\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})))^p}{x^2} \, dx\) [547]

Optimal result

Integrand size = 22, antiderivative size = 175 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x^2} \, dx=-\frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^2 e^2}+\frac {2 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x^2} \, dx=\frac {2^{-p} e^{-\frac {2 a}{b}} \left (-\Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^2 e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x^2} \, dx=\text {too large to display} \] Input:

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

Output:

( - 2*sqrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d*e*p**2 - 2*s 
qrt(x)*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*b*d*e*p + 2*(log((sqrt( 
x)*c*d + c*e)/sqrt(x))*b + a)**p*log((sqrt(x)*c*d + c*e)/sqrt(x))*b*d**2*p 
*x + 2*(log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p*a*d**2*p*x - 2*(log((sq 
rt(x)*c*d + c*e)/sqrt(x))*b + a)**p*a*e**2*p - 2*(log((sqrt(x)*c*d + c*e)/ 
sqrt(x))*b + a)**p*a*e**2 - 2*int((log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a) 
**p/(2*sqrt(x)*log((sqrt(x)*c*d + c*e)/sqrt(x))*a*b*e*x + sqrt(x)*log((sqr 
t(x)*c*d + c*e)/sqrt(x))*b**2*e*p*x + 2*sqrt(x)*a**2*e*x + sqrt(x)*a*b*e*p 
*x + 2*log((sqrt(x)*c*d + c*e)/sqrt(x))*a*b*d*x**2 + log((sqrt(x)*c*d + c* 
e)/sqrt(x))*b**2*d*p*x**2 + 2*a**2*d*x**2 + a*b*d*p*x**2),x)*a*b**2*d*e**2 
*p**3*x - 2*int((log((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p/(2*sqrt(x)*log 
((sqrt(x)*c*d + c*e)/sqrt(x))*a*b*e*x + sqrt(x)*log((sqrt(x)*c*d + c*e)/sq 
rt(x))*b**2*e*p*x + 2*sqrt(x)*a**2*e*x + sqrt(x)*a*b*e*p*x + 2*log((sqrt(x 
)*c*d + c*e)/sqrt(x))*a*b*d*x**2 + log((sqrt(x)*c*d + c*e)/sqrt(x))*b**2*d 
*p*x**2 + 2*a**2*d*x**2 + a*b*d*p*x**2),x)*a*b**2*d*e**2*p**2*x - int((log 
((sqrt(x)*c*d + c*e)/sqrt(x))*b + a)**p/(2*sqrt(x)*log((sqrt(x)*c*d + c*e) 
/sqrt(x))*a*b*e*x + sqrt(x)*log((sqrt(x)*c*d + c*e)/sqrt(x))*b**2*e*p*x + 
2*sqrt(x)*a**2*e*x + sqrt(x)*a*b*e*p*x + 2*log((sqrt(x)*c*d + c*e)/sqrt(x) 
)*a*b*d*x**2 + log((sqrt(x)*c*d + c*e)/sqrt(x))*b**2*d*p*x**2 + 2*a**2*d*x 
**2 + a*b*d*p*x**2),x)*b**3*d*e**2*p**4*x - int((log((sqrt(x)*c*d + c*e...