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

Optimal. Leaf size=1141 \[ -\frac {5^{-1-p} e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\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}}{c^5 e^{10}}+\frac {2^{1+p} 9^{-p} d e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^9 \Gamma \left (1+p,-\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\ 4^{-p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\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^{3+p} 7^{-p} d^3 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^7 \Gamma \left (1+p,-\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 {14\ 3^{1-p} d^4 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\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^{2+p} 5^{-1-p} d^5 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^5 \Gamma \left (1+p,-\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^{1-p} d^6 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\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^{3+p} 3^{1-p} d^7 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^3 \Gamma \left (1+p,-\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 (1+p,-\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}}{c e^{10}}+\frac {2^{1+p} d^9 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \Gamma \left (1+p,-\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}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.19, antiderivative size = 1141, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \begin {gather*} -\frac {5^{-p-1} e^{-\frac {5 a}{b}} \text {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}}{c^5 e^{10}}+\frac {2^{p+1} 9^{-p} d e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^9 \text {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\ 4^{-p} d^2 e^{-\frac {4 a}{b}} \text {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+3} 7^{-p} d^3 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^7 \text {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 {14\ 3^{1-p} d^4 e^{-\frac {3 a}{b}} \text {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+2} 5^{-p-1} d^5 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^5 \text {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^{1-p} d^6 e^{-\frac {2 a}{b}} \text {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+3} 3^{1-p} d^7 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^3 \text {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}} \text {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}}{c e^{10}}+\frac {2^{p+1} d^9 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \text {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((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)/(c^5*e
^10*E^((5*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p)) + (2^(1 + 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)/(9^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*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)/(4^p*c^4*e^10*E^((4*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])
^2])/b))^p) + (3*2^(3 + 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) - (14*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^(2 + 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*2^(
1 - p)*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)/(c^2*e^10
*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) + (2^(3 + 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)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(e^10*E^((3*a)/(2*b))*(c*
(d + e/Sqrt[x])^2)^(3/2)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - (9*d^8*Gamma[1 + p, -((a + b*Log[c*(d +
e/Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(c*e^10*E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))
^p) + (2^(1 + p)*d^9*(d + e/Sqrt[x])*Gamma[1 + p, -1/2*(a + b*Log[c*(d + e/Sqrt[x])^2])/b]*(a + b*Log[c*(d + e
/Sqrt[x])^2])^p)/(e^10*E^(a/(2*b))*Sqrt[c*(d + e/Sqrt[x])^2]*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p)

Rule 2212

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

Rule 2337

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

Rule 2347

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

Rule 2436

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

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx &=-\left (2 \text {Subst}\left (\int x^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\left (2 \text {Subst}\left (\int \left (-\frac {d^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac {9 d^8 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac {36 d^7 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac {84 d^6 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac {126 d^5 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac {126 d^4 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac {84 d^3 (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac {36 d^2 (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac {9 d (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac {(d+e x)^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {2 \text {Subst}\left (\int (d+e x)^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {(18 d) \text {Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (72 d^2\right ) \text {Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (168 d^3\right ) \text {Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (252 d^4\right ) \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (252 d^5\right ) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (168 d^6\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (72 d^7\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}-\frac {\left (18 d^8\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}+\frac {\left (2 d^9\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^9}\\ &=-\frac {2 \text {Subst}\left (\int x^9 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {(18 d) \text {Subst}\left (\int x^8 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (72 d^2\right ) \text {Subst}\left (\int x^7 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (168 d^3\right ) \text {Subst}\left (\int x^6 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (252 d^4\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (252 d^5\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (168 d^6\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (72 d^7\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}-\frac {\left (18 d^8\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}+\frac {\left (2 d^9\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^{10}}\\ &=-\frac {\text {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{c^5 e^{10}}-\frac {\left (36 d^2\right ) \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{c^4 e^{10}}-\frac {\left (126 d^4\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{c^3 e^{10}}-\frac {\left (84 d^6\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{c^2 e^{10}}-\frac {\left (9 d^8\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{c e^{10}}+\frac {\left (9 d \left (d+\frac {e}{\sqrt {x}}\right )^9\right ) \text {Subst}\left (\int e^{9 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{9/2}}+\frac {\left (84 d^3 \left (d+\frac {e}{\sqrt {x}}\right )^7\right ) \text {Subst}\left (\int e^{7 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{7/2}}+\frac {\left (126 d^5 \left (d+\frac {e}{\sqrt {x}}\right )^5\right ) \text {Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}}+\frac {\left (36 d^7 \left (d+\frac {e}{\sqrt {x}}\right )^3\right ) \text {Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}+\frac {\left (d^9 \left (d+\frac {e}{\sqrt {x}}\right )\right ) \text {Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{e^{10} \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}\\ &=-\frac {5^{-1-p} e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\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}}{c^5 e^{10}}+\frac {2^{1+p} 9^{-p} d e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^9 \Gamma \left (1+p,-\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\ 4^{-p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\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^{3+p} 7^{-p} d^3 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^7 \Gamma \left (1+p,-\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 {14\ 3^{1-p} d^4 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\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^{2+p} 5^{-1-p} d^5 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^5 \Gamma \left (1+p,-\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^{1-p} d^6 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\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^{3+p} 3^{1-p} d^7 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^3 \Gamma \left (1+p,-\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 (1+p,-\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}}{c e^{10}}+\frac {2^{1+p} d^9 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \Gamma \left (1+p,-\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}}\\ \end {align*}

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Mathematica [F]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{2}\right )\right )^{p}}{x^{6}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^2\right )\right )}^p}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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