3.6.63 \(\int x^2 (a+b \log (c (d+e \sqrt [3]{x})^2))^p \, dx\) [563]

Optimal. Leaf size=1035 \[ \frac {2^p 3^{-1-2 p} 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}-\frac {3\ 4^{-p} d 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^9}+\frac {3\ 2^{2+p} 7^{-p} d^2 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}-\frac {28\ 3^{-p} d^3 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^9}+\frac {21\ 2^{1+p} 5^{-p} d^4 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}-\frac {21\ 2^{1-p} d^5 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^9}+\frac {7\ 2^{2+p} 3^{-p} d^6 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {12 d^7 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}}{c e^9}+\frac {3\ 2^p d^8 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^9 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}} \]

[Out]

2^p*3^(-1-2*p)*(d+e*x^(1/3))^9*GAMMA(1+p,-9/2*(a+b*ln(c*(d+e*x^(1/3))^2))/b)*(a+b*ln(c*(d+e*x^(1/3))^2))^p/e^9
/exp(9/2*a/b)/(c*(d+e*x^(1/3))^2)^(9/2)/(((-a-b*ln(c*(d+e*x^(1/3))^2))/b)^p)-3*d*GAMMA(1+p,-4*(a+b*ln(c*(d+e*x
^(1/3))^2))/b)*(a+b*ln(c*(d+e*x^(1/3))^2))^p/(4^p)/c^4/e^9/exp(4*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))^2))/b)^p)+3*2
^(2+p)*d^2*(d+e*x^(1/3))^7*GAMMA(1+p,-7/2*(a+b*ln(c*(d+e*x^(1/3))^2))/b)*(a+b*ln(c*(d+e*x^(1/3))^2))^p/(7^p)/e
^9/exp(7/2*a/b)/(c*(d+e*x^(1/3))^2)^(7/2)/(((-a-b*ln(c*(d+e*x^(1/3))^2))/b)^p)-28*d^3*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^9/exp(3*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))^2))/b)^p
)+21*2^(1+p)*d^4*(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/(
5^p)/e^9/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)-21*2^(1-p)*d^5*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^9/exp(2*a/b)/(((-a-b*ln(c*(d+e*x^(1/3))^
2))/b)^p)+7*2^(2+p)*d^6*(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^9/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)-12*d^7*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^9/exp(a/b)/(((-a-b*ln(c*(d+e*x^(1/3))^2))/b)^
p)+3*2^p*d^8*(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^9/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 = 1.02, antiderivative size = 1035, normalized size of antiderivative = 1.00, number of steps used = 30, 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 {2^p 3^{-2 p-1} e^{-\frac {9 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^9 \text {Gamma}\left (p+1,-\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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}-\frac {3\ 4^{-p} d 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 )^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^9}+\frac {3\ 2^{p+2} 7^{-p} d^2 e^{-\frac {7 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^7 \text {Gamma}\left (p+1,-\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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}-\frac {28\ 3^{-p} d^3 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 )^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^9}+\frac {21\ 2^{p+1} 5^{-p} d^4 e^{-\frac {5 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^5 \text {Gamma}\left (p+1,-\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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}-\frac {21\ 2^{1-p} d^5 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 )^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^9}+\frac {7\ 2^{p+2} 3^{-p} d^6 e^{-\frac {3 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^3 \text {Gamma}\left (p+1,-\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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {12 d^7 e^{-\frac {a}{b}} \text {Gamma}\left (p+1,-\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}}{c e^9}+\frac {3\ 2^p d^8 e^{-\frac {a}{2 b}} \left (d+e \sqrt [3]{x}\right ) \text {Gamma}\left (p+1,-\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^9 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^p*3^(-1 - 2*p)*(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^9*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
) - (3*d*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)/(4^p*c^4*e^
9*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (3*2^(2 + p)*d^2*(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^9*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) - (28*d^3*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^9*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))
^2])/b))^p) + (21*2^(1 + p)*d^4*(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^9*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) - (21*2^(1 - p)*d^5*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^9*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (7*2^(2 + p)*d^6*(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^9*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) - (12*d^7*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)/(c*e^9*E^(a/b)*(-((a + b*Log[c*
(d + e*x^(1/3))^2])/b))^p) + (3*2^p*d^8*(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^9*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 x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx &=3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {d^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {8 d^7 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {28 d^6 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {56 d^5 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {70 d^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {56 d^3 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {28 d^2 (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {8 d (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {(d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 \text {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^8}-\frac {(24 d) \text {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^8}+\frac {\left (84 d^2\right ) \text {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^8}-\frac {\left (168 d^3\right ) \text {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^8}+\frac {\left (210 d^4\right ) \text {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^8}-\frac {\left (168 d^5\right ) \text {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^8}+\frac {\left (84 d^6\right ) \text {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^8}-\frac {\left (24 d^7\right ) \text {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^8}+\frac {\left (3 d^8\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )}{e^8}\\ &=\frac {3 \text {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^9}-\frac {(24 d) \text {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^9}+\frac {\left (84 d^2\right ) \text {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^9}-\frac {\left (168 d^3\right ) \text {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^9}+\frac {\left (210 d^4\right ) \text {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^9}-\frac {\left (168 d^5\right ) \text {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^9}+\frac {\left (84 d^6\right ) \text {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^9}-\frac {\left (24 d^7\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (3 d^8\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}\\ &=-\frac {(12 d) \text {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^9}-\frac {\left (84 d^3\right ) \text {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^9}-\frac {\left (84 d^5\right ) \text {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 )}{c^2 e^9}-\frac {\left (12 d^7\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{c e^9}+\frac {\left (3 \left (d+e \sqrt [3]{x}\right )^9\right ) \text {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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}+\frac {\left (42 d^2 \left (d+e \sqrt [3]{x}\right )^7\right ) \text {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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}+\frac {\left (105 d^4 \left (d+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+e \sqrt [3]{x}\right )^2\right )\right )}{e^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}+\frac {\left (42 d^6 \left (d+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+e \sqrt [3]{x}\right )^2\right )\right )}{e^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}+\frac {\left (3 d^8 \left (d+e \sqrt [3]{x}\right )\right ) \text {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^9 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}}\\ &=\frac {2^p 3^{-1-2 p} 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}-\frac {3\ 4^{-p} d 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^9}+\frac {3\ 2^{2+p} 7^{-p} d^2 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}-\frac {28\ 3^{-p} d^3 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^9}+\frac {21\ 2^{1+p} 5^{-p} d^4 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}-\frac {21\ 2^{1-p} d^5 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^9}+\frac {7\ 2^{2+p} 3^{-p} d^6 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^9 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {12 d^7 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}}{c e^9}+\frac {3\ 2^p d^8 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^9 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}}\\ \end {align*}

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Mathematica [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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