3.6.85 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})))^p}{x^2} \, dx\) [585]

3.6.85.1 Optimal result

Integrand size = 22, antiderivative size = 267 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^2} \, dx=-\frac {3^{-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^3 e^3}+\frac {3\ 2^{-p} d e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^2 e^3}-\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c e^3} \]

output

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

3.6.85.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^2} \, dx=-\frac {6^{-p} e^{-\frac {3 a}{b}} \left (2^p \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+3^{1+p} c d e^{a/b} \left (-\Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^3 e^3} \]

input

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

output

-(((2^p*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 3^(1 + p)*c* 
d*E^(a/b)*(-Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 2^p*c*d* 
E^(a/b)*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)]))*(a + b*Log[c*( 
d + e/x^(1/3))])^p)/(6^p*c^3*e^3*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3 
))])/b))^p))
 

3.6.85.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2904, 2848, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.6.85.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2848

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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])
 

3.6.85.4 Maple [F]

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

input

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

output

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

3.6.85.5 Fricas [F]

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

input

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

output

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

3.6.85.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.85.7 Maxima [F]

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

input

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

output

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

3.6.85.8 Giac [F]

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

input

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

output

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

3.6.85.9 Mupad [F(-1)]

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

input

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

output

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