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

Optimal. Leaf size=907 \[ \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}} \]

[Out]

-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)+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+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^6-12/125*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)

________________________________________________________________________________________

Rubi [A]  time = 1.01, antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, 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 {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}} \]

Antiderivative was successfully verified.

[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 2295

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

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 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 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 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 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])

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^4} \, dx &=-\left (2 \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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 ) \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^5}+\frac {\left (20 d^3\right ) \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^5}-\frac {\left (10 d^4\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^5}+\frac {\left (2 d^5\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^5}\\ &=-\frac {2 \operatorname {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) \operatorname {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 ) \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^6}+\frac {\left (20 d^3\right ) \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^6}-\frac {\left (10 d^4\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^6}+\frac {\left (2 d^5\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^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) \operatorname {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) \operatorname {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 ) \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 )}{e^6}-\frac {\left (20 b d^3 n\right ) \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^6}+\frac {\left (15 b d^4 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^6}-\frac {\left (6 b d^5 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^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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [A]  time = 1.79, size = 950, normalized size = 1.05 \[ \frac {-72000 b^3 n^3 x^3 \log ^3\left (d+\frac {e}{\sqrt {x}}\right ) d^6+809340 b^3 n^3 x^3 \log \left (\sqrt {x} d+e\right ) d^6-529200 a b^2 n^2 x^3 \log \left (\sqrt {x} d+e\right ) d^6+108000 a^2 b n x^3 \log \left (\sqrt {x} d+e\right ) d^6+5400 b^2 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 (\sqrt {x} d+e\right )-\log (x)\right ) d^6-404670 b^3 n^3 x^3 \log (x) d^6+264600 a b^2 n^2 x^3 \log (x) d^6-54000 a^2 b n x^3 \log (x) d^6+5400 b^2 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 (\sqrt {x} d+e\right )-10 b n \log (x)\right ) d^6-809340 b^3 e n^3 x^{5/2} d^5+529200 a b^2 e n^2 x^{5/2} d^5-108000 a^2 b e n x^{5/2} d^5+140070 b^3 e^2 n^3 x^2 d^4-156600 a b^2 e^2 n^2 x^2 d^4+54000 a^2 b e^2 n x^2 d^4-41180 b^3 e^3 n^3 x^{3/2} d^3+68400 a b^2 e^3 n^2 x^{3/2} d^3-36000 a^2 b e^3 n x^{3/2} d^3+13785 b^3 e^4 n^3 x d^2-33300 a b^2 e^4 n^2 x d^2+27000 a^2 b e^4 n x d^2-4368 b^3 e^5 n^3 \sqrt {x} d+15840 a b^2 e^5 n^2 \sqrt {x} d-21600 a^2 b e^5 n \sqrt {x} d-36000 a^3 e^6+1000 b^3 e^6 n^3-36000 b^3 e^6 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-6000 a b^2 e^6 n^2+18000 a^2 b e^6 n+1800 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (60 b n x^3 \log \left (\sqrt {x} d+e\right ) d^6-30 b n x^3 \log (x) d^6+e \left (-60 b n x^{5/2} d^5+30 b e n x^2 d^4-20 b e^2 n x^{3/2} d^3+15 b e^3 n x d^2-12 b e^4 n \sqrt {x} d-60 a e^5+10 b e^5 n\right )\right )-60 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (180 b n (49 b n-20 a) x^3 \log \left (\sqrt {x} d+e\right ) d^6+90 b n (20 a-49 b n) x^3 \log (x) d^6+1800 a^2 e^6+b^2 e n^2 \left (-8820 x^{5/2} d^5+2610 e x^2 d^4-1140 e^2 x^{3/2} d^3+555 e^3 x d^2-264 e^4 \sqrt {x} d+100 e^5\right )-60 a b e n \left (-60 x^{5/2} d^5+30 e x^2 d^4-20 e^2 x^{3/2} d^3+15 e^3 x d^2-12 e^4 \sqrt {x} d+10 e^5\right )\right )}{108000 e^6 x^3} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.48, size = 1203, normalized size = 1.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.86, size = 3651, normalized size = 4.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108000*(216000*(d*sqrt(x) + e)*b^3*d^5*n^3*log((d*sqrt(x) + e)/sqrt(x))^3/sqrt(x) - 540000*(d*sqrt(x) + e)^2
*b^3*d^4*n^3*log((d*sqrt(x) + e)/sqrt(x))^3/x - 648000*(d*sqrt(x) + e)*b^3*d^5*n^3*log((d*sqrt(x) + e)/sqrt(x)
)^2/sqrt(x) + 648000*(d*sqrt(x) + e)*b^3*d^5*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))^2/sqrt(x) + 720000*(d*sqr
t(x) + e)^3*b^3*d^3*n^3*log((d*sqrt(x) + e)/sqrt(x))^3/x^(3/2) + 810000*(d*sqrt(x) + e)^2*b^3*d^4*n^3*log((d*s
qrt(x) + e)/sqrt(x))^2/x - 1620000*(d*sqrt(x) + e)^2*b^3*d^4*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))^2/x - 540
000*(d*sqrt(x) + e)^4*b^3*d^2*n^3*log((d*sqrt(x) + e)/sqrt(x))^3/x^2 + 1296000*(d*sqrt(x) + e)*b^3*d^5*n^3*log
((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 1296000*(d*sqrt(x) + e)*b^3*d^5*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))/sq
rt(x) + 648000*(d*sqrt(x) + e)*b^3*d^5*n*log(c)^2*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 720000*(d*sqrt(x) + e
)^3*b^3*d^3*n^3*log((d*sqrt(x) + e)/sqrt(x))^2/x^(3/2) + 648000*(d*sqrt(x) + e)*a*b^2*d^5*n^2*log((d*sqrt(x) +
e)/sqrt(x))^2/sqrt(x) + 2160000*(d*sqrt(x) + e)^3*b^3*d^3*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))^2/x^(3/2) +
216000*(d*sqrt(x) + e)^5*b^3*d*n^3*log((d*sqrt(x) + e)/sqrt(x))^3/x^(5/2) - 810000*(d*sqrt(x) + e)^2*b^3*d^4*
n^3*log((d*sqrt(x) + e)/sqrt(x))/x + 1620000*(d*sqrt(x) + e)^2*b^3*d^4*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))
/x - 1620000*(d*sqrt(x) + e)^2*b^3*d^4*n*log(c)^2*log((d*sqrt(x) + e)/sqrt(x))/x + 405000*(d*sqrt(x) + e)^4*b^
3*d^2*n^3*log((d*sqrt(x) + e)/sqrt(x))^2/x^2 - 1620000*(d*sqrt(x) + e)^2*a*b^2*d^4*n^2*log((d*sqrt(x) + e)/sqr
