3.6.92 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^2))^p}{x^3} \, dx\) [592]

Optimal. Leaf size=673 \[ -\frac {3^{-p} 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}}{2 c^3 e^6}+\frac {3 \left (\frac {2}{5}\right )^p d 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^6 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}-\frac {15\ 2^{-1-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 [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^6}+\frac {5\ 2^{1+p} 3^{-p} d^3 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^6 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac {15 d^4 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}}{2 c e^6}+\frac {3\ 2^p d^5 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^6 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.68, antiderivative size = 673, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \begin {gather*} -\frac {3^{-p} e^{-\frac {3 a}{b}} \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} \text {Gamma}\left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right )}{2 c^3 e^6}-\frac {15 d^2 2^{-p-1} e^{-\frac {2 a}{b}} \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} \text {Gamma}\left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right )}{c^2 e^6}+\frac {3 d^5 2^p e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\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} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}}-\frac {15 d^4 e^{-\frac {a}{b}} \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} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )}{2 c e^6}+\frac {5 d^3 2^{p+1} 3^{-p} e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \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} \text {Gamma}\left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac {3 d \left (\frac {2}{5}\right )^p e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \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} \text {Gamma}\left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [F]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{2}\right )\right )^{p}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x^3,x)

[Out]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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