\(\int (a+b \log (c (d+e \sqrt [3]{x})^2))^p \, dx\) [565]

Optimal result

Integrand size = 20, antiderivative size = 338 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\frac {\left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^3 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {3 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{c e^3}+\frac {3\ 2^p d^2 e^{-\frac {a}{2 b}} \left (d+e \sqrt [3]{x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p}}{e^3 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}} \]

[Out]

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

Time = 0.30 (sec) , antiderivative size = 338, 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 = {2501, 2448, 2436, 2337, 2212, 2437, 2347} \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\frac {3 d^2 2^p e^{-\frac {a}{2 b}} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{2 b}\right )}{e^3 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}}+\frac {\left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{2 b}\right )}{e^3 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {3 d e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )}{c e^3} \]

[In]

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

[Out]

((2/3)^p*(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^3*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) - (3*d*
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^3*E^(a/b)*(-((a +
b*Log[c*(d + e*x^(1/3))^2])/b))^p) + (3*2^p*d^2*(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^3*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 2501

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 \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^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^2}-\frac {2 d (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^2}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^2}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 \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^2}-\frac {(6 d) \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^2}+\frac {\left (3 d^2\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^2} \\ & = \frac {3 \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^3}-\frac {(6 d) \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^3}+\frac {\left (3 d^2\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^3} \\ & = -\frac {(3 d) \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^3}+\frac {\left (3 \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 )}{2 e^3 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}+\frac {\left (3 d^2 \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^3 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}} \\ & = \frac {\left (\frac {2}{3}\right )^p 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^3 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {3 d 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^3}+\frac {3\ 2^p d^2 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^3 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{2}\right )\right )}^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((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)

Giac [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((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)

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^2\right )\right )}^p \,d x \]

[In]

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

[Out]

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