Integrand size = 20, antiderivative size = 273 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\frac {3^{-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c^3 e^3}-\frac {3\ 2^{-1-p} d e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c e^3} \] Output:
1/2*GAMMA(p+1,(-3*a-3*b*ln(c*(d+e*x^(2/3))))/b)*(a+b*ln(c*(d+e*x^(2/3))))^ p/(3^p)/c^3/e^3/exp(3*a/b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)-3*2^(-1-p)*d *GAMMA(p+1,(-2*a-2*b*ln(c*(d+e*x^(2/3))))/b)*(a+b*ln(c*(d+e*x^(2/3))))^p/c ^2/e^3/exp(2*a/b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)+3/2*d^2*GAMMA(p+1,-(a +b*ln(c*(d+e*x^(2/3))))/b)*(a+b*ln(c*(d+e*x^(2/3))))^p/c/e^3/exp(a/b)/((-( a+b*ln(c*(d+e*x^(2/3))))/b)^p)
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx \] Input:
Integrate[x*(a + b*Log[c*(d + e*x^(2/3))])^p,x]
Output:
Integrate[x*(a + b*Log[c*(d + e*x^(2/3))])^p, x]
Time = 1.11 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.
\(\displaystyle \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {3}{2} \int x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^pdx^{2/3}\) |
\(\Big \downarrow \) 2848 |
\(\displaystyle \frac {3}{2} \int \left (\frac {\left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^2}-\frac {2 d \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^2}+\frac {d^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^2}\right )dx^{2/3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} \left (\frac {3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^3 e^3}-\frac {d 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^2 e^3}+\frac {d^2 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )}{c e^3}\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e*x^(2/3))])^p,x]
Output:
(3*((3^(-1 - p)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b *Log[c*(d + e*x^(2/3))])^p)/(c^3*e^3*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^ (2/3))])/b))^p) - (d*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))]))/b]*( a + b*Log[c*(d + e*x^(2/3))])^p)/(2^p*c^2*e^3*E^((2*a)/b)*(-((a + b*Log[c* (d + e*x^(2/3))])/b))^p) + (d^2*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(2/3) )])/b)]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(c*e^3*E^(a/b)*(-((a + b*Log[c*( d + e*x^(2/3))])/b))^p)))/2
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]
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])
\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )\right )\right )}^{p}d x\]
Input:
int(x*(a+b*ln(c*(d+e*x^(2/3))))^p,x)
Output:
int(x*(a+b*ln(c*(d+e*x^(2/3))))^p,x)
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e*x^(2/3))))^p,x, algorithm="fricas")
Output:
integral((b*log(c*e*x^(2/3) + c*d) + a)^p*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*ln(c*(d+e*x**(2/3))))**p,x)
Output:
Timed out
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e*x^(2/3))))^p,x, algorithm="maxima")
Output:
integrate((b*log((e*x^(2/3) + d)*c) + a)^p*x, x)
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e*x^(2/3))))^p,x, algorithm="giac")
Output:
integrate((b*log((e*x^(2/3) + d)*c) + a)^p*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{2/3}\right )\right )\right )}^p \,d x \] Input:
int(x*(a + b*log(c*(d + e*x^(2/3))))^p,x)
Output:
int(x*(a + b*log(c*(d + e*x^(2/3))))^p, x)
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x*(a+b*log(c*(d+e*x^(2/3))))^p,x)
Output:
( - 6*x**(2/3)*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**2*e*p**2 - 6*x**(2/ 3)*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**2*e*p + 3*x**(1/3)*(log(x**(2/3 )*c*e + c*d)*b + a)**p*b*d*e**2*p**2*x + 3*x**(1/3)*(log(x**(2/3)*c*e + c* d)*b + a)**p*b*d*e**2*p*x + 6*(log(x**(2/3)*c*e + c*d)*b + a)**p*log(x**(2 /3)*c*e + c*d)*b*d**3*p + 6*(log(x**(2/3)*c*e + c*d)*b + a)**p*a*d**3*p + 6*(log(x**(2/3)*c*e + c*d)*b + a)**p*a*e**3*p*x**2 + 6*(log(x**(2/3)*c*e + c*d)*b + a)**p*a*e**3*x**2 - 6*int(((log(x**(2/3)*c*e + c*d)*b + a)**p*x) /(3*x**(2/3)*log(x**(2/3)*c*e + c*d)*a*b*e + x**(2/3)*log(x**(2/3)*c*e + c *d)*b**2*e*p + 3*x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 3*log(x**(2/3)*c*e + c*d)*a*b*d + log(x**(2/3)*c*e + c*d)*b**2*d*p + 3*a**2*d + a*b*d*p),x)*a* b**2*d*e**3*p**3 - 6*int(((log(x**(2/3)*c*e + c*d)*b + a)**p*x)/(3*x**(2/3 )*log(x**(2/3)*c*e + c*d)*a*b*e + x**(2/3)*log(x**(2/3)*c*e + c*d)*b**2*e* p + 3*x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 3*log(x**(2/3)*c*e + c*d)*a*b*d + log(x**(2/3)*c*e + c*d)*b**2*d*p + 3*a**2*d + a*b*d*p),x)*a*b**2*d*e**3 *p**2 - 2*int(((log(x**(2/3)*c*e + c*d)*b + a)**p*x)/(3*x**(2/3)*log(x**(2 /3)*c*e + c*d)*a*b*e + x**(2/3)*log(x**(2/3)*c*e + c*d)*b**2*e*p + 3*x**(2 /3)*a**2*e + x**(2/3)*a*b*e*p + 3*log(x**(2/3)*c*e + c*d)*a*b*d + log(x**( 2/3)*c*e + c*d)*b**2*d*p + 3*a**2*d + a*b*d*p),x)*b**3*d*e**3*p**4 - 2*int (((log(x**(2/3)*c*e + c*d)*b + a)**p*x)/(3*x**(2/3)*log(x**(2/3)*c*e + c*d )*a*b*e + x**(2/3)*log(x**(2/3)*c*e + c*d)*b**2*e*p + 3*x**(2/3)*a**2*e...