\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3}{x^4} \, dx\) [441]

Optimal result

Integrand size = 24, antiderivative size = 907 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\frac {15 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^6}-\frac {40 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{27 e^6}+\frac {15 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{32 e^6}-\frac {12 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{108 e^6}+\frac {12 a b^2 d^5 n^2}{e^5 \sqrt {x}}-\frac {12 b^3 d^5 n^3}{e^5 \sqrt {x}}+\frac {12 b^3 d^5 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^6}-\frac {15 b^2 d^4 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^6}+\frac {40 b^2 d^3 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 )}{9 e^6}-\frac {15 b^2 d^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 )}{8 e^6}+\frac {12 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{18 e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 e^6}+\frac {2 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {5 d^4 \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^6}+\frac {20 d^3 \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}{3 e^6}-\frac {5 d^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 )^3}{e^6}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{3 e^6} \]

[Out]

2*d^5*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))/e^6-5*d^4*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^2/e^6+
20/3*d^3*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^3/e^6-5*d^2*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^4
/e^6+2*d*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^5/e^6+1/108*b^3*n^3*(d+e/x^(1/2))^6/e^6-12/125*b^3*d*n^3*
(d+e/x^(1/2))^5/e^6-1/18*b^2*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^6/e^6+1/6*b*n*(a+b*ln(c*(d+e/x^(1/2
))^n))^2*(d+e/x^(1/2))^6/e^6-12*b^3*d^5*n^3/e^5/x^(1/2)+15/4*b^3*d^4*n^3*(d+e/x^(1/2))^2/e^6-40/27*b^3*d^3*n^3
*(d+e/x^(1/2))^3/e^6+15/32*b^3*d^2*n^3*(d+e/x^(1/2))^4/e^6-1/3*(a+b*ln(c*(d+e/x^(1/2))^n))^3*(d+e/x^(1/2))^6/e
^6+12*b^3*d^5*n^2*ln(c*(d+e/x^(1/2))^n)*(d+e/x^(1/2))/e^6-6*b*d^5*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2)
)/e^6-15/2*b^2*d^4*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^2/e^6+15/2*b*d^4*n*(a+b*ln(c*(d+e/x^(1/2))^n)
)^2*(d+e/x^(1/2))^2/e^6+40/9*b^2*d^3*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^3/e^6-20/3*b*d^3*n*(a+b*ln(
c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))^3/e^6-15/8*b^2*d^2*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^4/e^6+15/
4*b*d^2*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))^4/e^6+12/25*b^2*d*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x
^(1/2))^5/e^6-6/5*b*d*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/2))^5/e^6+12*a*b^2*d^5*n^2/e^5/x^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, 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^4} \, dx=\frac {b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{108 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{3 e^6}+\frac {b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^6}{6 e^6}-\frac {b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^6}{18 e^6}-\frac {12 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{125 e^6}+\frac {2 d \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{e^6}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^5}{5 e^6}+\frac {12 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^5}{25 e^6}+\frac {15 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{32 e^6}-\frac {5 d^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{e^6}+\frac {15 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{4 e^6}-\frac {15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^4}{8 e^6}-\frac {40 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{27 e^6}+\frac {20 d^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{3 e^6}-\frac {20 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{3 e^6}+\frac {40 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^6}+\frac {15 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^6}-\frac {5 d^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{e^6}+\frac {15 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^6}-\frac {15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^6}+\frac {2 d^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {6 b d^5 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {12 b^3 d^5 n^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {12 b^3 d^5 n^3}{e^5 \sqrt {x}}+\frac {12 a b^2 d^5 n^2}{e^5 \sqrt {x}} \]

[In]

Int[(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4,x]

[Out]

