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

Optimal. Leaf size=1141 \[ \text{result too large to display} \]

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

________________________________________________________________________________________

Rubi [A]  time = 1.73843, antiderivative size = 1141, normalized size of antiderivative = 1., 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 = {2454, 2401, 2389, 2300, 2181, 2390, 2310} \[ \text{result too large to display} \]

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

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 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 2300

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 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 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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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 \operatorname{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 \operatorname{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 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [F]  time = 0.132578, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p}{x^6} \, dx \]

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 = 0.333, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{2} \right ) \right ) ^{p}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (\frac{c d^{2} x + 2 \, c d e \sqrt{x} + c e^{2}}{x}\right ) + a\right )}^{p}}{x^{6}}, x\right ) \end{align*}

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*e*sqrt(x) + c*e^2)/x) + a)^p/x^6, x)

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}}\,{d x} \end{align*}

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)