\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^2))^p}{x^6} \, dx\) [555]

Optimal result

Integrand size = 24, antiderivative size = 1141 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx =\text {Too large to display} \] Output:

-5^(-1-p)*GAMMA(p+1,(-5*a-5*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^( 
1/2))^2))^p/c^5/e^10/exp(5*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)+2^(p+ 
1)*d*(d+e/x^(1/2))^9*GAMMA(p+1,1/2*(-9*a-9*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+ 
b*ln(c*(d+e/x^(1/2))^2))^p/(9^p)/e^10/exp(9/2*a/b)/(c*(d+e/x^(1/2))^2)^(9/ 
2)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)-9*d^2*GAMMA(p+1,(-4*a-4*b*ln(c*(d+ 
e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2))^2))^p/(4^p)/c^4/e^10/exp(4*a/b)/ 
((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)+3*2^(3+p)*d^3*(d+e/x^(1/2))^7*GAMMA(p 
+1,1/2*(-7*a-7*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2))^2))^p/( 
7^p)/e^10/exp(7/2*a/b)/(c*(d+e/x^(1/2))^2)^(7/2)/((-(a+b*ln(c*(d+e/x^(1/2) 
)^2))/b)^p)-14*3^(1-p)*d^4*GAMMA(p+1,(-3*a-3*b*ln(c*(d+e/x^(1/2))^2))/b)*( 
a+b*ln(c*(d+e/x^(1/2))^2))^p/c^3/e^10/exp(3*a/b)/((-(a+b*ln(c*(d+e/x^(1/2) 
)^2))/b)^p)+63*2^(2+p)*5^(-1-p)*d^5*(d+e/x^(1/2))^5*GAMMA(p+1,1/2*(-5*a-5* 
b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2))^2))^p/e^10/exp(5/2*a/b 
)/(c*(d+e/x^(1/2))^2)^(5/2)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)-21*2^(1-p 
)*d^6*GAMMA(p+1,(-2*a-2*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2) 
)^2))^p/c^2/e^10/exp(2*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)+2^(3+p)*3 
^(1-p)*d^7*(d+e/x^(1/2))^3*GAMMA(p+1,1/2*(-3*a-3*b*ln(c*(d+e/x^(1/2))^2))/ 
b)*(a+b*ln(c*(d+e/x^(1/2))^2))^p/e^10/exp(3/2*a/b)/(c*(d+e/x^(1/2))^2)^(3/ 
2)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)-9*d^8*GAMMA(p+1,-(a+b*ln(c*(d+e/x^ 
(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2))^2))^p/c/e^10/exp(a/b)/((-(a+b*ln(...
 

Mathematica [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx \] Input:

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^6,x]
 

Output:

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^6, x]
 

Rubi [A] (verified)

Time = 3.63 (sec) , antiderivative size = 1143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2904, 2848, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle -2 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^{9/2}}d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2848

\(\displaystyle -2 \int \left (\frac {\left (d+\frac {e}{\sqrt {x}}\right )^9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}-\frac {9 d \left (d+\frac {e}{\sqrt {x}}\right )^8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}+\frac {36 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}-\frac {84 d^3 \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}+\frac {126 d^4 \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}-\frac {126 d^5 \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}+\frac {84 d^6 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}-\frac {36 d^7 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}+\frac {9 d^8 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}-\frac {d^9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^9}\right )d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {5^{-p-1} e^{-\frac {5 a}{b}} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{2 c^5 e^{10}}-\frac {\left (\frac {2}{9}\right )^p d e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^9 \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{9/2}}+\frac {9\ 2^{-2 p-1} d^2 e^{-\frac {4 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{c^4 e^{10}}-\frac {3\ 2^{p+2} 7^{-p} d^3 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^7 \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{7/2}}+\frac {7\ 3^{1-p} d^4 e^{-\frac {3 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{c^3 e^{10}}-\frac {63\ 2^{p+1} 5^{-p-1} d^5 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^5 \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}}+\frac {21\ 2^{-p} d^6 e^{-\frac {2 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{c^2 e^{10}}-\frac {2^{p+2} 3^{1-p} d^7 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^3 \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}+\frac {9 d^8 e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{2 c e^{10}}-\frac {2^p d^9 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}\right )\)

Input:

Int[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^6,x]
 

Output:

-2*((5^(-1 - p)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b]*(a + 
 b*Log[c*(d + e/Sqrt[x])^2])^p)/(2*c^5*e^10*E^((5*a)/b)*(-((a + b*Log[c*(d 
 + e/Sqrt[x])^2])/b))^p) - ((2/9)^p*d*(d + e/Sqrt[x])^9*Gamma[1 + p, (-9*( 
a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p 
)/(e^10*E^((9*a)/(2*b))*(c*(d + e/Sqrt[x])^2)^(9/2)*(-((a + b*Log[c*(d + e 
/Sqrt[x])^2])/b))^p) + (9*2^(-1 - 2*p)*d^2*Gamma[1 + p, (-4*(a + b*Log[c*( 
d + e/Sqrt[x])^2]))/b]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(c^4*e^10*E^((4 
*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - (3*2^(2 + p)*d^3*(d + 
e/Sqrt[x])^7*Gamma[1 + p, (-7*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b)]*(a 
+ b*Log[c*(d + e/Sqrt[x])^2])^p)/(7^p*e^10*E^((7*a)/(2*b))*(c*(d + e/Sqrt[ 
x])^2)^(7/2)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) + (7*3^(1 - p)*d^4 
*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b]*(a + b*Log[c*(d + e 
/Sqrt[x])^2])^p)/(c^3*e^10*E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2]) 
/b))^p) - (63*2^(1 + p)*5^(-1 - p)*d^5*(d + e/Sqrt[x])^5*Gamma[1 + p, (-5* 
(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^ 
p)/(e^10*E^((5*a)/(2*b))*(c*(d + e/Sqrt[x])^2)^(5/2)*(-((a + b*Log[c*(d + 
e/Sqrt[x])^2])/b))^p) + (21*d^6*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/Sqrt[ 
x])^2]))/b]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(2^p*c^2*e^10*E^((2*a)/b)* 
(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - (2^(2 + p)*3^(1 - p)*d^7*(d + 
 e/Sqrt[x])^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b)]...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2848

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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])
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{2}\right )\right )}^{p}}{x^{6}}d x\]

Input:

int((a+b*ln(c*(d+e/x^(1/2))^2))^p/x^6,x)
 

Output:

int((a+b*ln(c*(d+e/x^(1/2))^2))^p/x^6,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^2))^p/x^6,x, algorithm="fricas")
 

Output:

integral((b*log((c*d^2*x + 2*c*d*e*sqrt(x) + c*e^2)/x) + a)^p/x^6, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*(d+e/x**(1/2))**2))**p/x**6,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^2))^p/x^6,x, algorithm="maxima")
 

Output:

integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x^6, x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/2))^2))^p/x^6,x, algorithm="giac")
 

Output:

integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^2\right )\right )}^p}{x^6} \,d x \] Input:

int((a + b*log(c*(d + e/x^(1/2))^2))^p/x^6,x)
 

Output:

int((a + b*log(c*(d + e/x^(1/2))^2))^p/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx=\text {too large to display} \] Input:

int((a+b*log(c*(d+e/x^(1/2))^2))^p/x^6,x)
 

Output:

( - 2520*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b 
*d**9*e*p**2*x**4 - 2520*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2 
)/x)*b + a)**p*b*d**9*e*p*x**4 - 840*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d** 
2*x + c*e**2)/x)*b + a)**p*b*d**7*e**3*p**2*x**3 - 840*sqrt(x)*(log((2*sqr 
t(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**7*e**3*p*x**3 - 504*sqrt 
(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**5*e**5*p* 
*2*x**2 - 504*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a) 
**p*b*d**5*e**5*p*x**2 - 360*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c* 
e**2)/x)*b + a)**p*b*d**3*e**7*p**2*x - 360*sqrt(x)*(log((2*sqrt(x)*c*d*e 
+ c*d**2*x + c*e**2)/x)*b + a)**p*b*d**3*e**7*p*x - 280*sqrt(x)*(log((2*sq 
rt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d*e**9*p**2 - 280*sqrt(x)* 
(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d*e**9*p + 1260* 
(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*log((2*sqrt(x)*c*d 
*e + c*d**2*x + c*e**2)/x)*b*d**10*p*x**5 + 1260*(log((2*sqrt(x)*c*d*e + c 
*d**2*x + c*e**2)/x)*b + a)**p*a*d**10*p*x**5 - 1260*(log((2*sqrt(x)*c*d*e 
 + c*d**2*x + c*e**2)/x)*b + a)**p*a*e**10*p - 1260*(log((2*sqrt(x)*c*d*e 
+ c*d**2*x + c*e**2)/x)*b + a)**p*a*e**10 + 1260*(log((2*sqrt(x)*c*d*e + c 
*d**2*x + c*e**2)/x)*b + a)**p*b*d**8*e**2*p**2*x**4 + 1260*(log((2*sqrt(x 
)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**8*e**2*p*x**4 + 630*(log((2 
*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**6*e**4*p**2*x**3 ...