(15*b^3*d^4*n^3*(d + e/Sqrt[x])^2)/(4*e^6) - (40*b^3*d^3*n^3*(d + e/Sqrt[x])^3)/(27*e^6) + (15*b^3*d^2*n^3*(d
+ e/Sqrt[x])^4)/(32*e^6) - (12*b^3*d*n^3*(d + e/Sqrt[x])^5)/(125*e^6) + (b^3*n^3*(d + e/Sqrt[x])^6)/(108*e^6)
+ (12*a*b^2*d^5*n^2)/(e^5*Sqrt[x]) - (12*b^3*d^5*n^3)/(e^5*Sqrt[x]) + (12*b^3*d^5*n^2*(d + e/Sqrt[x])*Log[c*(d
+ e/Sqrt[x])^n])/e^6 - (15*b^2*d^4*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e^6) + (40*b^2*
d^3*n^2*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) - (15*b^2*d^2*n^2*(d + e/Sqrt[x])^4*(a + b
*Log[c*(d + e/Sqrt[x])^n]))/(8*e^6) + (12*b^2*d*n^2*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(25*e^
6) - (b^2*n^2*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(18*e^6) - (6*b*d^5*n*(d + e/Sqrt[x])*(a + b
*Log[c*(d + e/Sqrt[x])^n])^2)/e^6 + (15*b*d^4*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^6)
- (20*b*d^3*n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(3*e^6) + (15*b*d^2*n*(d + e/Sqrt[x])^4*(a
+ b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*e^6) - (6*b*d*n*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(5*
e^6) + (b*n*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(6*e^6) + (2*d^5*(d + e/Sqrt[x])*(a + b*Log[
c*(d + e/Sqrt[x])^n])^3)/e^6 - (5*d^4*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 + (20*d^3*(d +
e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(3*e^6) - (5*d^2*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[
x])^n])^3)/e^6 + (2*d*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 - ((d + e/Sqrt[x])^6*(a + b*Lo
g[c*(d + e/Sqrt[x])^n])^3)/(3*e^6)