t(x))^2/x - 1620000*(d*sqrt(x) + e)^4*b^3*d^2*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))^2/x^2 - 36000*(d*sqrt(x)
+ e)^6*b^3*n^3*log((d*sqrt(x) + e)/sqrt(x))^3/x^3 - 1296000*(d*sqrt(x) + e)*b^3*d^5*n^3/sqrt(x) + 1296000*(d*
sqrt(x) + e)*b^3*d^5*n^2*log(c)/sqrt(x) - 648000*(d*sqrt(x) + e)*b^3*d^5*n*log(c)^2/sqrt(x) + 216000*(d*sqrt(x
) + e)*b^3*d^5*log(c)^3/sqrt(x) + 480000*(d*sqrt(x) + e)^3*b^3*d^3*n^3*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) -
1296000*(d*sqrt(x) + e)*a*b^2*d^5*n^2*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 1440000*(d*sqrt(x) + e)^3*b^3*d^3
*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 1296000*(d*sqrt(x) + e)*a*b^2*d^5*n*log(c)*log((d*sqrt(x) +
e)/sqrt(x))/sqrt(x) + 2160000*(d*sqrt(x) + e)^3*b^3*d^3*n*log(c)^2*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) - 129
600*(d*sqrt(x) + e)^5*b^3*d*n^3*log((d*sqrt(x) + e)/sqrt(x))^2/x^(5/2) + 2160000*(d*sqrt(x) + e)^3*a*b^2*d^3*n
^2*log((d*sqrt(x) + e)/sqrt(x))^2/x^(3/2) + 648000*(d*sqrt(x) + e)^5*b^3*d*n^2*log(c)*log((d*sqrt(x) + e)/sqrt
(x))^2/x^(5/2) + 405000*(d*sqrt(x) + e)^2*b^3*d^4*n^3/x - 810000*(d*sqrt(x) + e)^2*b^3*d^4*n^2*log(c)/x + 8100
00*(d*sqrt(x) + e)^2*b^3*d^4*n*log(c)^2/x - 540000*(d*sqrt(x) + e)^2*b^3*d^4*log(c)^3/x - 202500*(d*sqrt(x) +
e)^4*b^3*d^2*n^3*log((d*sqrt(x) + e)/sqrt(x))/x^2 + 1620000*(d*sqrt(x) + e)^2*a*b^2*d^4*n^2*log((d*sqrt(x) + e
)/sqrt(x))/x + 810000*(d*sqrt(x) + e)^4*b^3*d^2*n^2*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 3240000*(d*sqrt(
x) + e)^2*a*b^2*d^4*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x - 1620000*(d*sqrt(x) + e)^4*b^3*d^2*n*log(c)^2*log
((d*sqrt(x) + e)/sqrt(x))/x^2 + 18000*(d*sqrt(x) + e)^6*b^3*n^3*log((d*sqrt(x) + e)/sqrt(x))^2/x^3 - 1620000*(
d*sqrt(x) + e)^4*a*b^2*d^2*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/x^2 - 108000*(d*sqrt(x) + e)^6*b^3*n^2*log(c)*lo
g((d*sqrt(x) + e)/sqrt(x))^2/x^3 - 160000*(d*sqrt(x) + e)^3*b^3*d^3*n^3/x^(3/2) + 1296000*(d*sqrt(x) + e)*a*b^
2*d^5*n^2/sqrt(x) + 480000*(d*sqrt(x) + e)^3*b^3*d^3*n^2*log(c)/x^(3/2) - 1296000*(d*sqrt(x) + e)*a*b^2*d^5*n*
log(c)/sqrt(x) - 720000*(d*sqrt(x) + e)^3*b^3*d^3*n*log(c)^2/x^(3/2) + 648000*(d*sqrt(x) + e)*a*b^2*d^5*log(c)
^2/sqrt(x) + 720000*(d*sqrt(x) + e)^3*b^3*d^3*log(c)^3/x^(3/2) + 51840*(d*sqrt(x) + e)^5*b^3*d*n^3*log((d*sqrt
(x) + e)/sqrt(x))/x^(5/2) - 1440000*(d*sqrt(x) + e)^3*a*b^2*d^3*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 648
000*(d*sqrt(x) + e)*a^2*b*d^5*n*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 259200*(d*sqrt(x) + e)^5*b^3*d*n^2*log(
c)*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) + 4320000*(d*sqrt(x) + e)^3*a*b^2*d^3*n*log(c)*log((d*sqrt(x) + e)/sqr
t(x))/x^(3/2) + 648000*(d*sqrt(x) + e)^5*b^3*d*n*log(c)^2*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) + 648000*(d*sqr
t(x) + e)^5*a*b^2*d*n^2*log((d*sqrt(x) + e)/sqrt(x))^2/x^(5/2) + 50625*(d*sqrt(x) + e)^4*b^3*d^2*n^3/x^2 - 810
000*(d*sqrt(x) + e)^2*a*b^2*d^4*n^2/x - 202500*(d*sqrt(x) + e)^4*b^3*d^2*n^2*log(c)/x^2 + 1620000*(d*sqrt(x) +
e)^2*a*b^2*d^4*n*log(c)/x + 405000*(d*sqrt(x) + e)^4*b^3*d^2*n*log(c)^2/x^2 - 1620000*(d*sqrt(x) + e)^2*a*b^2
*d^4*log(c)^2/x - 540000*(d*sqrt(x) + e)^4*b^3*d^2*log(c)^3/x^2 - 6000*(d*sqrt(x) + e)^6*b^3*n^3*log((d*sqrt(x
