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

Optimal. Leaf size=1363 \[ \text{result too large to display} \]

[Out]

(2^(-2 - p)*Gamma[1 + p, (-6*(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^6
*e^12*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (3*(2/11)^p*d*(d + e*x^(1/3))^11*Gamma[1 + p, (
-11*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^12*E^((11*a)/(2*b))*(c*(d
+ e*x^(1/3))^2)^(11/2)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (33*d^2*Gamma[1 + p, (-5*(a + b*Log[c*(d +
e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(2*5^p*c^5*e^12*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(
1/3))^2])/b))^p) - (55*(2/9)^p*d^3*(d + e*x^(1/3))^9*Gamma[1 + p, (-9*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]
*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^12*E^((9*a)/(2*b))*(c*(d + e*x^(1/3))^2)^(9/2)*(-((a + b*Log[c*(d + e*
x^(1/3))^2])/b))^p) + (495*d^4*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/
3))^2])^p)/(2^(2*(1 + p))*c^4*e^12*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (99*2^(1 + p)*d^5*
(d + e*x^(1/3))^7*Gamma[1 + p, (-7*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p
)/(7^p*e^12*E^((7*a)/(2*b))*(c*(d + e*x^(1/3))^2)^(7/2)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (77*3^(1
- p)*d^6*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)/(c^3*e^12*E
^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (99*2^(1 + p)*d^7*(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)/(5^p*e^12*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) + (495*2^(-2 - p)*d^8*Gamma[1 + p, (-2*(a + b*L
og[c*(d + e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(c^2*e^12*E^((2*a)/b)*(-((a + b*Log[c*(d + e*
x^(1/3))^2])/b))^p) - (55*(2/3)^p*d^9*(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)/(e^12*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) + (33*d^10*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^12*E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (3*2^p*d^11*(d + e*x^(1/3))*Gamma[1
+ p, -(a + b*Log[c*(d + e*x^(1/3))^2])/(2*b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^12*E^(a/(2*b))*Sqrt[c*(d
+ e*x^(1/3))^2]*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p)

________________________________________________________________________________________

Rubi [A]  time = 2.13277, antiderivative size = 1363, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2454, 2401, 2389, 2300, 2181, 2390, 2310} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-2 - p)*Gamma[1 + p, (-6*(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^6
*e^12*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (3*(2/11)^p*d*(d + e*x^(1/3))^11*Gamma[1 + p, (
-11*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^12*E^((11*a)/(2*b))*(c*(d
+ e*x^(1/3))^2)^(11/2)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (33*d^2*Gamma[1 + p, (-5*(a + b*Log[c*(d +
e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(2*5^p*c^5*e^12*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(
1/3))^2])/b))^p) - (55*(2/9)^p*d^3*(d + e*x^(1/3))^9*Gamma[1 + p, (-9*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]
*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^12*E^((9*a)/(2*b))*(c*(d + e*x^(1/3))^2)^(9/2)*(-((a + b*Log[c*(d + e*
x^(1/3))^2])/b))^p) + (495*d^4*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/
3))^2])^p)/(2^(2*(1 + p))*c^4*e^12*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (99*2^(1 + p)*d^5*
(d + e*x^(1/3))^7*Gamma[1 + p, (-7*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p
)/(7^p*e^12*E^((7*a)/(2*b))*(c*(d + e*x^(1/3))^2)^(7/2)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (77*3^(1
- p)*d^6*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)/(c^3*e^12*E
^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (99*2^(1 + p)*d^7*(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)/(5^p*e^12*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) + (495*2^(-2 - p)*d^8*Gamma[1 + p, (-2*(a + b*L
og[c*(d + e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(c^2*e^12*E^((2*a)/b)*(-((a + b*Log[c*(d + e*
x^(1/3))^2])/b))^p) - (55*(2/3)^p*d^9*(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)/(e^12*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) + (33*d^10*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^12*E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (3*2^p*d^11*(d + e*x^(1/3))*Gamma[1
+ p, -(a + b*Log[c*(d + e*x^(1/3))^2])/(2*b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^12*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 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 2300