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^5 \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^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {2 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}+\frac {(10 d) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}-\frac {\left (20 d^2\right ) \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^5}+\frac {\left (20 d^3\right ) \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^5}-\frac {\left (10 d^4\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^5}+\frac {\left (2 d^5\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^5} \\ & = -\frac {2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {(10 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {\left (20 d^2\right ) \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^6}+\frac {\left (20 d^3\right ) \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^6}-\frac {\left (10 d^4\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^6}+\frac {\left (2 d^5\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^6} \\ & = \frac {2 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {5 d^4 \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^6}+\frac {20 d^3 \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}{3 e^6}-\frac {5 d^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 )^3}{e^6}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{3 e^6}+\frac {(b n) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {(6 b d n) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {\left (15 b d^2 n\right ) \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 )}{e^6}-\frac {\left (20 b d^3 n\right ) \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^6}+\frac {\left (15 b d^4 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^6}-\frac {\left (6 b d^5 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^6} \\ & = -\frac {6 b d^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 e^6}+\frac {2 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {5 d^4 \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^6}+\frac {20 d^3 \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}{3 e^6}-\frac {5 d^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 )^3}{e^6}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{3 e^6}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{3 e^6}+\frac {\left (12 b^2 d n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{5 e^6}-\frac {\left (15 b^2 d^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 )}{2 e^6}+\frac {\left (40 b^2 d^3 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 )}{3 e^6}-\frac {\left (15 b^2 d^4 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^6}+\frac {\left (12 b^2 d^5 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^6} \\ & = \frac {15 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^6}-\frac {40 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{27 e^6}+\frac {15 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{32 e^6}-\frac {12 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{108 e^6}+\frac {12 a b^2 d^5 n^2}{e^5 \sqrt {x}}-\frac {15 b^2 d^4 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^6}+\frac {40 b^2 d^3 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 )}{9 e^6}-\frac {15 b^2 d^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 )}{8 e^6}+\frac {12 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{18 e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 e^6}+\frac {2 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {5 d^4 \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^6}+\frac {20 d^3 \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}{3 e^6}-\frac {5 d^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 )^3}{e^6}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{3 e^6}+\frac {\left (12 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6} \\ & = \frac {15 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^6}-\frac {40 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{27 e^6}+\frac {15 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{32 e^6}-\frac {12 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^6}{108 e^6}+\frac {12 a b^2 d^5 n^2}{e^5 \sqrt {x}}-\frac {12 b^3 d^5 n^3}{e^5 \sqrt {x}}+\frac {12 b^3 d^5 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^6}-\frac {15 b^2 d^4 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^6}+\frac {40 b^2 d^3 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 )}{9 e^6}-\frac {15 b^2 d^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 )}{8 e^6}+\frac {12 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{18 e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 e^6}+\frac {2 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {5 d^4 \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^6}+\frac {20 d^3 \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}{3 e^6}-\frac {5 d^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 )^3}{e^6}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{3 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 950, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\frac {-36000 a^3 e^6+18000 a^2 b e^6 n-6000 a b^2 e^6 n^2+1000 b^3 e^6 n^3-21600 a^2 b d e^5 n \sqrt {x}+15840 a b^2 d e^5 n^2 \sqrt {x}-4368 b^3 d e^5 n^3 \sqrt {x}+27000 a^2 b d^2 e^4 n x-33300 a b^2 d^2 e^4 n^2 x+13785 b^3 d^2 e^4 n^3 x-36000 a^2 b d^3 e^3 n x^{3/2}+68400 a b^2 d^3 e^3 n^2 x^{3/2}-41180 b^3 d^3 e^3 n^3 x^{3/2}+54000 a^2 b d^4 e^2 n x^2-156600 a b^2 d^4 e^2 n^2 x^2+140070 b^3 d^4 e^2 n^3 x^2-108000 a^2 b d^5 e n x^{5/2}+529200 a b^2 d^5 e n^2 x^{5/2}-809340 b^3 d^5 e n^3 x^{5/2}-72000 b^3 d^6 n^3 x^3 \log ^3\left (d+\frac {e}{\sqrt {x}}\right )-36000 b^3 e^6 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+108000 a^2 b d^6 n x^3 \log \left (e+d \sqrt {x}\right )-529200 a b^2 d^6 n^2 x^3 \log \left (e+d \sqrt {x}\right )+809340 b^3 d^6 n^3 x^3 \log \left (e+d \sqrt {x}\right )+5400 b^2 d^6 n^2 x^3 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (-20 a+49 b n-20 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 )-54000 a^2 b d^6 n x^3 \log (x)+264600 a b^2 d^6 n^2 x^3 \log (x)-404670 b^3 d^6 n^3 x^3 \log (x)+5400 b^2 d^6 n^2 x^3 \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (20 a-49 b n+20 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+20 b n \log \left (e+d \sqrt {x}\right )-10 b n \log (x)\right )+1800 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (-60 a e^5+10 b e^5 n-12 b d e^4 n \sqrt {x}+15 b d^2 e^3 n x-20 b d^3 e^2 n x^{3/2}+30 b d^4 e n x^2-60 b d^5 n x^{5/2}\right )+60 b d^6 n x^3 \log \left (e+d \sqrt {x}\right )-30 b d^6 n x^3 \log (x)\right )-60 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (1800 a^2 e^6+b^2 e n^2 \left (100 e^5-264 d e^4 \sqrt {x}+555 d^2 e^3 x-1140 d^3 e^2 x^{3/2}+2610 d^4 e x^2-8820 d^5 x^{5/2}\right )-60 a b e n \left (10 e^5-12 d e^4 \sqrt {x}+15 d^2 e^3 x-20 d^3 e^2 x^{3/2}+30 d^4 e x^2-60 d^5 x^{5/2}\right )+180 b d^6 n (-20 a+49 b n) x^3 \log \left (e+d \sqrt {x}\right )+90 b d^6 n (20 a-49 b n) x^3 \log (x)\right )}{108000 e^6 x^3} \]

[In]

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4,x]

[Out]

