\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^2))^p}{x^2} \, dx\) [553]

Optimal result

Integrand size = 24, antiderivative size = 216 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^2} \, dx=\frac {2^{1+p} d e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \Gamma \left (1+p,\frac {-a-b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{e^2 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}-\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p}}{c e^2} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d* 
e*p**2 - 2*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p 
*b*d*e*p + (log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*log((2* 
sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b*d**2*p*x + (log((2*sqrt(x)*c*d*e + 
 c*d**2*x + c*e**2)/x)*b + a)**p*a*d**2*p*x - (log((2*sqrt(x)*c*d*e + c*d* 
*2*x + c*e**2)/x)*b + a)**p*a*e**2*p - (log((2*sqrt(x)*c*d*e + c*d**2*x + 
c*e**2)/x)*b + a)**p*a*e**2 - 2*int((log((2*sqrt(x)*c*d*e + c*d**2*x + c*e 
**2)/x)*b + a)**p/(sqrt(x)*log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*a* 
b*e*x + sqrt(x)*log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b**2*e*p*x + 
sqrt(x)*a**2*e*x + sqrt(x)*a*b*e*p*x + log((2*sqrt(x)*c*d*e + c*d**2*x + c 
*e**2)/x)*a*b*d*x**2 + log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b**2*d 
*p*x**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 
((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p/(sqrt(x)*log((2*sqrt(x 
)*c*d*e + c*d**2*x + c*e**2)/x)*a*b*e*x + sqrt(x)*log((2*sqrt(x)*c*d*e + c 
*d**2*x + c*e**2)/x)*b**2*e*p*x + sqrt(x)*a**2*e*x + sqrt(x)*a*b*e*p*x + l 
og((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*a*b*d*x**2 + log((2*sqrt(x)*c* 
d*e + c*d**2*x + c*e**2)/x)*b**2*d*p*x**2 + a**2*d*x**2 + a*b*d*p*x**2),x) 
*a*b**2*d*e**2*p**2*x - 2*int((log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x 
)*b + a)**p/(sqrt(x)*log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*a*b*e*x 
+ sqrt(x)*log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b**2*e*p*x + sqr...