\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^2))^p}{x^2} \, dx\) [591]

Optimal result

Integrand size = 24, antiderivative size = 342 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=-\frac {3\ 2^p d^2 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \Gamma \left (1+p,\frac {-a-b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^3 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}}-\frac {\left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^3 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac {3 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^3} \]

[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 = 0.32 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=-\frac {3 d^2 2^p e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right )}{e^3 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}}-\frac {\left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^3 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac {3 d e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )}{c e^3} \]

[In]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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