Integrand size = 22, antiderivative size = 557 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\frac {2^{-2-p} 3^{-p} e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^6 e^6}-\frac {3\ 5^{-p} d e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c^5 e^6}+\frac {15\ 2^{-2 (1+p)} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^4 e^6}-\frac {5\ 3^{-p} d^3 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^3 e^6}+\frac {15\ 2^{-2-p} d^4 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{c^2 e^6}-\frac {3 d^5 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c e^6} \] Output:
2^(-2-p)*GAMMA(p+1,(-6*a-6*b*ln(c*(d+e*x^(2/3))))/b)*(a+b*ln(c*(d+e*x^(2/3 ))))^p/(3^p)/c^6/e^6/exp(6*a/b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)-3/2*d*G AMMA(p+1,(-5*a-5*b*ln(c*(d+e*x^(2/3))))/b)*(a+b*ln(c*(d+e*x^(2/3))))^p/(5^ p)/c^5/e^6/exp(5*a/b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)+15*d^2*GAMMA(p+1, (-4*a-4*b*ln(c*(d+e*x^(2/3))))/b)*(a+b*ln(c*(d+e*x^(2/3))))^p/(2^(2*p+2))/ c^4/e^6/exp(4*a/b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)-5*d^3*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^6/ex p(3*a/b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)+15*2^(-2-p)*d^4*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^6/exp(2*a/ b)/((-(a+b*ln(c*(d+e*x^(2/3))))/b)^p)-3/2*d^5*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^6/exp(a/b)/((-(a+b*ln(c*(d+e*x ^(2/3))))/b)^p)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx \] Input:
Integrate[x^3*(a + b*Log[c*(d + e*x^(2/3))])^p,x]
Output:
Integrate[x^3*(a + b*Log[c*(d + e*x^(2/3))])^p, x]
Time = 2.13 (sec) , antiderivative size = 554, 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.
| \(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {3}{2} \int x^{10/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^pdx^{2/3}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \frac {3}{2} \int \left (\frac {\left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^5}-\frac {5 d \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p}{e^5}\right )dx^{2/3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} \left (\frac {6^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^6 e^6}-\frac {d 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^5 e^6}+\frac {5 d^2 2^{-2 p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^4 e^6}-\frac {10 d^3 3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^3 e^6}+\frac {5 d^4 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )}{c e^6}\right )\) |
Input:
Int[x^3*(a + b*Log[c*(d + e*x^(2/3))])^p,x]
Output:
(3*((6^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b *Log[c*(d + e*x^(2/3))])^p)/(c^6*e^6*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^ (2/3))])/b))^p) - (d*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(2/3))]))/b]*( a + b*Log[c*(d + e*x^(2/3))])^p)/(5^p*c^5*e^6*E^((5*a)/b)*(-((a + b*Log[c* (d + e*x^(2/3))])/b))^p) + (5*2^(-1 - 2*p)*d^2*Gamma[1 + p, (-4*(a + b*Log [c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(c^4*e^6*E^((4* a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) - (10*3^(-1 - p)*d^3*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^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) + (5*2 ^(-1 - p)*d^4*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*L og[c*(d + e*x^(2/3))])^p)/(c^2*e^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(2 /3))])/b))^p) - (d^5*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^6*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])
\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )\right )\right )}^{p}d x\]
Input:
int(x^3*(a+b*ln(c*(d+e*x^(2/3))))^p,x)
Output:
int(x^3*(a+b*ln(c*(d+e*x^(2/3))))^p,x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*ln(c*(d+e*x**(2/3))))**p,x)
Output:
Timed out
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{2/3}\right )\right )\right )}^p \,d x \] Input:
int(x^3*(a + b*log(c*(d + e*x^(2/3))))^p,x)
Output:
int(x^3*(a + b*log(c*(d + e*x^(2/3))))^p, x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x^3*(a+b*log(c*(d+e*x^(2/3))))^p,x)
Output:
(180*x**(2/3)*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**5*e*p**2 + 180*x**(2 /3)*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**5*e*p - 45*x**(2/3)*(log(x**(2 /3)*c*e + c*d)*b + a)**p*b*d**2*e**4*p**2*x**2 - 45*x**(2/3)*(log(x**(2/3) *c*e + c*d)*b + a)**p*b*d**2*e**4*p*x**2 - 90*x**(1/3)*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**4*e**2*p**2*x - 90*x**(1/3)*(log(x**(2/3)*c*e + c*d)* b + a)**p*b*d**4*e**2*p*x + 36*x**(1/3)*(log(x**(2/3)*c*e + c*d)*b + a)**p *b*d*e**5*p**2*x**3 + 36*x**(1/3)*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d*e **5*p*x**3 - 180*(log(x**(2/3)*c*e + c*d)*b + a)**p*log(x**(2/3)*c*e + c*d )*b*d**6*p - 180*(log(x**(2/3)*c*e + c*d)*b + a)**p*a*d**6*p + 180*(log(x* *(2/3)*c*e + c*d)*b + a)**p*a*e**6*p*x**4 + 180*(log(x**(2/3)*c*e + c*d)*b + a)**p*a*e**6*x**4 + 60*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**3*e**3*p **2*x**2 + 60*(log(x**(2/3)*c*e + c*d)*b + a)**p*b*d**3*e**3*p*x**2 - 144* int(((log(x**(2/3)*c*e + c*d)*b + a)**p*x**3)/(6*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 + 6*x**(2/3)*a** 2*e + x**(2/3)*a*b*e*p + 6*log(x**(2/3)*c*e + c*d)*a*b*d + log(x**(2/3)*c* e + c*d)*b**2*d*p + 6*a**2*d + a*b*d*p),x)*a*b**2*d*e**6*p**3 - 144*int((( log(x**(2/3)*c*e + c*d)*b + a)**p*x**3)/(6*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 + 6*x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 6*log(x**(2/3)*c*e + c*d)*a*b*d + log(x**(2/3)*c*e + c* d)*b**2*d*p + 6*a**2*d + a*b*d*p),x)*a*b**2*d*e**6*p**2 - 24*int(((log(...