(-36000*a^3*e^6 + 18000*a^2*b*e^6*n - 6000*a*b^2*e^6*n^2 + 1000*b^3*e^6*n^3 - 21600*a^2*b*d*e^5*n*Sqrt[x] + 15
840*a*b^2*d*e^5*n^2*Sqrt[x] - 4368*b^3*d*e^5*n^3*Sqrt[x] + 27000*a^2*b*d^2*e^4*n*x - 33300*a*b^2*d^2*e^4*n^2*x
+ 13785*b^3*d^2*e^4*n^3*x - 36000*a^2*b*d^3*e^3*n*x^(3/2) + 68400*a*b^2*d^3*e^3*n^2*x^(3/2) - 41180*b^3*d^3*e
^3*n^3*x^(3/2) + 54000*a^2*b*d^4*e^2*n*x^2 - 156600*a*b^2*d^4*e^2*n^2*x^2 + 140070*b^3*d^4*e^2*n^3*x^2 - 10800
0*a^2*b*d^5*e*n*x^(5/2) + 529200*a*b^2*d^5*e*n^2*x^(5/2) - 809340*b^3*d^5*e*n^3*x^(5/2) - 72000*b^3*d^6*n^3*x^
3*Log[d + e/Sqrt[x]]^3 - 36000*b^3*e^6*Log[c*(d + e/Sqrt[x])^n]^3 + 108000*a^2*b*d^6*n*x^3*Log[e + d*Sqrt[x]]
- 529200*a*b^2*d^6*n^2*x^3*Log[e + d*Sqrt[x]] + 809340*b^3*d^6*n^3*x^3*Log[e + d*Sqrt[x]] + 5400*b^2*d^6*n^2*x
^3*Log[d + e/Sqrt[x]]*(-20*a + 49*b*n - 20*b*Log[c*(d + e/Sqrt[x])^n])*(2*Log[e + d*Sqrt[x]] - Log[x]) - 54000
*a^2*b*d^6*n*x^3*Log[x] + 264600*a*b^2*d^6*n^2*x^3*Log[x] - 404670*b^3*d^6*n^3*x^3*Log[x] + 5400*b^2*d^6*n^2*x
^3*Log[d + e/Sqrt[x]]^2*(20*a - 49*b*n + 20*b*Log[c*(d + e/Sqrt[x])^n] + 20*b*n*Log[e + d*Sqrt[x]] - 10*b*n*Lo
g[x]) + 1800*b^2*Log[c*(d + e/Sqrt[x])^n]^2*(e*(-60*a*e^5 + 10*b*e^5*n - 12*b*d*e^4*n*Sqrt[x] + 15*b*d^2*e^3*n
*x - 20*b*d^3*e^2*n*x^(3/2) + 30*b*d^4*e*n*x^2 - 60*b*d^5*n*x^(5/2)) + 60*b*d^6*n*x^3*Log[e + d*Sqrt[x]] - 30*
b*d^6*n*x^3*Log[x]) - 60*b*Log[c*(d + e/Sqrt[x])^n]*(1800*a^2*e^6 + b^2*e*n^2*(100*e^5 - 264*d*e^4*Sqrt[x] + 5
55*d^2*e^3*x - 1140*d^3*e^2*x^(3/2) + 2610*d^4*e*x^2 - 8820*d^5*x^(5/2)) - 60*a*b*e*n*(10*e^5 - 12*d*e^4*Sqrt[
x] + 15*d^2*e^3*x - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x^2 - 60*d^5*x^(5/2)) + 180*b*d^6*n*(-20*a + 49*b*n)*x^3*Log
[e + d*Sqrt[x]] + 90*b*d^6*n*(20*a - 49*b*n)*x^3*Log[x]))/(108000*e^6*x^3)

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}}{x^{4}}d x\]

[In]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3/x^4,x)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 1203, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="fricas")

[Out]

