\(\int x^3 (a+b \log (c (d+e \sqrt [3]{x})))^p \, dx\) [556]

Optimal result

Integrand size = 22, antiderivative size = 1121 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} 4^{-1-p} e^{-\frac {12 a}{b}} \Gamma \left (1+p,-\frac {12 \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^{12} e^{12}}-\frac {3\ 11^{-p} d e^{-\frac {11 a}{b}} \Gamma \left (1+p,-\frac {11 \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^{11} e^{12}}+\frac {33\ 2^{-1-p} 5^{-p} d^2 e^{-\frac {10 a}{b}} \Gamma \left (1+p,-\frac {10 \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^{10} e^{12}}-\frac {55\ 9^{-p} d^3 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^{12}}+\frac {495\ 2^{-2-3 p} d^4 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^{12}}-\frac {198\ 7^{-p} d^5 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^{12}}+\frac {77\ 2^{-p} 3^{1-p} d^6 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^{12}}-\frac {198\ 5^{-p} d^7 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^{12}}+\frac {495\ 4^{-1-p} d^8 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^{12}}-\frac {55\ 3^{-p} d^9 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^{12}}+\frac {33\ 2^{-1-p} d^{10} 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^{12}}-\frac {3 d^{11} 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^{12}} \]

[Out]

4^(-1-p)*GAMMA(p+1,-12*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(3^p)/c^12/e^12/exp(12*a/b)/((
(-a-b*ln(c*(d+e*x^(1/3))))/b)^p)-3*d*GAMMA(p+1,-11*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(1
1^p)/c^11/e^12/exp(11*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+33*2^(-1-p)*d^2*GAMMA(p+1,-10*(a+b*ln(c*(d+e*x^(
1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(5^p)/c^10/e^12/exp(10*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)-55*d^3*G
AMMA(p+1,-9*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(9^p)/c^9/e^12/exp(9*a/b)/(((-a-b*ln(c*(d
+e*x^(1/3))))/b)^p)+495*2^(-2-3*p)*d^4*GAMMA(p+1,-8*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/c
^8/e^12/exp(8*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)-198*d^5*GAMMA(p+1,-7*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*l
n(c*(d+e*x^(1/3))))^p/(7^p)/c^7/e^12/exp(7*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+77*3^(1-p)*d^6*GAMMA(p+1,-6
*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(2^p)/c^6/e^12/exp(6*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))
))/b)^p)-198*d^7*GAMMA(p+1,-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^12/exp(5*a/
b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+495*4^(-1-p)*d^8*GAMMA(p+1,-4*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+
e*x^(1/3))))^p/c^4/e^12/exp(4*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)-55*d^9*GAMMA(p+1,-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^12/exp(3*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)+33*2^(-1-p)*
d^10*GAMMA(p+1,-2*(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/c^2/e^12/exp(2*a/b)/(((-a-b*ln(c*(d
+e*x^(1/3))))/b)^p)-3*d^11*GAMMA(p+1,(-a-b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/c/e^12/exp(a/b)
/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2504, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} 4^{-p-1} e^{-\frac {12 a}{b}} \Gamma \left (p+1,-\frac {12 \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^{12} e^{12}}-\frac {3\ 11^{-p} d e^{-\frac {11 a}{b}} \Gamma \left (p+1,-\frac {11 \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^{11} e^{12}}+\frac {33\ 2^{-p-1} 5^{-p} d^2 e^{-\frac {10 a}{b}} \Gamma \left (p+1,-\frac {10 \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^{10} e^{12}}-\frac {55\ 9^{-p} d^3 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^{12}}+\frac {495\ 2^{-3 p-2} d^4 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^{12}}-\frac {198\ 7^{-p} d^5 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^{12}}+\frac {77\ 2^{-p} 3^{1-p} d^6 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^{12}}-\frac {198\ 5^{-p} d^7 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^{12}}+\frac {495\ 4^{-p-1} d^8 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^{12}}-\frac {55\ 3^{-p} d^9 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^{12}}+\frac {33\ 2^{-p-1} d^{10} 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^{12}}-\frac {3 d^{11} 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^{12}} \]

[In]

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

[Out]

(4^(-1 - p)*Gamma[1 + p, (-12*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*c^12*e
^12*E^((12*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (3*d*Gamma[1 + p, (-11*(a + b*Log[c*(d + e*x^(1/3)
)]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(11^p*c^11*e^12*E^((11*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p
) + (33*2^(-1 - p)*d^2*Gamma[1 + p, (-10*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/
(5^p*c^10*e^12*E^((10*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (55*d^3*Gamma[1 + p, (-9*(a + b*Log[c*(
d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(9^p*c^9*e^12*E^((9*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3
))])/b))^p) + (495*2^(-2 - 3*p)*d^4*Gamma[1 + p, (-8*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^
(1/3))])^p)/(c^8*e^12*E^((8*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (198*d^5*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^12*E^((7*a)/b)*(-((a + b*Log[c*(d + e
*x^(1/3))])/b))^p) + (77*3^(1 - p)*d^6*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e
*x^(1/3))])^p)/(2^p*c^6*e^12*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (198*d^7*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^12*E^((5*a)/b)*(-((a + b*Log[c
*(d + e*x^(1/3))])/b))^p) + (495*4^(-1 - p)*d^8*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^12*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (55*d^9*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^12*E^((3*a)/b)*(-((a + b*L
og[c*(d + e*x^(1/3))])/b))^p) + (33*2^(-1 - p)*d^10*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^12*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (3*d^11*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^12*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*} \text {integral}& = 3 \text {Subst}\left (\int x^{11} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (-\frac {d^{11} (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {11 d^{10} (d+e x) (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {55 d^9 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {165 d^8 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {330 d^7 (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {462 d^6 (d+e x)^5 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {462 d^5 (d+e x)^6 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {330 d^4 (d+e x)^7 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {165 d^3 (d+e x)^8 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {55 d^2 (d+e x)^9 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {11 d (d+e x)^{10} (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {(d+e x)^{11} (a+b \log (c (d+e x)))^p}{e^{11}}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^{11} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {(33 d) \text {Subst}\left (\int (d+e x)^{10} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int (d+e x)^9 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int (d+e x)^8 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int (d+e x)^7 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int (d+e x)^6 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}} \\ & = \frac {3 \text {Subst}\left (\int x^{11} (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {(33 d) \text {Subst}\left (\int x^{10} (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int x^9 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int x^8 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int x^7 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int x^6 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}} \\ & = \frac {3 \text {Subst}\left (\int e^{12 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^{12} e^{12}}-\frac {(33 d) \text {Subst}\left (\int e^{11 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^{11} e^{12}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int e^{10 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^{10} e^{12}}-\frac {\left (495 d^3\right ) \text {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^{12}}+\frac {\left (990 d^4\right ) \text {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^{12}}-\frac {\left (1386 d^5\right ) \text {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^{12}}+\frac {\left (1386 d^6\right ) \text {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^{12}}-\frac {\left (990 d^7\right ) \text {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^{12}}+\frac {\left (495 d^8\right ) \text {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^{12}}-\frac {\left (165 d^9\right ) \text {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^{12}}+\frac {\left (33 d^{10}\right ) \text {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^{12}}-\frac {\left (3 d^{11}\right ) \text {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^{12}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )\right )\right )}^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]

[In]

integrate(x^3*(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^3, x)

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]

[In]

integrate(x^3*(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^3, x)

Giac [F]

\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]

[In]

integrate(x^3*(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^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{1/3}\right )\right )\right )}^p \,d x \]

[In]

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

[Out]

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