\(\int x (a+b \log (c (d+e \sqrt {x})))^p \, dx\) [534]

Optimal result

Integrand size = 20, antiderivative size = 360 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-1-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^4}-\frac {2\ 3^{-p} d e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^3 e^4}+\frac {3\ 2^{-p} d^2 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^4}-\frac {2 d^3 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^4} \] Output:

2^(-1-2*p)*GAMMA(p+1,(-4*a-4*b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1 
/2))))^p/c^4/e^4/exp(4*a/b)/((-(a+b*ln(c*(d+e*x^(1/2))))/b)^p)-2*d*GAMMA(p 
+1,(-3*a-3*b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(3^p)/c^3 
/e^4/exp(3*a/b)/((-(a+b*ln(c*(d+e*x^(1/2))))/b)^p)+3*d^2*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^4/exp(2 
*a/b)/((-(a+b*ln(c*(d+e*x^(1/2))))/b)^p)-2*d^3*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^4/exp(a/b)/((-(a+b*ln(c*(d+e* 
x^(1/2))))/b)^p)
 

Mathematica [A] (verified)

Time = 6.14 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.64 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-1-2 p} 3^{-p} e^{-\frac {4 a}{b}} \left (3^p \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )-2^{1+p} c d e^{a/b} \left (2^{1+p} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )+3^p c d e^{a/b} \left (-3 \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )+2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )\right )\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^4} \] Input:

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

Output:

(2^(-1 - 2*p)*(3^p*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*Sqrt[x])]))/b] - 2 
^(1 + p)*c*d*E^(a/b)*(2^(1 + p)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*Sqrt[ 
x])]))/b] + 3^p*c*d*E^(a/b)*(-3*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)/(3^p*c^4*e^4*E^((4*a)/b)*(- 
((a + b*Log[c*(d + e*Sqrt[x])])/b))^p)
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 2 \int x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^pd\sqrt {x}\)

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {4^{-p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^4 e^4}-\frac {d 3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^3 e^4}+\frac {3 d^2 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^2 e^4}-\frac {d^3 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )}{c e^4}\right )\)

Input:

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

Output:

2*((4^(-1 - p)*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b* 
Log[c*(d + e*Sqrt[x])])^p)/(c^4*e^4*E^((4*a)/b)*(-((a + b*Log[c*(d + e*Sqr 
t[x])])/b))^p) - (d*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a 
 + b*Log[c*(d + e*Sqrt[x])])^p)/(3^p*c^3*e^4*E^((3*a)/b)*(-((a + b*Log[c*( 
d + e*Sqrt[x])])/b))^p) + (3*2^(-1 - p)*d^2*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^4*E^((2*a)/ 
b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (d^3*Gamma[1 + p, -((a + b*L 
og[c*(d + e*Sqrt[x])])/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(c*e^4*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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*(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, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x*(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, x)
 

Giac [F]

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

integrate(x*(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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(12*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**3*e*p**2 + 12*sqrt(x)*( 
log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**3*e*p + 4*sqrt(x)*(log(sqrt(x)*c*e + 
 c*d)*b + a)**p*b*d*e**3*p**2*x + 4*sqrt(x)*(log(sqrt(x)*c*e + c*d)*b + a) 
**p*b*d*e**3*p*x - 12*(log(sqrt(x)*c*e + c*d)*b + a)**p*log(sqrt(x)*c*e + 
c*d)*b*d**4*p - 12*(log(sqrt(x)*c*e + c*d)*b + a)**p*a*d**4*p + 12*(log(sq 
rt(x)*c*e + c*d)*b + a)**p*a*e**4*p*x**2 + 12*(log(sqrt(x)*c*e + c*d)*b + 
a)**p*a*e**4*x**2 - 6*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**2*e**2*p**2*x 
 - 6*(log(sqrt(x)*c*e + c*d)*b + a)**p*b*d**2*e**2*p*x - 24*int((log(sqrt( 
x)*c*e + c*d)*b + a)**p/(4*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 + 4*sqrt(x)*a**2*e + sqrt(x)*a*b*e*p + 4*l 
og(sqrt(x)*c*e + c*d)*a*b*d + log(sqrt(x)*c*e + c*d)*b**2*d*p + 4*a**2*d + 
 a*b*d*p),x)*a*b**2*d**3*e**2*p**3 - 24*int((log(sqrt(x)*c*e + c*d)*b + a) 
**p/(4*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 + 4*sqrt(x)*a**2*e + sqrt(x)*a*b*e*p + 4*log(sqrt(x)*c*e + c*d 
)*a*b*d + log(sqrt(x)*c*e + c*d)*b**2*d*p + 4*a**2*d + a*b*d*p),x)*a*b**2* 
d**3*e**2*p**2 - 6*int((log(sqrt(x)*c*e + c*d)*b + a)**p/(4*sqrt(x)*log(sq 
rt(x)*c*e + c*d)*a*b*e + sqrt(x)*log(sqrt(x)*c*e + c*d)*b**2*e*p + 4*sqrt( 
x)*a**2*e + sqrt(x)*a*b*e*p + 4*log(sqrt(x)*c*e + c*d)*a*b*d + log(sqrt(x) 
*c*e + c*d)*b**2*d*p + 4*a**2*d + a*b*d*p),x)*b**3*d**3*e**2*p**4 - 6*int( 
(log(sqrt(x)*c*e + c*d)*b + a)**p/(4*sqrt(x)*log(sqrt(x)*c*e + c*d)*a*b...