Integrand size = 22, antiderivative size = 832 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, 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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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*l n(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* 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)
Time = 0.92 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx=-\frac {3^{-1-2 p} 280^{-p} e^{-\frac {9 a}{b}} \left (280^p \Gamma \left (1+p,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-9^{1+p} 35^p c d e^{a/b} \Gamma \left (1+p,-\frac {8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^{2+3 p} 5^p 9^{1+p} c^2 d^2 e^{\frac {2 a}{b}} \Gamma \left (1+p,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-5^p 84^{1+p} c^3 d^3 e^{\frac {3 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^{1+3 p} 63^{1+p} c^4 d^4 e^{\frac {4 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-5^p 126^{1+p} c^5 d^5 e^{\frac {5 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^{2+3 p} 5^p 21^{1+p} c^6 d^6 e^{\frac {6 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )-35^p 36^{1+p} c^7 d^7 e^{\frac {7 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+9^{1+p} 280^p c^8 d^8 e^{\frac {8 a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^9 e^9} \]
-((3^(-1 - 2*p)*(280^p*Gamma[1 + p, (-9*(a + b*Log[c*(d + e/x^(1/3))]))/b] - 9^(1 + p)*35^p*c*d*E^(a/b)*Gamma[1 + p, (-8*(a + b*Log[c*(d + e/x^(1/3) )]))/b] + 2^(2 + 3*p)*5^p*9^(1 + p)*c^2*d^2*E^((2*a)/b)*Gamma[1 + p, (-7*( a + b*Log[c*(d + e/x^(1/3))]))/b] - 5^p*84^(1 + p)*c^3*d^3*E^((3*a)/b)*Gam ma[1 + p, (-6*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 2^(1 + 3*p)*63^(1 + p)* c^4*d^4*E^((4*a)/b)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))]))/b] - 5^p*126^(1 + p)*c^5*d^5*E^((5*a)/b)*Gamma[1 + p, (-4*(a + b*Log[c*(d + e/x ^(1/3))]))/b] + 2^(2 + 3*p)*5^p*21^(1 + p)*c^6*d^6*E^((6*a)/b)*Gamma[1 + p , (-3*(a + b*Log[c*(d + e/x^(1/3))]))/b] - 35^p*36^(1 + p)*c^7*d^7*E^((7*a )/b)*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 9^(1 + p)*280^p *c^8*d^8*E^((8*a)/b)*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)])*(a + b*Log[c*(d + e/x^(1/3))])^p)/(280^p*c^9*e^9*E^((9*a)/b)*(-((a + b*Log[c *(d + e/x^(1/3))])/b))^p))
Time = 1.64 (sec) , antiderivative size = 836, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^{8/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -3 \int \left (\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}-\frac {8 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}+\frac {28 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}-\frac {56 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}+\frac {70 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}-\frac {56 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}+\frac {28 d^6 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}-\frac {8 d^7 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}+\frac {d^8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^8}\right )d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {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*Lo g[c*(d + e/x^(1/3))])/b))^p) + (28*3^(-1 - p)*d^6*Gamma[1 + p, (-3*(a + b* Log[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* Gamma[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))
3.6.87.3.1 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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )\right )\right )}^{p}}{x^{4}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (d+\frac {e}{x^{1/3}}\right )\right )\right )}^p}{x^4} \,d x \]