\(\int \frac {(a+b \log (c \sqrt {x}))^p}{x^2} \, dx\) [188]

Optimal result

Integrand size = 18, antiderivative size = 73 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x^2} \, dx=-2^{-p} c^2 e^{\frac {2 a}{b}} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x^2} \, dx=-2^{-p} c^2 e^{\frac {2 a}{b}} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2747, 2612}

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.

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2612

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2612

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 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*log(c*sqrt(x)))**p/x**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x^2} \, dx=-\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p + 1} c^{2} e^{\left (\frac {2 \, a}{b}\right )} E_{-p}\left (\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}}{b}\right )}{b} \] Input:

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

Output:

-2*(b*log(c*sqrt(x)) + a)^(p + 1)*c^2*e^(2*a/b)*exp_integral_e(-p, 2*(b*lo 
g(c*sqrt(x)) + a)/b)/b
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*(log(sqrt(x)*c)*b + a)**p*a - 2*int(((log(sqrt(x)*c)*b + a)**p*log(s 
qrt(x)*c))/(2*log(sqrt(x)*c)*a*b*x**2 - log(sqrt(x)*c)*b**2*p*x**2 + 2*a** 
2*x**2 - a*b*p*x**2),x)*a*b**2*p*x + int(((log(sqrt(x)*c)*b + a)**p*log(sq 
rt(x)*c))/(2*log(sqrt(x)*c)*a*b*x**2 - log(sqrt(x)*c)*b**2*p*x**2 + 2*a**2 
*x**2 - a*b*p*x**2),x)*b**3*p**2*x)/(x*(2*a - b*p))