1/108000*(1000*b^3*e^6*n^3 - 36000*b^3*e^6*log(c)^3 - 6000*a*b^2*e^6*n^2 + 18000*a^2*b*e^6*n - 36000*a^3*e^6 +
36000*(b^3*d^6*n^3*x^3 - b^3*e^6*n^3)*log((d*x + e*sqrt(x))/x)^3 + 30*(4669*b^3*d^4*e^2*n^3 - 5220*a*b^2*d^4*
e^2*n^2 + 1800*a^2*b*d^4*e^2*n)*x^2 + 9000*(6*b^3*d^4*e^2*n*x^2 + 3*b^3*d^2*e^4*n*x + 2*b^3*e^6*n - 12*a*b^2*e
^6)*log(c)^2 + 1800*(30*b^3*d^4*e^2*n^3*x^2 + 15*b^3*d^2*e^4*n^3*x + 10*b^3*e^6*n^3 - 60*a*b^2*e^6*n^2 - 3*(49
*b^3*d^6*n^3 - 20*a*b^2*d^6*n^2)*x^3 + 60*(b^3*d^6*n^2*x^3 - b^3*e^6*n^2)*log(c) - 4*(15*b^3*d^5*e*n^3*x^2 + 5
*b^3*d^3*e^3*n^3*x + 3*b^3*d*e^5*n^3)*sqrt(x))*log((d*x + e*sqrt(x))/x)^2 + 15*(919*b^3*d^2*e^4*n^3 - 2220*a*b
^2*d^2*e^4*n^2 + 1800*a^2*b*d^2*e^4*n)*x - 300*(20*b^3*e^6*n^2 - 120*a*b^2*e^6*n + 360*a^2*b*e^6 + 18*(29*b^3*
d^4*e^2*n^2 - 20*a*b^2*d^4*e^2*n)*x^2 + 3*(37*b^3*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n)*x)*log(c) - 60*(100*b^3*e^
6*n^3 - 600*a*b^2*e^6*n^2 + 1800*a^2*b*e^6*n - (13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^6*n)*x^3
+ 90*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)*x^2 - 1800*(b^3*d^6*n*x^3 - b^3*e^6*n)*log(c)^2 + 15*(37*b^3
*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x - 60*(30*b^3*d^4*e^2*n^2*x^2 + 15*b^3*d^2*e^4*n^2*x + 10*b^3*e^6*n^2 -
60*a*b^2*e^6*n - 3*(49*b^3*d^6*n^2 - 20*a*b^2*d^6*n)*x^3)*log(c) - 12*(22*b^3*d*e^5*n^3 - 60*a*b^2*d*e^5*n^2 +
15*(49*b^3*d^5*e*n^3 - 20*a*b^2*d^5*e*n^2)*x^2 + 5*(19*b^3*d^3*e^3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x - 20*(15*b^3
*d^5*e*n^2*x^2 + 5*b^3*d^3*e^3*n^2*x + 3*b^3*d*e^5*n^2)*log(c))*sqrt(x))*log((d*x + e*sqrt(x))/x) - 4*(1092*b^
3*d*e^5*n^3 - 3960*a*b^2*d*e^5*n^2 + 5400*a^2*b*d*e^5*n + 15*(13489*b^3*d^5*e*n^3 - 8820*a*b^2*d^5*e*n^2 + 180
0*a^2*b*d^5*e*n)*x^2 + 1800*(15*b^3*d^5*e*n*x^2 + 5*b^3*d^3*e^3*n*x + 3*b^3*d*e^5*n)*log(c)^2 + 5*(2059*b^3*d^
3*e^3*n^3 - 3420*a*b^2*d^3*e^3*n^2 + 1800*a^2*b*d^3*e^3*n)*x - 180*(22*b^3*d*e^5*n^2 - 60*a*b^2*d*e^5*n + 15*(
49*b^3*d^5*e*n^2 - 20*a*b^2*d^5*e*n)*x^2 + 5*(19*b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x)*log(c))*sqrt(x))/(e^
6*x^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(d+e/x**(1/2))**n))**3/x**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 864, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="maxima")

[Out]

1/60*a^2*b*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^
2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3)) + 1/1800*(60*e*n*(60*d^6*log(d*sqrt(x) + e)/e
^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x)
- 10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt(x))^n) - (1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4
410*d^6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x - 264*d*e^5*
sqrt(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e))*n^2/(e^6*x^3))*a*b^2 + 1/108000*(
1800*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3
/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt(x))^n)^2 + e*n*((36000*d^6*x^3*lo
g(d*sqrt(x) + e)^3 - 4500*d^6*x^3*log(x)^3 + 66150*d^6*x^3*log(x)^2 - 404670*d^6*x^3*log(x) - 809340*d^5*e*x^(
5/2) + 140070*d^4*e^2*x^2 - 41180*d^3*e^3*x^(3/2) + 13785*d^2*e^4*x - 4368*d*e^5*sqrt(x) + 1000*e^6 - 5400*(10
*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e)^2 + 60*(450*d^6*x^3*log(x)^2 - 4410*d^6*x^3*log(x) + 13489*d^
6*x^3)*log(d*sqrt(x) + e))*n^2/(e^7*x^3) - 60*(1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4410
*d^6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x - 264*d*e^5*sqr
t(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e))*n*log(c*(d + e/sqrt(x))^n)/(e^7*x^3)
))*b^3 - 1/3*b^3*log(c*(d + e/sqrt(x))^n)^3/x^3 - a*b^2*log(c*(d + e/sqrt(x))^n)^2/x^3 - a^2*b*log(c*(d + e/sq
rt(x))^n)/x^3 - 1/3*a^3/x^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1747 vs. \(2 (787) = 1574\).

