3.593 \(\int \frac{(a+b \log (c (d+\frac{e}{\sqrt [3]{x}})^2))^p}{x^4} \, dx\)

Optimal. Leaf size=1036 \[ \text{result too large to display} \]

[Out]

-((2^p*3^(-1 - 2*p)*(d + e/x^(1/3))^9*Gamma[1 + p, (-9*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(
d + e/x^(1/3))^2])^p)/(e^9*E^((9*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(9/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))
^p)) + (3*d*Gamma[1 + p, (-4*(a + b*Log[c*(d + e/x^(1/3))^2]))/b]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(4^p*c^4
*e^9*E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) - (3*2^(2 + p)*d^2*(d + e/x^(1/3))^7*Gamma[1 + p,
(-7*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(7^p*e^9*E^((7*a)/(2*b))*(c*(
d + e/x^(1/3))^2)^(7/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) + (28*d^3*Gamma[1 + p, (-3*(a + b*Log[c*(d
+ e/x^(1/3))^2]))/b]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3^p*c^3*e^9*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/
3))^2])/b))^p) - (21*2^(1 + p)*d^4*(d + e/x^(1/3))^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]
*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(5^p*e^9*E^((5*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(5/2)*(-((a + b*Log[c*(d +
e/x^(1/3))^2])/b))^p) + (21*2^(1 - p)*d^5*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))^2]))/b]*(a + b*Log[c*
(d + e/x^(1/3))^2])^p)/(c^2*e^9*E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) - (7*2^(2 + p)*d^6*(d +
e/x^(1/3))^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3
^p*e^9*E^((3*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(3/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) + (12*d^7*Gamma[
1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(c*e^9*E^(a/b)*(-((a + b*Log
[c*(d + e/x^(1/3))^2])/b))^p) - (3*2^p*d^8*(d + e/x^(1/3))*Gamma[1 + p, -(a + b*Log[c*(d + e/x^(1/3))^2])/(2*b
)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(e^9*E^(a/(2*b))*Sqrt[c*(d + e/x^(1/3))^2]*(-((a + b*Log[c*(d + e/x^(1/
3))^2])/b))^p)

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Rubi [A]  time = 1.51902, antiderivative size = 1036, normalized size of antiderivative = 1., number of steps used = 30, 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/x^(1/3))^2])^p/x^4,x]

[Out]

