Integrand size = 24, antiderivative size = 678 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=-\frac {3\ 2^{-1+p} d^5 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^6 \sqrt {c \left (d+e x^{2/3}\right )^2}}+\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 )^2\right )\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}}{4 c^3 e^6}-\frac {3\ 2^{-1+p} 5^{-p} d e^{-\frac {5 a}{2 b}} \left (d+e x^{2/3}\right )^5 \Gamma \left (1+p,-\frac {5 \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^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{5/2}}+\frac {15\ 2^{-2-p} d^2 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\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}}{c^2 e^6}-\frac {5 \left (\frac {2}{3}\right )^p d^3 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^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}+\frac {15 d^4 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}}{4 c e^6} \]
1/4*GAMMA(p+1,-3*(a+b*ln(c*(d+e*x^(2/3))^2))/b)*(a+b*ln(c*(d+e*x^(2/3))^2) )^p/(3^p)/c^3/e^6/exp(3*a/b)/(((-a-b*ln(c*(d+e*x^(2/3))^2))/b)^p)-3*2^(-1+ p)*d*(d+e*x^(2/3))^5*GAMMA(p+1,-5/2*(a+b*ln(c*(d+e*x^(2/3))^2))/b)*(a+b*ln (c*(d+e*x^(2/3))^2))^p/(5^p)/e^6/exp(5/2*a/b)/(c*(d+e*x^(2/3))^2)^(5/2)/(( (-a-b*ln(c*(d+e*x^(2/3))^2))/b)^p)+15*2^(-2-p)*d^2*GAMMA(p+1,-2*(a+b*ln(c* (d+e*x^(2/3))^2))/b)*(a+b*ln(c*(d+e*x^(2/3))^2))^p/c^2/e^6/exp(2*a/b)/(((- a-b*ln(c*(d+e*x^(2/3))^2))/b)^p)-5*(2/3)^p*d^3*(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/e^6/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)+15/ 4*d^4*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^6/exp(a/b)/(((-a-b*ln(c*(d+e*x^(2/3))^2))/b)^p)-3*2^(-1+p)*d^5*(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^6/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)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx \]
Time = 1.22 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
| \(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {3}{2} \int x^{10/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^pdx^{2/3}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \frac {3}{2} \int \left (\frac {\left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p}{e^5}-\frac {5 d \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p}{e^5}\right )dx^{2/3}\) |
| \(\Big \downarrow \) 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 )^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 )}{b}\right )}{2 c^3 e^6}+\frac {5 d^2 2^{-p-1} e^{-\frac {2 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 {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 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^6 \sqrt {c \left (d+e x^{2/3}\right )^2}}+\frac {5 d^4 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 )}{2 c e^6}-\frac {5 d^3 2^{p+1} 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^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}-\frac {d \left (\frac {2}{5}\right )^p e^{-\frac {5 a}{2 b}} \left (d+e x^{2/3}\right )^5 \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 {5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{5/2}}\right )\) |
(3*((3^(-1 - p)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))^2]))/b]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(2*c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - ((2/5)^p*d*(d + e*x^(2/3))^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(2/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p) /(e^6*E^((5*a)/(2*b))*(c*(d + e*x^(2/3))^2)^(5/2)*(-((a + b*Log[c*(d + e*x ^(2/3))^2])/b))^p) + (5*2^(-1 - p)*d^2*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))^2]))/b]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(c^2*e^6*E^((2*a)/b )*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - (5*2^(1 + p)*3^(-1 - p)*d^3 *(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^6*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) + (5*d^4*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)/(2*c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - (2^p*d ^5*(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^6*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
3.6.75.3.1 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])
\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{2}\right )\right )}^{p}d x\]
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\text {Timed out} \]
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \]
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^2\right )\right )}^p \,d x \]