Time = 0.41 (sec) , antiderivative size = 1747, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="giac")

[Out]

1/108000*(36000*(6*(d*sqrt(x) + e)*b^3*d^5*n^3/(e^5*sqrt(x)) - 15*(d*sqrt(x) + e)^2*b^3*d^4*n^3/(e^5*x) + 20*(
d*sqrt(x) + e)^3*b^3*d^3*n^3/(e^5*x^(3/2)) - 15*(d*sqrt(x) + e)^4*b^3*d^2*n^3/(e^5*x^2) + 6*(d*sqrt(x) + e)^5*
b^3*d*n^3/(e^5*x^(5/2)) - (d*sqrt(x) + e)^6*b^3*n^3/(e^5*x^3))*log((d*sqrt(x) + e)/sqrt(x))^3 + 1800*(10*(b^3*
n^3 - 6*b^3*n^2*log(c) - 6*a*b^2*n^2)*(d*sqrt(x) + e)^6/(e^5*x^3) - 72*(b^3*d*n^3 - 5*b^3*d*n^2*log(c) - 5*a*b
^2*d*n^2)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) + 225*(b^3*d^2*n^3 - 4*b^3*d^2*n^2*log(c) - 4*a*b^2*d^2*n^2)*(d*sqrt
(x) + e)^4/(e^5*x^2) - 400*(b^3*d^3*n^3 - 3*b^3*d^3*n^2*log(c) - 3*a*b^2*d^3*n^2)*(d*sqrt(x) + e)^3/(e^5*x^(3/
2)) + 450*(b^3*d^4*n^3 - 2*b^3*d^4*n^2*log(c) - 2*a*b^2*d^4*n^2)*(d*sqrt(x) + e)^2/(e^5*x) - 360*(b^3*d^5*n^3
- b^3*d^5*n^2*log(c) - a*b^2*d^5*n^2)*(d*sqrt(x) + e)/(e^5*sqrt(x)))*log((d*sqrt(x) + e)/sqrt(x))^2 - 60*(100*
(b^3*n^3 - 6*b^3*n^2*log(c) + 18*b^3*n*log(c)^2 - 6*a*b^2*n^2 + 36*a*b^2*n*log(c) + 18*a^2*b*n)*(d*sqrt(x) + e
)^6/(e^5*x^3) - 432*(2*b^3*d*n^3 - 10*b^3*d*n^2*log(c) + 25*b^3*d*n*log(c)^2 - 10*a*b^2*d*n^2 + 50*a*b^2*d*n*l
og(c) + 25*a^2*b*d*n)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) + 3375*(b^3*d^2*n^3 - 4*b^3*d^2*n^2*log(c) + 8*b^3*d^2*n
*log(c)^2 - 4*a*b^2*d^2*n^2 + 16*a*b^2*d^2*n*log(c) + 8*a^2*b*d^2*n)*(d*sqrt(x) + e)^4/(e^5*x^2) - 4000*(2*b^3
*d^3*n^3 - 6*b^3*d^3*n^2*log(c) + 9*b^3*d^3*n*log(c)^2 - 6*a*b^2*d^3*n^2 + 18*a*b^2*d^3*n*log(c) + 9*a^2*b*d^3
*n)*(d*sqrt(x) + e)^3/(e^5*x^(3/2)) + 13500*(b^3*d^4*n^3 - 2*b^3*d^4*n^2*log(c) + 2*b^3*d^4*n*log(c)^2 - 2*a*b
^2*d^4*n^2 + 4*a*b^2*d^4*n*log(c) + 2*a^2*b*d^4*n)*(d*sqrt(x) + e)^2/(e^5*x) - 10800*(2*b^3*d^5*n^3 - 2*b^3*d^