-((2^p*3^(-1 - 2*p)*(d + e/x^(1/3))^9*Gamma[1 + p, (-9*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(
d + e/x^(1/3))^2])^p)/(e^9*E^((9*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(9/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))
^p)) + (3*d*Gamma[1 + p, (-4*(a + b*Log[c*(d + e/x^(1/3))^2]))/b]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(4^p*c^4
*e^9*E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) - (3*2^(2 + p)*d^2*(d + e/x^(1/3))^7*Gamma[1 + p,
(-7*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(7^p*e^9*E^((7*a)/(2*b))*(c*(
d + e/x^(1/3))^2)^(7/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) + (28*d^3*Gamma[1 + p, (-3*(a + b*Log[c*(d
+ e/x^(1/3))^2]))/b]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3^p*c^3*e^9*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/
3))^2])/b))^p) - (21*2^(1 + p)*d^4*(d + e/x^(1/3))^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]
*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(5^p*e^9*E^((5*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(5/2)*(-((a + b*Log[c*(d +
e/x^(1/3))^2])/b))^p) + (21*2^(1 - p)*d^5*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))^2]))/b]*(a + b*Log[c*
(d + e/x^(1/3))^2])^p)/(c^2*e^9*E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) - (7*2^(2 + p)*d^6*(d +
e/x^(1/3))^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3
^p*e^9*E^((3*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(3/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) + (12*d^7*Gamma[
1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(c*e^9*E^(a/b)*(-((a + b*Log
[c*(d + e/x^(1/3))^2])/b))^p) - (3*2^p*d^8*(d + e/x^(1/3))*Gamma[1 + p, -(a + b*Log[c*(d + e/x^(1/3))^2])/(2*b
)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(e^9*E^(a/(2*b))*Sqrt[c*(d + e/x^(1/3))^2]*(-((a + b*Log[c*(d + e/x^(1/
3))^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 [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx &=-\left (3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{d^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac{8 d^7 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac{28 d^6 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac{56 d^5 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac{70 d^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac{56 d^3 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac{28 d^2 (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac{8 d (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac{(d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac{3 \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 [3]{x}}\right )}{e^8}+\frac{(24 d) \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 [3]{x}}\right )}{e^8}-\frac{\left (84 d^2\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 [3]{x}}\right )}{e^8}+\frac{\left (168 d^3\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 [3]{x}}\right )}{e^8}-\frac{\left (210 d^4\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 [3]{x}}\right )}{e^8}+\frac{\left (168 d^5\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 [3]{x}}\right )}{e^8}-\frac{\left (84 d^6\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 [3]{x}}\right )}{e^8}+\frac{\left (24 d^7\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 [3]{x}}\right )}{e^8}-\frac{\left (3 d^8\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^8}\\ &=-\frac{3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}+\frac{(24 d) \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}-\frac{\left (84 d^2\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}+\frac{\left (168 d^3\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}-\frac{\left (210 d^4\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}+\frac{\left (168 d^5\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}-\frac{\left (84 d^6\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}+\frac{\left (24 d^7\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}-\frac{\left (3 d^8\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^9}\\ &=\frac{(12 d) \operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^4 e^9}+\frac{\left (84 d^3\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^3 e^9}+\frac{\left (84 d^5\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^2 e^9}+\frac{\left (12 d^7\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{c e^9}-\frac{\left (3 \left (d+\frac{e}{\sqrt [3]{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 [3]{x}}\right )^2\right )\right )}{2 e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{9/2}}-\frac{\left (42 d^2 \left (d+\frac{e}{\sqrt [3]{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 [3]{x}}\right )^2\right )\right )}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{7/2}}-\frac{\left (105 d^4 \left (d+\frac{e}{\sqrt [3]{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 [3]{x}}\right )^2\right )\right )}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}-\frac{\left (42 d^6 \left (d+\frac{e}{\sqrt [3]{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 [3]{x}}\right )^2\right )\right )}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac{\left (3 d^8 \left (d+\frac{e}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 e^9 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}\\ &=-\frac{2^p 3^{-1-2 p} e^{-\frac{9 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^9 \Gamma \left (1+p,-\frac{9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{9/2}}+\frac{3\ 4^{-p} d e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\frac{4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^4 e^9}-\frac{3\ 2^{2+p} 7^{-p} d^2 e^{-\frac{7 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^7 \Gamma \left (1+p,-\frac{7 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{7/2}}+\frac{28\ 3^{-p} d^3 e^{-\frac{3 a}{b}} \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^3 e^9}-\frac{21\ 2^{1+p} 5^{-p} d^4 e^{-\frac{5 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^5 \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}+\frac{21\ 2^{1-p} d^5 e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^2 e^9}-\frac{7\ 2^{2+p} 3^{-p} d^6 e^{-\frac{3 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac{12 d^7 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^9}-\frac{3\ 2^p d^8 e^{-\frac{a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}\\ \end{align*}

Mathematica [F]  time = 0.132336, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^4,x]

[Out]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^4, x]

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Maple [F]  time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{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^(1/3))^2))^p/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x^4,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}{x^{\frac{1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x, algorithm="maxima")

[Out]

integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x^4, 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 x^{\frac{2}{3}} + c e^{2} x^{\frac{1}{3}}}{x}\right ) + a\right )}^{p}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x, algorithm="fricas")

[Out]

integral((b*log((c*d^2*x + 2*c*d*e*x^(2/3) + c*e^2*x^(1/3))/x) + a)^p/x^4, 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**(1/3))**2))**p/x**4,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}{x^{\frac{1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x^4, x)