\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^2))^p}{x^4} \, dx\) [554]

Optimal result

Integrand size = 24, antiderivative size = 676 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=-\frac {3^{-1-p} 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^6}+\frac {2^{1+p} 5^{-p} d 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^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}}-\frac {5\ 2^{-p} d^2 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^6}+\frac {5\ 2^{2+p} 3^{-1-p} d^3 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^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}-\frac {5 d^4 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^6}+\frac {2^{1+p} d^5 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^6 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}} \]

[Out]

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

Time = 0.68 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=-\frac {3^{-p-1} e^{-\frac {3 a}{b}} \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} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right )}{c^3 e^6}-\frac {5 d^2 2^{-p} e^{-\frac {2 a}{b}} \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} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right )}{c^2 e^6}+\frac {d^5 2^{p+1} e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\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} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}-\frac {5 d^4 e^{-\frac {a}{b}} \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} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )}{c e^6}+\frac {5 d^3 2^{p+2} 3^{-p-1} e^{-\frac {3 a}{2 b}} \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 \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \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 )}{e^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}+\frac {d 2^{p+1} 5^{-p} e^{-\frac {5 a}{2 b}} \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 \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \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 )}{e^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}} \]

[In]

Int[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4,x]

[Out]

-((3^(-1 - p)*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
^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p)) + (2^(1 + p)*d*(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)/(5^p*e^6*E^((5*a)/(2*b))*(c*(d + e
/Sqrt[x])^2)^(5/2)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - (5*d^2*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/Sq
rt[x])^2]))/b]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(2^p*c^2*e^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2]
)/b))^p) + (5*2^(2 + p)*3^(-1 - p)*d^3*(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^6*E^((3*a)/(2*b))*(c*(d + e/Sqrt[x])^2)^(3/2)*(-((a + b*Log[c*(d +
e/Sqrt[x])^2])/b))^p) - (5*d^4*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^6*E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) + (2^(1 + p)*d^5*(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^6*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*} \text {integral}& = -\left (2 \text {Subst}\left (\int x^5 \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^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {2 \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^5}+\frac {(10 d) \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^5}-\frac {\left (20 d^2\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^5}+\frac {\left (20 d^3\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^5}-\frac {\left (10 d^4\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^5}+\frac {\left (2 d^5\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^5} \\ & = -\frac {2 \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^6}+\frac {(10 d) \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^6}-\frac {\left (20 d^2\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^6}+\frac {\left (20 d^3\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^6}-\frac {\left (10 d^4\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^6}+\frac {\left (2 d^5\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^6} \\ & = -\frac {\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^6}-\frac {\left (10 d^2\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^6}-\frac {\left (5 d^4\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^6}+\frac {\left (5 d \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^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}}+\frac {\left (10 d^3 \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^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}+\frac {\left (d^5 \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^6 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}} \\ & = -\frac {3^{-1-p} 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^6}+\frac {2^{1+p} 5^{-p} d 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^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}}-\frac {5\ 2^{-p} d^2 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^6}+\frac {5\ 2^{2+p} 3^{-1-p} d^3 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^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}-\frac {5 d^4 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^6}+\frac {2^{1+p} d^5 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^6 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx \]

[In]

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4,x]

[Out]

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4, x]

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{2}\right )\right )}^{p}}{x^{4}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]

[In]

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

[Out]

integral((b*log((c*d^2*x + 2*c*d*e*sqrt(x) + c*e^2)/x) + a)^p/x^4, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x^4, x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^2\right )\right )}^p}{x^4} \,d x \]

[In]

int((a + b*log(c*(d + e/x^(1/2))^2))^p/x^4,x)

[Out]

int((a + b*log(c*(d + e/x^(1/2))^2))^p/x^4, x)