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=\frac {3^{-1-2 p} e^{-\frac {9 a}{b}} \Gamma \left (1+p,-\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 {3\ 8^{-p} d e^{-\frac {8 a}{b}} \Gamma \left (1+p,-\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 {12\ 7^{-p} d^2 e^{-\frac {7 a}{b}} \Gamma \left (1+p,-\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} d^3 e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\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 {42\ 5^{-p} d^4 e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\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 {21\ 2^{1-2 p} d^5 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\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} d^6 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\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 {3\ 2^{2-p} d^7 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\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 {3 d^8 e^{-\frac {a}{b}} \Gamma \left (1+p,-\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} \]
3^(-1-2*p)*GAMMA(p+1,-9*(a+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+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+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+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+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+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*ln(c*(d+e*x^( 1/3))))/b)^p)+28*d^6*GAMMA(p+1,-3*(a+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+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*G AMMA(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/e xp(a/b)/(((-a-b*ln(c*(d+e*x^(1/3))))/b)^p)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx \]
Time = 1.69 (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 )\) |
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))
3.6.57.3.1 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])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )\right )\right )}^{p}d x\]
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\text {Timed out} \]
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \]
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{1/3}\right )\right )\right )}^p \,d x \]