Integrand size = 22, antiderivative size = 831 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx =\text {Too large to display} \] Output:
3^(-1-2*p)*GAMMA(p+1,(-9*a-9*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1 /3))))^p/c^9/e^9/exp(9*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-3*d*GAMMA(p +1,(-8*a-8*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(8^p)/c^8 /e^9/exp(8*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+12*d^2*GAMMA(p+1,(-7*a- 7*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(7^p)/c^7/e^9/exp( 7*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-7*2^(2-p)*d^3*GAMMA(p+1,(-6*a-6* b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(3^p)/c^6/e^9/exp(6* a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+42*d^4*GAMMA(p+1,(-5*a-5*b*ln(c*(d +e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(5^p)/c^5/e^9/exp(5*a/b)/((-( a+b*ln(c*(d+e*x^(1/3))))/b)^p)-21*2^(1-2*p)*d^5*GAMMA(p+1,(-4*a-4*b*ln(c*( d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/c^4/e^9/exp(4*a/b)/((-(a+b*l n(c*(d+e*x^(1/3))))/b)^p)+28*d^6*GAMMA(p+1,(-3*a-3*b*ln(c*(d+e*x^(1/3))))/ b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(3^p)/c^3/e^9/exp(3*a/b)/((-(a+b*ln(c*(d+e* x^(1/3))))/b)^p)-3*2^(2-p)*d^7*GAMMA(p+1,(-2*a-2*b*ln(c*(d+e*x^(1/3))))/b) *(a+b*ln(c*(d+e*x^(1/3))))^p/c^2/e^9/exp(2*a/b)/((-(a+b*ln(c*(d+e*x^(1/3)) ))/b)^p)+3*d^8*GAMMA(p+1,-(a+b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1 /3))))^p/c/e^9/exp(a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx \] Input:
Integrate[x^2*(a + b*Log[c*(d + e*x^(1/3))])^p,x]
Output:
Integrate[x^2*(a + b*Log[c*(d + e*x^(1/3))])^p, x]
Time = 3.03 (sec) , antiderivative size = 836, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 3 \int x^{8/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^pd\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 3 \int \left (\frac {\left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}-\frac {8 d \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}+\frac {28 d^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}-\frac {56 d^3 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}+\frac {70 d^4 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}-\frac {56 d^5 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}+\frac {28 d^6 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}-\frac {8 d^7 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}+\frac {d^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^8}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {9^{-p-1} e^{-\frac {9 a}{b}} \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^9 e^9}-\frac {8^{-p} d e^{-\frac {8 a}{b}} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^8 e^9}+\frac {4\ 7^{-p} d^2 e^{-\frac {7 a}{b}} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^7 e^9}-\frac {7\ 2^{2-p} 3^{-p-1} d^3 e^{-\frac {6 a}{b}} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^6 e^9}+\frac {14\ 5^{-p} d^4 e^{-\frac {5 a}{b}} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^5 e^9}-\frac {7\ 2^{1-2 p} d^5 e^{-\frac {4 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^4 e^9}+\frac {28\ 3^{-p-1} d^6 e^{-\frac {3 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^9}-\frac {2^{2-p} d^7 e^{-\frac {2 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^2 e^9}+\frac {d^8 e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^9}\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e*x^(1/3))])^p,x]
Output:
