4\,n^3\right )}{48\,e^4} \]

[In]

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

[Out]

((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/(3*e) - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2)
)/(36*e))/x^(3/2) - log(c*(d + e/x^(1/2))^n)^3*(b^3/(2*x^2) - (b^3*d^4)/(2*e^4)) + ((d*((d*((d*(2*a^3 - (3*b^3
*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(12*e)))/e + (b^2*d^2
*n^2*(12*a - 13*b*n))/(8*e^2)))/e + (b^2*d^3*n^2*(12*a - 25*b*n))/(4*e^3))/x^(1/2) + log(c*(d + e/x^(1/2))^n)^
2*(((b^2*d*(4*a - b*n))/e - (4*a*b^2*d)/e)/(2*x^(3/2)) - (3*b^2*(4*a - b*n))/(8*x^2) + (d*(12*a*b^2*d^3 - 25*b
^3*d^3*n))/(8*e^4) - (d*((6*b^2*d*(4*a - b*n))/e - (24*a*b^2*d)/e))/(8*e*x) + (d^2*((6*b^2*d*(4*a - b*n))/e -
(24*a*b^2*d)/e))/(4*e^2*x^(1/2))) - ((d*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d
*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(12*e)))/(2*e) + (b^2*d^2*n^2*(12*a - 13*b*n))/(16*e^2))/x - (a^3/2 - (3
*b^3*n^3)/64 + (3*a*b^2*n^2)/16 - (3*a^2*b*n)/8)/x^2 - (log(c*(d + e/x^(1/2))^n)*((16*b*d*e^3*(6*a^2 - b^2*n^2
) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n))/(12*e^2*x^(3/2)) + ((d*((d*(16*b*d*e^3*(6*a^2 - b^2*n^2) - 12*b*d*
e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2))/e - 48*b^3*d^3*e*n^2)/(4*e^2*x^(1/2)) - ((d*(16*b*d
*e^3*(6*a^2 - b^2*n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2)/(8*e^2*x) + (3*b*e^2
*(8*a^2 + b^2*n^2 - 4*a*b*n))/(4*x^2)))/(4*e^2) + (log(d + e/x^(1/2))*(415*b^3*d^4*n^3 - 300*a*b^2*d^4*n^2 + 7
2*a^2*b*d^4*n))/(48*e^4)