\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})))^p}{x^6} \, dx\) [549]

Optimal result

Integrand size = 22, antiderivative size = 926 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x^6} \, dx =\text {Too large to display} \] Output:

-5^(-1-p)*GAMMA(p+1,(-10*a-10*b*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^( 
1/2))))^p/(2^p)/c^10/e^10/exp(10*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)+2 
*d*GAMMA(p+1,(-9*a-9*b*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p 
/(9^p)/c^9/e^10/exp(9*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)-9*d^2*GAMMA( 
p+1,(-8*a-8*b*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p/(8^p)/c^ 
8/e^10/exp(8*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)+24*d^3*GAMMA(p+1,(-7* 
a-7*b*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p/(7^p)/c^7/e^10/e 
xp(7*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)-7*6^(1-p)*d^4*GAMMA(p+1,(-6*a 
-6*b*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p/c^6/e^10/exp(6*a/ 
b)/((-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)+252*5^(-1-p)*d^5*GAMMA(p+1,(-5*a-5*b 
*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p/c^5/e^10/exp(5*a/b)/( 
(-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)-21*2^(1-2*p)*d^6*GAMMA(p+1,(-4*a-4*b*ln( 
c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p/c^4/e^10/exp(4*a/b)/((-(a 
+b*ln(c*(d+e/x^(1/2))))/b)^p)+8*3^(1-p)*d^7*GAMMA(p+1,(-3*a-3*b*ln(c*(d+e/ 
x^(1/2))))/b)*(a+b*ln(c*(d+e/x^(1/2))))^p/c^3/e^10/exp(3*a/b)/((-(a+b*ln(c 
*(d+e/x^(1/2))))/b)^p)-9*d^8*GAMMA(p+1,(-2*a-2*b*ln(c*(d+e/x^(1/2))))/b)*( 
a+b*ln(c*(d+e/x^(1/2))))^p/(2^p)/c^2/e^10/exp(2*a/b)/((-(a+b*ln(c*(d+e/x^( 
1/2))))/b)^p)+2*d^9*GAMMA(p+1,-(a+b*ln(c*(d+e/x^(1/2))))/b)*(a+b*ln(c*(d+e 
/x^(1/2))))^p/c/e^10/exp(a/b)/((-(a+b*ln(c*(d+e/x^(1/2))))/b)^p)
 

Mathematica [A] (verified)

Time = 5.95 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.57 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x^6} \, dx=\frac {5^{-1-p} 504^{-p} e^{-\frac {10 a}{b}} \left (-252^p \Gamma \left (1+p,-\frac {10 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d e^{a/b} \left (2^{1+3 p} 5^{1+p} 7^p \Gamma \left (1+p,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d e^{a/b} \left (-7^p 45^{1+p} \Gamma \left (1+p,-\frac {8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \left (2^{3+2 p} 3^{1+2 p} 5^{1+p} \Gamma \left (1+p,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+7^p c d e^{a/b} \left (-7 30^{1+p} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d e^{a/b} \left (7\ 36^{1+p} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+3^p 5^{1+p} c d e^{a/b} \left (-14 3^{1+p} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \left (3\ 2^{3+p} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+3^p c d e^{a/b} \left (-9 \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )\right )\right )\right )\right )\right )\right )\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^{10} e^{10}} \] Input:

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

Output:

(5^(-1 - p)*(-(252^p*Gamma[1 + p, (-10*(a + b*Log[c*(d + e/Sqrt[x])]))/b]) 
 + c*d*E^(a/b)*(2^(1 + 3*p)*5^(1 + p)*7^p*Gamma[1 + p, (-9*(a + b*Log[c*(d 
 + e/Sqrt[x])]))/b] + c*d*E^(a/b)*(-(7^p*45^(1 + p)*Gamma[1 + p, (-8*(a + 
b*Log[c*(d + e/Sqrt[x])]))/b]) + 2^p*c*d*E^(a/b)*(2^(3 + 2*p)*3^(1 + 2*p)* 
5^(1 + p)*Gamma[1 + p, (-7*(a + b*Log[c*(d + e/Sqrt[x])]))/b] + 7^p*c*d*E^ 
(a/b)*(-7*30^(1 + p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/Sqrt[x])]))/b] + 
 c*d*E^(a/b)*(7*36^(1 + p)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/Sqrt[x])]) 
)/b] + 3^p*5^(1 + p)*c*d*E^(a/b)*(-14*3^(1 + p)*Gamma[1 + p, (-4*(a + b*Lo 
g[c*(d + e/Sqrt[x])]))/b] + 2^p*c*d*E^(a/b)*(3*2^(3 + p)*Gamma[1 + p, (-3* 
(a + b*Log[c*(d + e/Sqrt[x])]))/b] + 3^p*c*d*E^(a/b)*(-9*Gamma[1 + p, (-2* 
(a + b*Log[c*(d + e/Sqrt[x])]))/b] + 2^(1 + p)*c*d*E^(a/b)*Gamma[1 + p, -( 
(a + b*Log[c*(d + e/Sqrt[x])])/b)])))))))))*(a + b*Log[c*(d + e/Sqrt[x])]) 
^p)/(504^p*c^10*e^10*E^((10*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p)
 

