\(\int (a+b \log (c (d+e \sqrt {x})))^p \, dx\) [535]

Optimal result

Integrand size = 18, antiderivative size = 174 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^2}-\frac {2 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^2} \] 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 = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.75 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a}{b}} \left (\Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )-2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^2} \] Input:

Integrate[(a + b*Log[c*(d + e*Sqrt[x])])^p,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 = 0.80 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2901, 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.

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

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*Log[c*(d + e*Sqrt[x])])^p,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*Sqr 
t[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 2901

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k - 1)*(a + b*Log[c* 
(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, 
 x] && FractionQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d + e*sqrt(x))))**p, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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