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

Optimal. Leaf size=831 \[ \frac {3^{-2 p-1} e^{-\frac {9 a}{b}} \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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 (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^2 e^9}+\frac {3 d^8 e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^9} \]

[Out]

3^(-1-2*p)*GAMMA(1+p,-9*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/c^9/e^9/exp(9*a/b)/(((-a-b*ln
(c*(d+e*x^(1/3))))/b)^p)-3*d*GAMMA(1+p,-8*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(8^p)/c^8/e
^9/exp(8*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+12*d^2*GAMMA(1+p,-7*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d
+e*x^(1/3))))^p/(7^p)/c^7/e^9/exp(7*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)-7*2^(2-p)*d^3*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^9/exp(6*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+
42*d^4*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^9/exp(5*a/b)/(((-a-b*
ln(c*(d+e*x^(1/3))))/b)^p)-21*2^(1-2*p)*d^5*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^9/exp(4*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+28*d^6*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^9/exp(3*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)-3*2^(2-p)*d^7*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^9/exp(2*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^
p)+3*d^8*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^9/exp(a/b)/(((-a-b*ln(c*(d+e*
x^(1/3))))/b)^p)

________________________________________________________________________________________

Rubi [A]  time = 1.35, antiderivative size = 831, normalized size of antiderivative = 1.00, 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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^9} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*(d + e*x^(1/3))])^p,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) + (2
8*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 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 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 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]

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 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 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 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])

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx &=3 \operatorname {Subst}\left (\int x^8 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )\\ &=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,\sqrt [3]{x}\right )\\ &=\frac {3 \operatorname {Subst}\left (\int (d+e x)^8 (a+b \log (c (d+e x)))^p \, dx,x,\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,\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,\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,\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,\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,\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,\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,\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,\sqrt [3]{x}\right )}{e^8}\\ &=\frac {3 \operatorname {Subst}\left (\int x^8 (a+b \log (c x))^p \, dx,x,d+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+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+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+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+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+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+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+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+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+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+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+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+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+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+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+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+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+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^9}\\ \end {align*}

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Mathematica [A]  time = 1.00, size = 501, normalized size = 0.60 \[ \frac {3^{-2 p-1} 280^{-p} e^{-\frac {9 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \left (c^8 d^8 9^{p+1} 280^p e^{\frac {8 a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )-c^7 d^7 35^p 36^{p+1} e^{\frac {7 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+c^6 d^6 2^{3 p+2} 5^p 21^{p+1} e^{\frac {6 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )-c^5 d^5 5^p 126^{p+1} e^{\frac {5 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+c^4 d^4 2^{3 p+1} 63^{p+1} e^{\frac {4 a}{b}} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )-c^3 d^3 5^p 84^{p+1} e^{\frac {3 a}{b}} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+c^2 d^2 2^{3 p+2} 5^p 9^{p+1} e^{\frac {2 a}{b}} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )-c d 9^{p+1} 35^p e^{a/b} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+280^p \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )\right )}{c^9 e^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*(d + e*x^(1/3))])^p,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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fricas [F]  time = 1.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c e x^{\frac {1}{3}} + c d\right ) + a\right )}^{p} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{2} \left (b \ln \left (\left (e \,x^{\frac {1}{3}}+d \right ) c \right )+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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