\(\int x (a+b \log (c (d+e x^{2/3})))^p \, dx\) [569]

Optimal result

Integrand size = 20, antiderivative size = 273 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\frac {3^{-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c^3 e^3}-\frac {3\ 2^{-1-p} d e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2504, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\frac {3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{2 c^3 e^3}-\frac {3 d 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )}{2 c e^3} \]

[In]

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

[Out]

(Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2*3^p*c^3*e^3*E^((3*a)
/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) - (3*2^(-1 - p)*d*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))])
)/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(c^2*e^3*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) + (3*d^2*
Gamma[1 + p, -((a + b*Log[c*(d + e*x^(2/3))])/b)]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2*c*e^3*E^(a/b)*(-((a + b
*Log[c*(d + e*x^(2/3))])/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}& = \frac {3}{2} \text {Subst}\left (\int x^2 (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right ) \\ & = \frac {3}{2} \text {Subst}\left (\int \left (\frac {d^2 (a+b \log (c (d+e x)))^p}{e^2}-\frac {2 d (d+e x) (a+b \log (c (d+e x)))^p}{e^2}+\frac {(d+e x)^2 (a+b \log (c (d+e x)))^p}{e^2}\right ) \, dx,x,x^{2/3}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )}{2 e^2}-\frac {(3 d) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )}{e^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )}{2 e^2} \\ & = \frac {3 \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e x^{2/3}\right )}{2 e^3}-\frac {(3 d) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e x^{2/3}\right )}{e^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e x^{2/3}\right )}{2 e^3} \\ & = \frac {3 \text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )\right )\right )}{2 c^3 e^3}-\frac {(3 d) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )\right )\right )}{c^2 e^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )\right )\right )}{2 c e^3} \\ & = \frac {3^{-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c^3 e^3}-\frac {3\ 2^{-1-p} d e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c e^3} \\ \end{align*}

Mathematica [F]

\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )\right )\right )}^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{2/3}\right )\right )\right )}^p \,d x \]

[In]

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

[Out]

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