Integrand size = 22, antiderivative size = 350 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\frac {3\ 2^{-1+p} d^2 e^{-\frac {a}{2 b}} \left (d+e x^{2/3}\right ) \Gamma \left (1+p,\frac {-a-b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^3 \sqrt {c \left (d+e x^{2/3}\right )^2}}+\frac {2^{-1+p} 3^{-p} e^{-\frac {3 a}{2 b}} \left (d+e x^{2/3}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^3 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}-\frac {3 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{2 c e^3} \]
2^(-1+p)*(d+e*x^(2/3))^3*GAMMA(p+1,-3/2*(a+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)+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)/(((-a -b*ln(c*(d+e*x^(2/3))^2))/b)^p)/(c*(d+e*x^(2/3))^2)^(1/2)
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx \]
Time = 0.71 (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 |
\(\displaystyle \frac {3}{2} \left (\frac {d^2 2^p e^{-\frac {a}{2 b}} \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{2 b}\right )}{e^3 \sqrt {c \left (d+e x^{2/3}\right )^2}}+\frac {2^p 3^{-p-1} e^{-\frac {3 a}{2 b}} \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right )}{e^3 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}-\frac {d e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )}{c e^3}\right )\) |
(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
3.6.76.3.1 Defintions of rubi rules used
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])
\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{2}\right )\right )}^{p}d x\]
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\text {Timed out} \]
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \]
\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^2\right )\right )}^p \,d x \]