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

Optimal. Leaf size=926 \[ -\frac {2^{-p} 5^{-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 {2\ 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\ 8^{-p} 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 {24\ 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\ 6^{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 {252\ 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\ 2^{1-2 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 {8\ 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} 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 {2 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}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.56, antiderivative size = 926, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2454, 2401, 2389, 2299, 2181, 2390, 2309} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((5^(-1 - p)*Gamma[1 + p, (-10*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(2^p*c^10
*e^10*E^((10*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p)) + (2*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*d^2*Gamma[1 + p, (-8*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(8^p*c^8*e^10*
E^((8*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) + (24*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
*6^(1 - p)*d^4*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(c^6*e^10
*E^((6*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) + (252*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*2^(1 - 2*p)*d^6*Gamma[1 + p, (-4*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^
p)/(c^4*e^10*E^((4*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) + (8*3^(1 - p)*d^7*Gamma[1 + p, (-3*(a + b*L
og[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*d^8*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(
2^p*c^2*e^10*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p) + (2*d^9*Gamma[1 + p, -((a + b*Log[c*(d + e/
Sqrt[x])])/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(c*e^10*E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p)

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

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 )\right )\right )^p}{x^6} \, dx &=-\left (2 \operatorname {Subst}\left (\int x^9 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \left (-\frac {d^9 (a+b \log (c (d+e x)))^p}{e^9}+\frac {9 d^8 (d+e x) (a+b \log (c (d+e x)))^p}{e^9}-\frac {36 d^7 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^9}+\frac {84 d^6 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^9}-\frac {126 d^5 (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^9}+\frac {126 d^4 (d+e x)^5 (a+b \log (c (d+e x)))^p}{e^9}-\frac {84 d^3 (d+e x)^6 (a+b \log (c (d+e x)))^p}{e^9}+\frac {36 d^2 (d+e x)^7 (a+b \log (c (d+e x)))^p}{e^9}-\frac {9 d (d+e x)^8 (a+b \log (c (d+e x)))^p}{e^9}+\frac {(d+e x)^9 (a+b \log (c (d+e x)))^p}{e^9}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int (d+e x)^9 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {(18 d) \operatorname {Subst}\left (\int (d+e x)^8 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (72 d^2\right ) \operatorname {Subst}\left (\int (d+e x)^7 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (168 d^3\right ) \operatorname {Subst}\left (\int (d+e x)^6 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (252 d^4\right ) \operatorname {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (252 d^5\right ) \operatorname {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (168 d^6\right ) \operatorname {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (72 d^7\right ) \operatorname {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (18 d^8\right ) \operatorname {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (2 d^9\right ) \operatorname {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^9 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {(18 d) \operatorname {Subst}\left (\int x^8 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (72 d^2\right ) \operatorname {Subst}\left (\int x^7 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (168 d^3\right ) \operatorname {Subst}\left (\int x^6 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (252 d^4\right ) \operatorname {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (252 d^5\right ) \operatorname {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (168 d^6\right ) \operatorname {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (72 d^7\right ) \operatorname {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (18 d^8\right ) \operatorname {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (2 d^9\right ) \operatorname {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}\\ &=-\frac {2 \operatorname {Subst}\left (\int e^{10 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^{10} e^{10}}+\frac {(18 d) \operatorname {Subst}\left (\int e^{9 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^9 e^{10}}-\frac {\left (72 d^2\right ) \operatorname {Subst}\left (\int e^{8 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^8 e^{10}}+\frac {\left (168 d^3\right ) \operatorname {Subst}\left (\int e^{7 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^7 e^{10}}-\frac {\left (252 d^4\right ) \operatorname {Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^6 e^{10}}+\frac {\left (252 d^5\right ) \operatorname {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^5 e^{10}}-\frac {\left (168 d^6\right ) \operatorname {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^4 e^{10}}+\frac {\left (72 d^7\right ) \operatorname {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^3 e^{10}}-\frac {\left (18 d^8\right ) \operatorname {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^2 e^{10}}+\frac {\left (2 d^9\right ) \operatorname {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c e^{10}}\\ &=-\frac {2^{-p} 5^{-1-p} e^{-\frac {10 a}{b}} \Gamma \left (1+p,-\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 {2\ 9^{-p} d e^{-\frac {9 a}{b}} \Gamma \left (1+p,-\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\ 8^{-p} d^2 e^{-\frac {8 a}{b}} \Gamma \left (1+p,-\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 {24\ 7^{-p} d^3 e^{-\frac {7 a}{b}} \Gamma \left (1+p,-\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\ 6^{1-p} d^4 e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\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 {252\ 5^{-1-p} d^5 e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\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\ 2^{1-2 p} d^6 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\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 {8\ 3^{1-p} d^7 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\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} d^8 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\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 {2 d^9 e^{-\frac {a}{b}} \Gamma \left (1+p,-\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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 3.31, size = 525, normalized size = 0.57 \[ \frac {5^{-p-1} 504^{-p} e^{-\frac {10 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \left (c d e^{a/b} \left (2^{3 p+1} 5^{p+1} 7^p \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d e^{a/b} \left (c d 2^p e^{a/b} \left (2^{2 p+3} 3^{2 p+1} 5^{p+1} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d 7^p e^{a/b} \left (c d e^{a/b} \left (7\ 36^{p+1} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d 3^p 5^{p+1} e^{a/b} \left (c d 2^p e^{a/b} \left (3\ 2^{p+3} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d 3^p e^{a/b} \left (c d 2^{p+1} e^{a/b} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )-9 \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )\right )-14\ 3^{p+1} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )\right )-7\ 30^{p+1} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )\right )-7^p 45^{p+1} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )\right )-252^p \Gamma \left (p+1,-\frac {10 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )}{c^{10} e^{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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*Log[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/Sq
rt[x])])^p)/(504^p*c^10*e^10*E^((10*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p)

________________________________________________________________________________________

fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (\frac {c d x + c e \sqrt {x}}{x}\right ) + a\right )}^{p}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (d +\frac {e}{\sqrt {x}}\right ) c \right )+a \right )^{p}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,\left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}^p}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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))))**p/x**6,x)

[Out]

Timed out

________________________________________________________________________________________