3*((9^(-1 - p)*Gamma[1 + p, (-9*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b* Log[c*(d + e*x^(1/3))])^p)/(c^9*e^9*E^((9*a)/b)*(-((a + b*Log[c*(d + e*x^( 1/3))])/b))^p) - (d*Gamma[1 + p, (-8*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(8^p*c^8*e^9*E^((8*a)/b)*(-((a + b*Log[c*( d + e*x^(1/3))])/b))^p) + (4*d^2*Gamma[1 + p, (-7*(a + b*Log[c*(d + e*x^(1 /3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(7^p*c^7*e^9*E^((7*a)/b)*(-(( a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (7*2^(2 - p)*3^(-1 - p)*d^3*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))] )^p)/(c^6*e^9*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) + (14*d ^4*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e *x^(1/3))])^p)/(5^p*c^5*e^9*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/ b))^p) - (7*2^(1 - 2*p)*d^5*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*x^(1/3))] ))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(c^4*e^9*E^((4*a)/b)*(-((a + b*Log [c*(d + e*x^(1/3))])/b))^p) + (28*3^(-1 - p)*d^6*Gamma[1 + p, (-3*(a + b*L og[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(c^3*e^9*E^(( 3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (2^(2 - p)*d^7*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))]) ^p)/(c^2*e^9*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) + (d^8*G amma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))])/b)]*(a + b*Log[c*(d + e*x^(1/ 3))])^p)/(c*e^9*E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p))
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])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )\right )\right )}^{p}d x\]
Input:
int(x^2*(a+b*ln(c*(d+e*x^(1/3))))^p,x)
Output:
int(x^2*(a+b*ln(c*(d+e*x^(1/3))))^p,x)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(1/3))))^p,x, algorithm="fricas")
Output:
integral((b*log(c*e*x^(1/3) + c*d) + a)^p*x^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**2*(a+b*ln(c*(d+e*x**(1/3))))**p,x)
Output:
Timed out
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(1/3))))^p,x, algorithm="maxima")
Output:
integrate((b*log((e*x^(1/3) + d)*c) + a)^p*x^2, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(1/3))))^p,x, algorithm="giac")
Output:
integrate((b*log((e*x^(1/3) + d)*c) + a)^p*x^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{1/3}\right )\right )\right )}^p \,d x \] Input:
int(x^2*(a + b*log(c*(d + e*x^(1/3))))^p,x)
Output:
int(x^2*(a + b*log(c*(d + e*x^(1/3))))^p, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x^2*(a+b*log(c*(d+e*x^(1/3))))^p,x)
Output:
(1260*x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**7*e**2*p**2 + 1260* x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**7*e**2*p - 504*x**(2/3)*( log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**4*e**5*p**2*x - 504*x**(2/3)*(log(x **(1/3)*c*e + c*d)*b + a)**p*b*d**4*e**5*p*x + 315*x**(2/3)*(log(x**(1/3)* c*e + c*d)*b + a)**p*b*d*e**8*p**2*x**2 + 315*x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d*e**8*p*x**2 - 2520*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**8*e*p**2 - 2520*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p* b*d**8*e*p + 630*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**5*e**4*p **2*x + 630*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**5*e**4*p*x - 360*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**2*e**7*p**2*x**2 - 36 0*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**2*e**7*p*x**2 + 2520*(l og(x**(1/3)*c*e + c*d)*b + a)**p*log(x**(1/3)*c*e + c*d)*b*d**9*p + 2520*( log(x**(1/3)*c*e + c*d)*b + a)**p*a*d**9*p + 2520*(log(x**(1/3)*c*e + c*d) *b + a)**p*a*e**9*p*x**3 + 2520*(log(x**(1/3)*c*e + c*d)*b + a)**p*a*e**9* x**3 - 840*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**6*e**3*p**2*x - 840*(lo g(x**(1/3)*c*e + c*d)*b + a)**p*b*d**6*e**3*p*x + 420*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**3*e**6*p**2*x**2 + 420*(log(x**(1/3)*c*e + c*d)*b + a) **p*b*d**3*e**6*p*x**2 + 7560*int((log(x**(1/3)*c*e + c*d)*b + a)**p/(9*x* *(2/3)*log(x**(1/3)*c*e + c*d)*a*b*e + x**(2/3)*log(x**(1/3)*c*e + c*d)*b* *2*e*p + 9*x**(2/3)*a**2*e + x**(2/3)*a*b*e*p + 9*x**(1/3)*log(x**(1/3)...