Rubi [A] (verified)

Time = 3.34 (sec) , antiderivative size = 929, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {10^{-p-1} e^{-\frac {10 a}{b}} \Gamma \left (p+1,-\frac {10 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^{10} e^{10}}-\frac {9^{-p} d e^{-\frac {9 a}{b}} \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^9 e^{10}}+\frac {9\ 2^{-3 p-1} d^2 e^{-\frac {8 a}{b}} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^8 e^{10}}-\frac {12\ 7^{-p} d^3 e^{-\frac {7 a}{b}} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^7 e^{10}}+\frac {7\ 2^{-p} 3^{1-p} d^4 e^{-\frac {6 a}{b}} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^6 e^{10}}-\frac {126\ 5^{-p-1} d^5 e^{-\frac {5 a}{b}} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^5 e^{10}}+\frac {21\ 4^{-p} d^6 e^{-\frac {4 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^4 e^{10}}-\frac {4\ 3^{1-p} d^7 e^{-\frac {3 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^3 e^{10}}+\frac {9\ 2^{-p-1} d^8 e^{-\frac {2 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^2 e^{10}}-\frac {d^9 e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c e^{10}}\right )\)

Input:

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

Output:

-2*((10^(-1 - p)*Gamma[1 + p, (-10*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + 
 b*Log[c*(d + e/Sqrt[x])])^p)/(c^10*e^10*E^((10*a)/b)*(-((a + b*Log[c*(d + 
 e/Sqrt[x])])/b))^p) - (d*Gamma[1 + p, (-9*(a + b*Log[c*(d + e/Sqrt[x])])) 
/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(9^p*c^9*e^10*E^((9*a)/b)*(-((a + b* 
Log[c*(d + e/Sqrt[x])])/b))^p) + (9*2^(-1 - 3*p)*d^2*Gamma[1 + p, (-8*(a + 
 b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(c^8*e^10 
*E^((8*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) - (12*d^3*Gamma[1 + 
p, (-7*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p 
)/(7^p*c^7*e^10*E^((7*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) + (7* 
3^(1 - p)*d^4*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*L 
og[c*(d + e/Sqrt[x])])^p)/(2^p*c^6*e^10*E^((6*a)/b)*(-((a + b*Log[c*(d + e 
/Sqrt[x])])/b))^p) - (126*5^(-1 - p)*d^5*Gamma[1 + p, (-5*(a + b*Log[c*(d 
+ e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(c^5*e^10*E^((5*a)/b) 
*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) + (21*d^6*Gamma[1 + p, (-4*(a + 
b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(4^p*c^4*e 
^10*E^((4*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) - (4*3^(1 - p)*d^ 
7*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/ 
Sqrt[x])])^p)/(c^3*e^10*E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^ 
p) + (9*2^(-1 - p)*d^8*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/Sqrt[x])]))/b] 
*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(c^2*e^10*E^((2*a)/b)*(-((a + b*Log[...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 5040*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**9*e*p**2 
*x**4 - 5040*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**9*e* 
p*x**4 - 1680*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**7*e 
**3*p**2*x**3 - 1680*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b 
*d**7*e**3*p*x**3 - 1008*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)* 
*p*b*d**5*e**5*p**2*x**2 - 1008*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))* 
b + a)**p*b*d**5*e**5*p*x**2 - 720*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x 
))*b + a)**p*b*d**3*e**7*p**2*x - 720*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqr 
t(x))*b + a)**p*b*d**3*e**7*p*x - 560*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqr 
t(x))*b + a)**p*b*d*e**9*p**2 - 560*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt( 
x))*b + a)**p*b*d*e**9*p + 5040*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)** 
p*log((sqrt(x)*c*d + c*e)/sqrt(x))*b*d**10*p*x**5 + 5040*(log((sqrt(x)*c*d 
 + c*e)/sqrt(x))*b + a)**p*a*d**10*p*x**5 - 5040*(log((sqrt(x)*c*d + c*e)/ 
sqrt(x))*b + a)**p*a*e**10*p - 5040*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + 
a)**p*a*e**10 + 2520*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**8*e* 
*2*p**2*x**4 + 2520*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**8*e** 
2*p*x**4 + 1260*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**6*e**4*p* 
*2*x**3 + 1260*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**6*e**4*p*x 
**3 + 840*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**4*e**6*p**2*x** 
2 + 840*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d**4*e**6*p*x**2 ...