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 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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx &=3 \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{d^{11} \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}+\frac{11 d^{10} (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}-\frac{55 d^9 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}+\frac{165 d^8 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}-\frac{330 d^7 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}+\frac{462 d^6 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}-\frac{462 d^5 (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}+\frac{330 d^4 (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}-\frac{165 d^3 (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}+\frac{55 d^2 (d+e x)^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}-\frac{11 d (d+e x)^{10} \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}+\frac{(d+e x)^{11} \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^{11}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int (d+e x)^{11} \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{(33 d) \operatorname{Subst}\left (\int (d+e x)^{10} \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (165 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (495 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (990 d^4\right ) \operatorname{Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (1386 d^5\right ) \operatorname{Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (1386 d^6\right ) \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (990 d^7\right ) \operatorname{Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (495 d^8\right ) \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (165 d^9\right ) \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (33 d^{10}\right ) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (3 d^{11}\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}\\ &=\frac{3 \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{(33 d) \operatorname{Subst}\left (\int x^{10} \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (165 d^2\right ) \operatorname{Subst}\left (\int x^9 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (495 d^3\right ) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (990 d^4\right ) \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (1386 d^5\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (1386 d^6\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (990 d^7\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (495 d^8\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (165 d^9\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (33 d^{10}\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (3 d^{11}\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}\\ &=\frac{3 \operatorname{Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 c^6 e^{12}}+\frac{\left (165 d^2\right ) \operatorname{Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 c^5 e^{12}}+\frac{\left (495 d^4\right ) \operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{c^4 e^{12}}+\frac{\left (693 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 )^2\right )\right )}{c^3 e^{12}}+\frac{\left (495 d^8\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 c^2 e^{12}}+\frac{\left (33 d^{10}\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 c e^{12}}-\frac{\left (33 d \left (d+e \sqrt [3]{x}\right )^{11}\right ) \operatorname{Subst}\left (\int e^{11 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{11/2}}-\frac{\left (495 d^3 \left (d+e \sqrt [3]{x}\right )^9\right ) \operatorname{Subst}\left (\int e^{9 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}-\frac{\left (693 d^5 \left (d+e \sqrt [3]{x}\right )^7\right ) \operatorname{Subst}\left (\int e^{7 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}-\frac{\left (495 d^7 \left (d+e \sqrt [3]{x}\right )^5\right ) \operatorname{Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}-\frac{\left (165 d^9 \left (d+e \sqrt [3]{x}\right )^3\right ) \operatorname{Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac{\left (3 d^{11} \left (d+e \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 e^{12} \sqrt{c \left (d+e \sqrt [3]{x}\right )^2}}\\ &=\frac{2^{-2-p} 3^{-p} e^{-\frac{6 a}{b}} \Gamma \left (1+p,-\frac{6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{c^6 e^{12}}-\frac{3 \left (\frac{2}{11}\right )^p d e^{-\frac{11 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^{11} \Gamma \left (1+p,-\frac{11 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{11/2}}+\frac{33\ 5^{-p} d^2 e^{-\frac{5 a}{b}} \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{2 c^5 e^{12}}-\frac{55 \left (\frac{2}{9}\right )^p d^3 e^{-\frac{9 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^9 \Gamma \left (1+p,-\frac{9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}+\frac{495\ 4^{-1-p} d^4 e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{c^4 e^{12}}-\frac{99\ 2^{1+p} 7^{-p} d^5 e^{-\frac{7 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^7 \Gamma \left (1+p,-\frac{7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}+\frac{77\ 3^{1-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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{c^3 e^{12}}-\frac{99\ 2^{1+p} 5^{-p} d^7 e^{-\frac{5 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^5 \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}+\frac{495\ 2^{-2-p} d^8 e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{c^2 e^{12}}-\frac{55 \left (\frac{2}{3}\right )^p d^9 e^{-\frac{3 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^3 \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}+\frac{33 d^{10} e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{2 c e^{12}}-\frac{3\ 2^p d^{11} e^{-\frac{a}{2 b}} \left (d+e \sqrt [3]{x}\right ) \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^{12} \sqrt{c \left (d+e \sqrt [3]{x}\right )^2}}\\ \end{align*}

Mathematica [F]  time = 0.649893, size = 0, normalized size = 0. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.576, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{2} \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))^2))^p,x)

[Out]

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

[Out]

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

[Out]

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

[Out]

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