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

Optimal. Leaf size=832 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.3026, antiderivative size = 832, normalized size of antiderivative = 1., number of steps used = 30, 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} \[ -\frac{3^{-2 p-1} e^{-\frac{9 a}{b}} \text{Gamma}\left (p+1,-\frac{9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^9 e^9}+\frac{3\ 8^{-p} d e^{-\frac{8 a}{b}} \text{Gamma}\left (p+1,-\frac{8 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^8 e^9}-\frac{12\ 7^{-p} d^2 e^{-\frac{7 a}{b}} \text{Gamma}\left (p+1,-\frac{7 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^7 e^9}+\frac{7\ 2^{2-p} 3^{-p} d^3 e^{-\frac{6 a}{b}} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^6 e^9}-\frac{42\ 5^{-p} d^4 e^{-\frac{5 a}{b}} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^5 e^9}+\frac{21\ 2^{1-2 p} d^5 e^{-\frac{4 a}{b}} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^4 e^9}-\frac{28\ 3^{-p} d^6 e^{-\frac{3 a}{b}} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^3 e^9}+\frac{3\ 2^{2-p} d^7 e^{-\frac{2 a}{b}} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^2 e^9}-\frac{3 d^8 e^{-\frac{a}{b}} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.830324, size = 502, normalized size = 0.6 \[ -\frac{3^{-2 p-1} 280^{-p} e^{-\frac{9 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \left (c^8 d^8 9^{p+1} 280^p e^{\frac{8 a}{b}} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )-c^7 d^7 35^p 36^{p+1} e^{\frac{7 a}{b}} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+c^6 d^6 2^{3 p+2} 5^p 21^{p+1} e^{\frac{6 a}{b}} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-c^5 d^5 5^p 126^{p+1} e^{\frac{5 a}{b}} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+c^4 d^4 2^{3 p+1} 63^{p+1} e^{\frac{4 a}{b}} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-c^3 d^3 5^p 84^{p+1} e^{\frac{3 a}{b}} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+c^2 d^2 2^{3 p+2} 5^p 9^{p+1} e^{\frac{2 a}{b}} \text{Gamma}\left (p+1,-\frac{7 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-c d 9^{p+1} 35^p e^{a/b} \text{Gamma}\left (p+1,-\frac{8 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+280^p \text{Gamma}\left (p+1,-\frac{9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )\right )}{c^9 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3^(-1 - 2*p)*(280^p*Gamma[1 + p, (-9*(a + b*Log[c*(d + e/x^(1/3))]))/b] - 9^(1 + p)*35^p*c*d*E^(a/b)*Gamma[
1 + p, (-8*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 2^(2 + 3*p)*5^p*9^(1 + p)*c^2*d^2*E^((2*a)/b)*Gamma[1 + p, (-7
*(a + b*Log[c*(d + e/x^(1/3))]))/b] - 5^p*84^(1 + p)*c^3*d^3*E^((3*a)/b)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/
x^(1/3))]))/b] + 2^(1 + 3*p)*63^(1 + p)*c^4*d^4*E^((4*a)/b)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))]))/b
] - 5^p*126^(1 + p)*c^5*d^5*E^((5*a)/b)*Gamma[1 + p, (-4*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 2^(2 + 3*p)*5^p*
21^(1 + p)*c^6*d^6*E^((6*a)/b)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))]))/b] - 35^p*36^(1 + p)*c^7*d^7*E
^((7*a)/b)*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 9^(1 + p)*280^p*c^8*d^8*E^((8*a)/b)*Gamma[1 +
p, -((a + b*Log[c*(d + e/x^(1/3))])/b)])*(a + b*Log[c*(d + e/x^(1/3))])^p)/(280^p*c^9*e^9*E^((9*a)/b)*(-((a +
b*Log[c*(d + e/x^(1/3))])/b))^p))

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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 ) \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))))^p/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))))^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 )}\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))))^p/x^4,x, algorithm="maxima")

[Out]

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

[Out]

integral((b*log((c*d*x + c*e*x^(2/3))/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))))**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 )}\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))))^p/x^4,x, algorithm="giac")

[Out]

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