\(\int x (a+b \log (c (d+e x^{2/3})^2))^p \, dx\) [576]

Optimal result

Integrand size = 22, antiderivative size = 350 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\frac {3\ 2^{-1+p} d^2 e^{-\frac {a}{2 b}} \left (d+e x^{2/3}\right ) \Gamma \left (1+p,\frac {-a-b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^3 \sqrt {c \left (d+e x^{2/3}\right )^2}}+\frac {2^{-1+p} 3^{-p} e^{-\frac {3 a}{2 b}} \left (d+e x^{2/3}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^3 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}-\frac {3 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{2 c e^3} \] Output:

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

Mathematica [F]

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

Integrate[x*(a + b*Log[c*(d + e*x^(2/3))^2])^p,x]
 

Output:

Integrate[x*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x]
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle \frac {3}{2} \int x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^pdx^{2/3}\)

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

Int[x*(a + b*Log[c*(d + e*x^(2/3))^2])^p,x]
 

Output:

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

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]

\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{2}\right )\right )}^{p}d x\]

Input:

int(x*(a+b*ln(c*(d+e*x^(2/3))^2))^p,x)
 

Output:

int(x*(a+b*ln(c*(d+e*x^(2/3))^2))^p,x)
 

Fricas [F]

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

integrate(x*(a+b*log(c*(d+e*x^(2/3))^2))^p,x, algorithm="fricas")
 

Output:

integral((b*log(c*e^2*x^(4/3) + 2*c*d*e*x^(2/3) + c*d^2) + a)^p*x, x)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:

integrate(x*(a+b*ln(c*(d+e*x**(2/3))**2))**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x*(a+b*log(c*(d+e*x^(2/3))^2))^p,x, algorithm="maxima")
 

Output:

integrate((b*log((e*x^(2/3) + d)^2*c) + a)^p*x, x)
 

Giac [F]

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

integrate(x*(a+b*log(c*(d+e*x^(2/3))^2))^p,x, algorithm="giac")
 

Output:

integrate((b*log((e*x^(2/3) + d)^2*c) + a)^p*x, x)
 

Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^2\right )\right )}^p \,d x \] Input:

int(x*(a + b*log(c*(d + e*x^(2/3))^2))^p,x)
 

Output:

int(x*(a + b*log(c*(d + e*x^(2/3))^2))^p, x)
 

Reduce [F]

\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:

int(x*(a+b*log(c*(d+e*x^(2/3))^2))^p,x)
 

Output:

( - 6*x**(2/3)*(log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)* 
*p*b*d**2*e*p**2 - 6*x**(2/3)*(log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + 
c*d**2)*b + a)**p*b*d**2*e*p + 3*x**(1/3)*(log(2*x**(2/3)*c*d*e + x**(1/3) 
*c*e**2*x + c*d**2)*b + a)**p*b*d*e**2*p**2*x + 3*x**(1/3)*(log(2*x**(2/3) 
*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*b*d*e**2*p*x + 3*(log(2*x** 
(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*log(2*x**(2/3)*c*d*e + 
 x**(1/3)*c*e**2*x + c*d**2)*b*d**3*p + 3*(log(2*x**(2/3)*c*d*e + x**(1/3) 
*c*e**2*x + c*d**2)*b + a)**p*a*d**3*p + 3*(log(2*x**(2/3)*c*d*e + x**(1/3 
)*c*e**2*x + c*d**2)*b + a)**p*a*e**3*p*x**2 + 3*(log(2*x**(2/3)*c*d*e + x 
**(1/3)*c*e**2*x + c*d**2)*b + a)**p*a*e**3*x**2 - 12*int(((log(2*x**(2/3) 
*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*x)/(3*x**(2/3)*log(2*x**(2/ 
3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*a*b*e + 2*x**(2/3)*log(2*x**(2/3)*c 
*d*e + x**(1/3)*c*e**2*x + c*d**2)*b**2*e*p + 3*x**(2/3)*a**2*e + 2*x**(2/ 
3)*a*b*e*p + 3*log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*a*b*d + 
2*log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b**2*d*p + 3*a**2*d + 
 2*a*b*d*p),x)*a*b**2*d*e**3*p**3 - 12*int(((log(2*x**(2/3)*c*d*e + x**(1/ 
3)*c*e**2*x + c*d**2)*b + a)**p*x)/(3*x**(2/3)*log(2*x**(2/3)*c*d*e + x**( 
1/3)*c*e**2*x + c*d**2)*a*b*e + 2*x**(2/3)*log(2*x**(2/3)*c*d*e + x**(1/3) 
*c*e**2*x + c*d**2)*b**2*e*p + 3*x**(2/3)*a**2*e + 2*x**(2/3)*a*b*e*p + 3* 
log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*a*b*d + 2*log(2*x**(...