5*n^2*log(c) + b^3*d^5*n*log(c)^2 - 2*a*b^2*d^5*n^2 + 2*a*b^2*d^5*n*log(c) + a^2*b*d^5*n)*(d*sqrt(x) + e)/(e^5
*sqrt(x)))*log((d*sqrt(x) + e)/sqrt(x)) + 1000*(b^3*n^3 - 6*b^3*n^2*log(c) + 18*b^3*n*log(c)^2 - 36*b^3*log(c)
^3 - 6*a*b^2*n^2 + 36*a*b^2*n*log(c) - 108*a*b^2*log(c)^2 + 18*a^2*b*n - 108*a^2*b*log(c) - 36*a^3)*(d*sqrt(x)
+ e)^6/(e^5*x^3) - 1728*(6*b^3*d*n^3 - 30*b^3*d*n^2*log(c) + 75*b^3*d*n*log(c)^2 - 125*b^3*d*log(c)^3 - 30*a*
b^2*d*n^2 + 150*a*b^2*d*n*log(c) - 375*a*b^2*d*log(c)^2 + 75*a^2*b*d*n - 375*a^2*b*d*log(c) - 125*a^3*d)*(d*sq
rt(x) + e)^5/(e^5*x^(5/2)) + 16875*(3*b^3*d^2*n^3 - 12*b^3*d^2*n^2*log(c) + 24*b^3*d^2*n*log(c)^2 - 32*b^3*d^2
*log(c)^3 - 12*a*b^2*d^2*n^2 + 48*a*b^2*d^2*n*log(c) - 96*a*b^2*d^2*log(c)^2 + 24*a^2*b*d^2*n - 96*a^2*b*d^2*l
og(c) - 32*a^3*d^2)*(d*sqrt(x) + e)^4/(e^5*x^2) - 80000*(2*b^3*d^3*n^3 - 6*b^3*d^3*n^2*log(c) + 9*b^3*d^3*n*lo
g(c)^2 - 9*b^3*d^3*log(c)^3 - 6*a*b^2*d^3*n^2 + 18*a*b^2*d^3*n*log(c) - 27*a*b^2*d^3*log(c)^2 + 9*a^2*b*d^3*n
- 27*a^2*b*d^3*log(c) - 9*a^3*d^3)*(d*sqrt(x) + e)^3/(e^5*x^(3/2)) + 135000*(3*b^3*d^4*n^3 - 6*b^3*d^4*n^2*log
(c) + 6*b^3*d^4*n*log(c)^2 - 4*b^3*d^4*log(c)^3 - 6*a*b^2*d^4*n^2 + 12*a*b^2*d^4*n*log(c) - 12*a*b^2*d^4*log(c
)^2 + 6*a^2*b*d^4*n - 12*a^2*b*d^4*log(c) - 4*a^3*d^4)*(d*sqrt(x) + e)^2/(e^5*x) - 216000*(6*b^3*d^5*n^3 - 6*b
^3*d^5*n^2*log(c) + 3*b^3*d^5*n*log(c)^2 - b^3*d^5*log(c)^3 - 6*a*b^2*d^5*n^2 + 6*a*b^2*d^5*n*log(c) - 3*a*b^2
*d^5*log(c)^2 + 3*a^2*b*d^5*n - 3*a^2*b*d^5*log(c) - a^3*d^5)*(d*sqrt(x) + e)/(e^5*sqrt(x)))/e

Mupad [B] (verification not implemented)

Time = 9.31 (sec) , antiderivative size = 989, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx=\text {Too large to display} \]

[In]

int((a + b*log(c*(d + e/x^(1/2))^n))^3/x^4,x)

[Out]

