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} \] Output:
-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)/(c*(d+e*x^(2/3))^2)^(1/2) /((-(a+b*ln(c*(d+e*x^(2/3))^2))/b)^p)+1/4*GAMMA(p+1,(-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)/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+1,1/2 *(-5*a-5*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-2*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+1,1/2*(-3*a-3*b*ln(c*(d+e*x^(2/ 3))^2))/b)*(a+b*ln(c*(d+e*x^(2/3))^2))^p/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)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx \] Input:
Integrate[x^3*(a + b*Log[c*(d + e*x^(2/3))^2])^p,x]
Output:
Integrate[x^3*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x]
Time = 2.20 (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 )\) |
Input:
Int[x^3*(a + b*Log[c*(d + e*x^(2/3))^2])^p,x]
Output:
(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
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])
\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{2}\right )\right )}^{p}d x\]
Input:
int(x^3*(a+b*ln(c*(d+e*x^(2/3))^2))^p,x)
Output:
int(x^3*(a+b*ln(c*(d+e*x^(2/3))^2))^p,x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*ln(c*(d+e*x**(2/3))**2))**p,x)
Output:
Timed out
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^2\right )\right )}^p \,d x \] Input:
int(x^3*(a + b*log(c*(d + e*x^(2/3))^2))^p,x)
Output:
int(x^3*(a + b*log(c*(d + e*x^(2/3))^2))^p, x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x^3*(a+b*log(c*(d+e*x^(2/3))^2))^p,x)
Output:
(180*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**5*e*p**2 + 180*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**5*e*p - 45*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**4*p**2*x**2 - 45*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**4*p*x**2 - 90*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**4*e**2*p**2*x - 90*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**4*e**2*p*x + 36*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**5*p**2*x**3 + 36*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**5*p* x**3 - 90*(log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*lo g(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b*d**6*p - 90*(log(2*x**( 2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*a*d**6*p + 90*(log(2*x* *(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*a*e**6*p*x**4 + 90*(l og(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*a*e**6*x**4 + 60*(log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*b*d**3*e* *3*p**2*x**2 + 60*(log(2*x**(2/3)*c*d*e + x**(1/3)*c*e**2*x + c*d**2)*b + a)**p*b*d**3*e**3*p*x**2 - 144*int(((log(2*x**(2/3)*c*d*e + x**(1/3)*c*e** 2*x + c*d**2)*b + a)**p*x**3)/(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 + x**(2/3)*log(2*x**(2/3)*c*d*e + x**(1/3)*c*e...