\(\int x (a+b \log (c (d+e \sqrt [3]{x})))^p \, dx\) [558]

Optimal result

Integrand size = 20, antiderivative size = 553 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {2^{-1-p} 3^{-p} e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \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^6 e^6}-\frac {3\ 5^{-p} d e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \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^5 e^6}+\frac {15\ 2^{-1-2 p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \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^4 e^6}-\frac {10\ 3^{-p} d^3 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^6}+\frac {15\ 2^{-1-p} d^4 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^6}-\frac {3 d^5 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^6} \] Output:

2^(-1-p)*GAMMA(p+1,(-6*a-6*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3 
))))^p/(3^p)/c^6/e^6/exp(6*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-3*d*GAM 
MA(p+1,(-5*a-5*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(5^p) 
/c^5/e^6/exp(5*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+15*2^(-1-2*p)*d^2*G 
AMMA(p+1,(-4*a-4*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/c^4 
/e^6/exp(4*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-10*d^3*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^6/exp( 
3*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+15*2^(-1-p)*d^4*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/c^2/e^6/exp(2*a/b) 
/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-3*d^5*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^6/exp(a/b)/((-(a+b*ln(c*(d+e*x^(1/ 
3))))/b)^p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 3 \int x^{5/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^pd\sqrt [3]{x}\)

\(\Big \downarrow \) 2848

\(\displaystyle 3 \int \left (\frac {\left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^5}-\frac {5 d \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^5}\right )d\sqrt [3]{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {6^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^6 e^6}-\frac {d 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^5 e^6}+\frac {5 d^2 2^{-2 p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^4 e^6}-\frac {10 d^3 3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^3 e^6}+\frac {5 d^4 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )}{c e^6}\right )\)

Input:

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

Output:

3*((6^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b* 
Log[c*(d + e*x^(1/3))])^p)/(c^6*e^6*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^( 
1/3))])/b))^p) - (d*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a 
 + b*Log[c*(d + e*x^(1/3))])^p)/(5^p*c^5*e^6*E^((5*a)/b)*(-((a + b*Log[c*( 
d + e*x^(1/3))])/b))^p) + (5*2^(-1 - 2*p)*d^2*Gamma[1 + p, (-4*(a + b*Log[ 
c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(c^4*e^6*E^((4*a 
)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (10*3^(-1 - p)*d^3*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^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) + (5*2^ 
(-1 - p)*d^4*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Lo 
g[c*(d + e*x^(1/3))])^p)/(c^2*e^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/ 
3))])/b))^p) - (d^5*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^6*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 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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*(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, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x*(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, x)
 

Giac [F]

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

integrate(x*(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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 90*x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**4*e**2*p**2 - 90*x 
**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**4*e**2*p + 36*x**(2/3)*(lo 
g(x**(1/3)*c*e + c*d)*b + a)**p*b*d*e**5*p**2*x + 36*x**(2/3)*(log(x**(1/3 
)*c*e + c*d)*b + a)**p*b*d*e**5*p*x + 180*x**(1/3)*(log(x**(1/3)*c*e + c*d 
)*b + a)**p*b*d**5*e*p**2 + 180*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)** 
p*b*d**5*e*p - 45*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**2*e**4* 
p**2*x - 45*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**2*e**4*p*x - 
180*(log(x**(1/3)*c*e + c*d)*b + a)**p*log(x**(1/3)*c*e + c*d)*b*d**6*p - 
180*(log(x**(1/3)*c*e + c*d)*b + a)**p*a*d**6*p + 180*(log(x**(1/3)*c*e + 
c*d)*b + a)**p*a*e**6*p*x**2 + 180*(log(x**(1/3)*c*e + c*d)*b + a)**p*a*e* 
*6*x**2 + 60*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**3*e**3*p**2*x + 60*(l 
og(x**(1/3)*c*e + c*d)*b + a)**p*b*d**3*e**3*p*x - 360*int((log(x**(1/3)*c 
*e + c*d)*b + a)**p/(6*x**(2/3)*log(x**(1/3)*c*e + c*d)*a*b*e + x**(2/3)*l 
og(x**(1/3)*c*e + c*d)*b**2*e*p + 6*x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 6 
*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 + 6*x**(1/3)*a**2*d + x**(1/3)*a*b*d*p),x)*a*b**2*d**5*e**2*p**3 
 - 360*int((log(x**(1/3)*c*e + c*d)*b + a)**p/(6*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 + 6*x**(2/3)*a** 
2*e + x**(2/3)*a*b*e*p + 6*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 + 6*x**(1/3)*a**2*d + x**(1/3)*a*b*...