3.6.86 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})))^p}{x^3} \, dx\) [586]

Optimal. Leaf size=554 \[ -\frac {2^{-1-p} 3^{-p} 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^6}+\frac {3\ 5^{-p} d 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^6}-\frac {15\ 2^{-1-2 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 [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^6}+\frac {10\ 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 )\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^6}-\frac {15\ 2^{-1-p} d^4 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^6}+\frac {3 d^5 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^6} \]

[Out]

-2^(-1-p)*GAMMA(1+p,-6*(a+b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/(3^p)/c^6/e^6/exp(6*a/b)/(((-a
-b*ln(c*(d+e/x^(1/3))))/b)^p)+3*d*GAMMA(1+p,-5*(a+b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/(5^p)/
c^5/e^6/exp(5*a/b)/(((-a-b*ln(c*(d+e/x^(1/3))))/b)^p)-15*2^(-1-2*p)*d^2*GAMMA(1+p,-4*(a+b*ln(c*(d+e/x^(1/3))))
/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/c^4/e^6/exp(4*a/b)/(((-a-b*ln(c*(d+e/x^(1/3))))/b)^p)+10*d^3*GAMMA(1+p,-3*(a+b
*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/(3^p)/c^3/e^6/exp(3*a/b)/(((-a-b*ln(c*(d+e/x^(1/3))))/b)^
p)-15*2^(-1-p)*d^4*GAMMA(1+p,-2*(a+b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/c^2/e^6/exp(2*a/b)/((
(-a-b*ln(c*(d+e/x^(1/3))))/b)^p)+3*d^5*GAMMA(1+p,(-a-b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/c/e
^6/exp(a/b)/(((-a-b*ln(c*(d+e/x^(1/3))))/b)^p)

________________________________________________________________________________________

Rubi [A]
time = 0.60, antiderivative size = 554, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2504, 2448, 2436, 2336, 2212, 2437, 2346} \begin {gather*} -\frac {2^{-p-1} 3^{-p} e^{-\frac {6 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} \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^6 e^6}+\frac {3 d 5^{-p} e^{-\frac {5 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} \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^5 e^6}-\frac {15 d^2 2^{-2 p-1} e^{-\frac {4 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} \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 e^6}+\frac {10 d^3 3^{-p} e^{-\frac {3 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} \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^3 e^6}-\frac {15 d^4 2^{-p-1} e^{-\frac {2 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} \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^2 e^6}+\frac {3 d^5 e^{-\frac {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} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )}{c e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(-1 - p)*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
^6*E^((6*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p)) + (3*d*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^6*E^((5*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) - (1
5*2^(-1 - 2*p)*d^2*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^6*E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) + (10*d^3*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^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) -
(15*2^(-1 - p)*d^4*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^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) + (3*d^5*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^6*E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/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 2336

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 2346

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]

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 [3]{x}}\right )\right )\right )^p}{x^3} \, dx &=-\left (3 \text {Subst}\left (\int x^5 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \text {Subst}\left (\int \left (-\frac {d^5 (a+b \log (c (d+e x)))^p}{e^5}+\frac {5 d^4 (d+e x) (a+b \log (c (d+e x)))^p}{e^5}-\frac {10 d^3 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^5}+\frac {10 d^2 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^5}-\frac {5 d (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^5}+\frac {(d+e x)^5 (a+b \log (c (d+e x)))^p}{e^5}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac {3 \text {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {(15 d) \text {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}-\frac {\left (30 d^2\right ) \text {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {\left (30 d^3\right ) \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}-\frac {\left (15 d^4\right ) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {\left (3 d^5\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}\\ &=-\frac {3 \text {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {(15 d) \text {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {\left (30 d^2\right ) \text {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (30 d^3\right ) \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {\left (15 d^4\right ) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (3 d^5\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\\ &=-\frac {3 \text {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^6}+\frac {(15 d) \text {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^6}-\frac {\left (30 d^2\right ) \text {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^6}+\frac {\left (30 d^3\right ) \text {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^6}-\frac {\left (15 d^4\right ) \text {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^6}+\frac {\left (3 d^5\right ) \text {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^6}\\ &=-\frac {2^{-1-p} 3^{-p} 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^6}+\frac {3\ 5^{-p} d 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^6}-\frac {15\ 2^{-1-2 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 [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^6}+\frac {10\ 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 )\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^6}-\frac {15\ 2^{-1-p} d^4 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^6}+\frac {3 d^5 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^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.94, size = 325, normalized size = 0.59 \begin {gather*} \frac {2^{-1-2 p} 15^{-p} e^{-\frac {6 a}{b}} \left (-10^p \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+c d e^{a/b} \left (2^{1+2 p} 3^{1+p} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+5^p c d e^{a/b} \left (-5 3^{1+p} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \left (5\ 2^{2+p} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+3^{1+p} c d e^{a/b} \left (-5 \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )\right )\right )\right )\right )\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^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-1 - 2*p)*(-(10^p*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/x^(1/3))]))/b]) + c*d*E^(a/b)*(2^(1 + 2*p)*3^(1 + p
)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 5^p*c*d*E^(a/b)*(-5*3^(1 + p)*Gamma[1 + p, (-4*(a + b*
Log[c*(d + e/x^(1/3))]))/b] + 2^p*c*d*E^(a/b)*(5*2^(2 + p)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))]))/b]
+ 3^(1 + p)*c*d*E^(a/b)*(-5*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 2^(1 + p)*c*d*E^(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)/(15^p*c^6*e^6*E^((6*a)/b)*
(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p)

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )\right )\right )^{p}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/3))))^p/x^3,x)

[Out]

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

[Out]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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