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

Optimal. Leaf size=1121 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.86769, antiderivative size = 1121, normalized size of antiderivative = 1., number of steps used = 39, 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} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[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 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 x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx &=3 \operatorname{Subst}\left (\int x^{11} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{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 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}\\ &=\frac{3 \operatorname{Subst}\left (\int x^{11} (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{(33 d) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}\\ &=\frac{3 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \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^{12}}+\frac{\left (990 d^4\right ) \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^{12}}-\frac{\left (1386 d^5\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^{12}}+\frac{\left (1386 d^6\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^{12}}-\frac{\left (990 d^7\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^{12}}+\frac{\left (495 d^8\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^{12}}-\frac{\left (165 d^9\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^{12}}+\frac{\left (33 d^{10}\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^{12}}-\frac{\left (3 d^{11}\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^{12}}\\ &=\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}}\\ \end{align*}

Mathematica [A]  time = 2.66171, size = 670, normalized size = 0.6 \[ -\frac{2^{-3 p-2} 3465^{-p} e^{-\frac{12 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^2 d^2 e^{\frac{2 a}{b}} \left (c^2 d^2 e^{\frac{2 a}{b}} \left (c^7 d^7 2^{3 p+2} 3^{2 p+1} 385^p e^{\frac{7 a}{b}} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )-c^6 d^6 6^{2 p+1} 11^{p+1} 35^p e^{\frac{6 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 )+c^5 d^5 2^{3 p+2} 21^p 55^{p+1} e^{\frac{5 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 )-c^4 d^4 14^p 495^{p+1} 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 )+c^3 d^3 7^p 792^{p+1} e^{\frac{3 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 )-c^2 d^2 5^p 924^{p+1} e^{\frac{2 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 )+c d 5^p 792^{p+1} e^{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 )-7^p 495^{p+1} \text{Gamma}\left (p+1,-\frac{8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )\right )+c d 2^{3 p+2} 7^p 55^{p+1} e^{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 )-6^{2 p+1} 7^p 11^{p+1} \text{Gamma}\left (p+1,-\frac{10 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )\right )+c d 2^{3 p+2} 3^{2 p+1} 35^p e^{a/b} \text{Gamma}\left (p+1,-\frac{11 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )-2310^p \text{Gamma}\left (p+1,-\frac{12 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )\right )}{c^{12} e^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(-2 - 3*p)*(-(2310^p*Gamma[1 + p, (-12*(a + b*Log[c*(d + e*x^(1/3))]))/b]) + 2^(2 + 3*p)*3^(1 + 2*p)*35^p
*c*d*E^(a/b)*Gamma[1 + p, (-11*(a + b*Log[c*(d + e*x^(1/3))]))/b] + c^2*d^2*E^((2*a)/b)*(-(6^(1 + 2*p)*7^p*11^
(1 + p)*Gamma[1 + p, (-10*(a + b*Log[c*(d + e*x^(1/3))]))/b]) + 2^(2 + 3*p)*7^p*55^(1 + p)*c*d*E^(a/b)*Gamma[1
+ p, (-9*(a + b*Log[c*(d + e*x^(1/3))]))/b] + c^2*d^2*E^((2*a)/b)*(-(7^p*495^(1 + p)*Gamma[1 + p, (-8*(a + b*
Log[c*(d + e*x^(1/3))]))/b]) + 5^p*792^(1 + p)*c*d*E^(a/b)*Gamma[1 + p, (-7*(a + b*Log[c*(d + e*x^(1/3))]))/b]
- 5^p*924^(1 + p)*c^2*d^2*E^((2*a)/b)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(1/3))]))/b] + 7^p*792^(1 + p)*c
^3*d^3*E^((3*a)/b)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(1/3))]))/b] - 14^p*495^(1 + p)*c^4*d^4*E^((4*a)/b)*
Gamma[1 + p, (-4*(a + b*Log[c*(d + e*x^(1/3))]))/b] + 2^(2 + 3*p)*21^p*55^(1 + p)*c^5*d^5*E^((5*a)/b)*Gamma[1
+ p, (-3*(a + b*Log[c*(d + e*x^(1/3))]))/b] - 6^(1 + 2*p)*11^(1 + p)*35^p*c^6*d^6*E^((6*a)/b)*Gamma[1 + p, (-2
*(a + b*Log[c*(d + e*x^(1/3))]))/b] + 2^(2 + 3*p)*3^(1 + 2*p)*385^p*c^7*d^7*E^((7*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)/(3465^p*c^12*e^12*E^((12*a)/b)*(-((a + b*Log[
c*(d + e*x^(1/3))])/b))^p))

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Maple [F]  time = 0.484, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c e x^{\frac{1}{3}} + c d\right ) + a\right )}^{p} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)