\(\int (a+b \log (c (d+e \sqrt [3]{x})))^p \, dx\) [559]

Optimal result

Integrand size = 18, antiderivative size = 266 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^3}-\frac {3\ 2^{-p} d e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.65 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {6^{-p} e^{-\frac {3 a}{b}} \left (2^p \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+3^{1+p} c d e^{a/b} \left (-\Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^3} \] Input:

Integrate[(a + b*Log[c*(d + e*x^(1/3))])^p,x]
 

Output:

((2^p*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(1/3))]))/b] + 3^(1 + p)*c*d* 
E^(a/b)*(-Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(1/3))]))/b] + 2^p*c*d*E^ 
(a/b)*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))])/b)]))*(a + b*Log[c*(d 
+ e*x^(1/3))])^p)/(6^p*c^3*e^3*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3)) 
])/b))^p)
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.

\(\displaystyle \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx\)

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*Log[c*(d + e*x^(1/3))])^p,x]
 

Output:

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

\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )\right )\right )}^{p}d x\]

Input:

int((a+b*ln(c*(d+e*x^(1/3))))^p,x)
 

Output:

int((a+b*ln(c*(d+e*x^(1/3))))^p,x)
 

Fricas [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} \,d x } \] Input:

integrate((a+b*log(c*(d+e*x^(1/3))))^p,x, algorithm="fricas")
 

Output:

integral((b*log(c*e*x^(1/3) + c*d) + a)^p, x)
 

Sympy [F]

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

integrate((a+b*ln(c*(d+e*x**(1/3))))**p,x)
 

Output:

Integral((a + b*log(c*(d + e*x**(1/3))))**p, x)
 

Maxima [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} \,d x } \] Input:

integrate((a+b*log(c*(d+e*x^(1/3))))^p,x, algorithm="maxima")
 

Output:

integrate((b*log((e*x^(1/3) + d)*c) + a)^p, x)
 

Giac [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} \,d x } \] Input:

integrate((a+b*log(c*(d+e*x^(1/3))))^p,x, algorithm="giac")
 

Output:

integrate((b*log((e*x^(1/3) + d)*c) + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int {\left (a+b\,\ln \left (c\,\left (d+e\,x^{1/3}\right )\right )\right )}^p \,d x \] Input:

int((a + b*log(c*(d + e*x^(1/3))))^p,x)
 

Output:

int((a + b*log(c*(d + e*x^(1/3))))^p, x)
 

Reduce [F]

\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\text {too large to display} \] Input:

int((a+b*log(c*(d+e*x^(1/3))))^p,x)
 

Output:

(3*x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d*e**2*p**2 + 3*x**(2/3)* 
(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d*e**2*p - 6*x**(1/3)*(log(x**(1/3)*c 
*e + c*d)*b + a)**p*b*d**2*e*p**2 - 6*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b 
+ a)**p*b*d**2*e*p + 6*(log(x**(1/3)*c*e + c*d)*b + a)**p*log(x**(1/3)*c*e 
 + c*d)*b*d**3*p + 6*(log(x**(1/3)*c*e + c*d)*b + a)**p*a*d**3*p + 6*(log( 
x**(1/3)*c*e + c*d)*b + a)**p*a*e**3*p*x + 6*(log(x**(1/3)*c*e + c*d)*b + 
a)**p*a*e**3*x + 6*int((log(x**(1/3)*c*e + c*d)*b + a)**p/(3*x**(2/3)*log( 
x**(1/3)*c*e + c*d)*a*b*e + x**(2/3)*log(x**(1/3)*c*e + c*d)*b**2*e*p + 3* 
x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 3*x**(1/3)*log(x**(1/3)*c*e + c*d)*a* 
b*d + x**(1/3)*log(x**(1/3)*c*e + c*d)*b**2*d*p + 3*x**(1/3)*a**2*d + x**( 
1/3)*a*b*d*p),x)*a*b**2*d**2*e**2*p**3 + 6*int((log(x**(1/3)*c*e + c*d)*b 
+ a)**p/(3*x**(2/3)*log(x**(1/3)*c*e + c*d)*a*b*e + x**(2/3)*log(x**(1/3)* 
c*e + c*d)*b**2*e*p + 3*x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 3*x**(1/3)*lo 
g(x**(1/3)*c*e + c*d)*a*b*d + x**(1/3)*log(x**(1/3)*c*e + c*d)*b**2*d*p + 
3*x**(1/3)*a**2*d + x**(1/3)*a*b*d*p),x)*a*b**2*d**2*e**2*p**2 + 2*int((lo 
g(x**(1/3)*c*e + c*d)*b + a)**p/(3*x**(2/3)*log(x**(1/3)*c*e + c*d)*a*b*e 
+ x**(2/3)*log(x**(1/3)*c*e + c*d)*b**2*e*p + 3*x**(2/3)*a**2*e + x**(2/3) 
*a*b*e*p + 3*x**(1/3)*log(x**(1/3)*c*e + c*d)*a*b*d + x**(1/3)*log(x**(1/3 
)*c*e + c*d)*b**2*d*p + 3*x**(1/3)*a**2*d + x**(1/3)*a*b*d*p),x)*b**3*d**2 
*e**2*p**4 + 2*int((log(x**(1/3)*c*e + c*d)*b + a)**p/(3*x**(2/3)*log(x...