(b^3*n^3)/(108*x^3) - (b^3*log(c*(d + e/x^(1/2))^n)^3)/(3*x^3) - a^3/(3*x^3) - (a*b^2*log(c*(d + e/x^(1/2))^n)
^2)/x^3 + (b^3*n*log(c*(d + e/x^(1/2))^n)^2)/(6*x^3) - (b^3*n^2*log(c*(d + e/x^(1/2))^n))/(18*x^3) - (a*b^2*n^
2)/(18*x^3) + (b^3*d^6*log(c*(d + e/x^(1/2))^n)^3)/(3*e^6) - (a^2*b*log(c*(d + e/x^(1/2))^n))/x^3 + (a^2*b*n)/
(6*x^3) + (a*b^2*n*log(c*(d + e/x^(1/2))^n))/(3*x^3) + (13489*b^3*d^6*n^3*log(d + e/x^(1/2)))/(1800*e^6) + (91
9*b^3*d^2*n^3)/(7200*e^2*x^2) + (4669*b^3*d^4*n^3)/(3600*e^4*x) - (2059*b^3*d^3*n^3)/(5400*e^3*x^(3/2)) - (134
89*b^3*d^5*n^3)/(1800*e^5*x^(1/2)) + (a*b^2*d^6*log(c*(d + e/x^(1/2))^n)^2)/e^6 - (49*b^3*d^6*n*log(c*(d + e/x
^(1/2))^n)^2)/(20*e^6) - (91*b^3*d*n^3)/(2250*e*x^(5/2)) + (a^2*b*d^6*n*log(d + e/x^(1/2)))/e^6 - (b^3*d*n*log
(c*(d + e/x^(1/2))^n)^2)/(5*e*x^(5/2)) + (11*b^3*d*n^2*log(c*(d + e/x^(1/2))^n))/(75*e*x^(5/2)) + (a^2*b*d^2*n
)/(4*e^2*x^2) + (a^2*b*d^4*n)/(2*e^4*x) + (11*a*b^2*d*n^2)/(75*e*x^(5/2)) - (a^2*b*d^3*n)/(3*e^3*x^(3/2)) - (a
^2*b*d^5*n)/(e^5*x^(1/2)) - (49*a*b^2*d^6*n^2*log(d + e/x^(1/2)))/(10*e^6) + (b^3*d^2*n*log(c*(d + e/x^(1/2))^
n)^2)/(4*e^2*x^2) - (37*b^3*d^2*n^2*log(c*(d + e/x^(1/2))^n))/(120*e^2*x^2) + (b^3*d^4*n*log(c*(d + e/x^(1/2))
^n)^2)/(2*e^4*x) - (29*b^3*d^4*n^2*log(c*(d + e/x^(1/2))^n))/(20*e^4*x) - (b^3*d^3*n*log(c*(d + e/x^(1/2))^n)^
2)/(3*e^3*x^(3/2)) + (19*b^3*d^3*n^2*log(c*(d + e/x^(1/2))^n))/(30*e^3*x^(3/2)) - (b^3*d^5*n*log(c*(d + e/x^(1
/2))^n)^2)/(e^5*x^(1/2)) + (49*b^3*d^5*n^2*log(c*(d + e/x^(1/2))^n))/(10*e^5*x^(1/2)) - (37*a*b^2*d^2*n^2)/(12
0*e^2*x^2) - (29*a*b^2*d^4*n^2)/(20*e^4*x) + (19*a*b^2*d^3*n^2)/(30*e^3*x^(3/2)) + (49*a*b^2*d^5*n^2)/(10*e^5*
x^(1/2)) - (a^2*b*d*n)/(5*e*x^(5/2)) - (2*a*b^2*d*n*log(c*(d + e/x^(1/2))^n))/(5*e*x^(5/2)) + (a*b^2*d^2*n*log
(c*(d + e/x^(1/2))^n))/(2*e^2*x^2) + (a*b^2*d^4*n*log(c*(d + e/x^(1/2))^n))/(e^4*x) - (2*a*b^2*d^3*n*log(c*(d
+ e/x^(1/2))^n))/(3*e^3*x^(3/2)) - (2*a*b^2*d^5*n*log(c*(d + e/x^(1/2))^n))/(e^5*x^(1/2))