) + e)/sqrt(x))/x^3 + 810000*(d*sqrt(x) + e)^4*a*b^2*d^2*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 1620000*(d*sqr
t(x) + e)^2*a^2*b*d^4*n*log((d*sqrt(x) + e)/sqrt(x))/x + 36000*(d*sqrt(x) + e)^6*b^3*n^2*log(c)*log((d*sqrt(x)
+ e)/sqrt(x))/x^3 - 3240000*(d*sqrt(x) + e)^4*a*b^2*d^2*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 108000*(d
*sqrt(x) + e)^6*b^3*n*log(c)^2*log((d*sqrt(x) + e)/sqrt(x))/x^3 - 108000*(d*sqrt(x) + e)^6*a*b^2*n^2*log((d*sq
rt(x) + e)/sqrt(x))^2/x^3 - 10368*(d*sqrt(x) + e)^5*b^3*d*n^3/x^(5/2) + 480000*(d*sqrt(x) + e)^3*a*b^2*d^3*n^2
/x^(3/2) - 648000*(d*sqrt(x) + e)*a^2*b*d^5*n/sqrt(x) + 51840*(d*sqrt(x) + e)^5*b^3*d*n^2*log(c)/x^(5/2) - 144
0000*(d*sqrt(x) + e)^3*a*b^2*d^3*n*log(c)/x^(3/2) + 648000*(d*sqrt(x) + e)*a^2*b*d^5*log(c)/sqrt(x) - 129600*(
d*sqrt(x) + e)^5*b^3*d*n*log(c)^2/x^(5/2) + 2160000*(d*sqrt(x) + e)^3*a*b^2*d^3*log(c)^2/x^(3/2) + 216000*(d*s
qrt(x) + e)^5*b^3*d*log(c)^3/x^(5/2) - 259200*(d*sqrt(x) + e)^5*a*b^2*d*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^(5/
2) + 2160000*(d*sqrt(x) + e)^3*a^2*b*d^3*n*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 1296000*(d*sqrt(x) + e)^5*a*
b^2*d*n*log(c)*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) + 1000*(d*sqrt(x) + e)^6*b^3*n^3/x^3 - 202500*(d*sqrt(x) +
e)^4*a*b^2*d^2*n^2/x^2 + 810000*(d*sqrt(x) + e)^2*a^2*b*d^4*n/x - 6000*(d*sqrt(x) + e)^6*b^3*n^2*log(c)/x^3 +
810000*(d*sqrt(x) + e)^4*a*b^2*d^2*n*log(c)/x^2 - 1620000*(d*sqrt(x) + e)^2*a^2*b*d^4*log(c)/x + 18000*(d*sqr
t(x) + e)^6*b^3*n*log(c)^2/x^3 - 1620000*(d*sqrt(x) + e)^4*a*b^2*d^2*log(c)^2/x^2 - 36000*(d*sqrt(x) + e)^6*b^
3*log(c)^3/x^3 + 36000*(d*sqrt(x) + e)^6*a*b^2*n^2*log((d*sqrt(x) + e)/sqrt(x))/x^3 - 1620000*(d*sqrt(x) + e)^
4*a^2*b*d^2*n*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 216000*(d*sqrt(x) + e)^6*a*b^2*n*log(c)*log((d*sqrt(x) + e)/s
qrt(x))/x^3 + 51840*(d*sqrt(x) + e)^5*a*b^2*d*n^2/x^(5/2) - 720000*(d*sqrt(x) + e)^3*a^2*b*d^3*n/x^(3/2) + 216
000*(d*sqrt(x) + e)*a^3*d^5/sqrt(x) - 259200*(d*sqrt(x) + e)^5*a*b^2*d*n*log(c)/x^(5/2) + 2160000*(d*sqrt(x) +
e)^3*a^2*b*d^3*log(c)/x^(3/2) + 648000*(d*sqrt(x) + e)^5*a*b^2*d*log(c)^2/x^(5/2) + 648000*(d*sqrt(x) + e)^5*
a^2*b*d*n*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) - 6000*(d*sqrt(x) + e)^6*a*b^2*n^2/x^3 + 405000*(d*sqrt(x) + e)
^4*a^2*b*d^2*n/x^2 - 540000*(d*sqrt(x) + e)^2*a^3*d^4/x + 36000*(d*sqrt(x) + e)^6*a*b^2*n*log(c)/x^3 - 1620000
*(d*sqrt(x) + e)^4*a^2*b*d^2*log(c)/x^2 - 108000*(d*sqrt(x) + e)^6*a*b^2*log(c)^2/x^3 - 108000*(d*sqrt(x) + e)
^6*a^2*b*n*log((d*sqrt(x) + e)/sqrt(x))/x^3 - 129600*(d*sqrt(x) + e)^5*a^2*b*d*n/x^(5/2) + 720000*(d*sqrt(x) +
e)^3*a^3*d^3/x^(3/2) + 648000*(d*sqrt(x) + e)^5*a^2*b*d*log(c)/x^(5/2) + 18000*(d*sqrt(x) + e)^6*a^2*b*n/x^3
- 540000*(d*sqrt(x) + e)^4*a^3*d^2/x^2 - 108000*(d*sqrt(x) + e)^6*a^2*b*log(c)/x^3 + 216000*(d*sqrt(x) + e)^5*
a^3*d/x^(5/2) - 36000*(d*sqrt(x) + e)^6*a^3/x^3)*e^(-6)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a \right )^{3}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 864, normalized size = 0.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 8.18, size = 989, normalized size = 1.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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