Integrand size = 24, antiderivative size = 1036 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=-\frac {2^p 3^{-1-2 p} e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^9 \Gamma \left (1+p,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{9/2}}+\frac {3\ 4^{-p} d 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^4 e^9}-\frac {3\ 2^{2+p} 7^{-p} d^2 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^7 \Gamma \left (1+p,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{7/2}}+\frac {28\ 3^{-p} d^3 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^3 e^9}-\frac {21\ 2^{1+p} 5^{-p} d^4 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}+\frac {21\ 2^{1-p} d^5 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^2 e^9}-\frac {7\ 2^{2+p} 3^{-p} d^6 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac {12 d^7 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^9}-\frac {3\ 2^p d^8 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}} \]
[Out]
Time = 1.07 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=-\frac {2^p 3^{-2 p-1} e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^9 \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{9/2}}+\frac {3\ 4^{-p} d 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^4 e^9}-\frac {3\ 2^{p+2} 7^{-p} d^2 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^7 \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{7/2}}+\frac {28\ 3^{-p} d^3 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^3 e^9}-\frac {21\ 2^{p+1} 5^{-p} d^4 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}+\frac {21\ 2^{1-p} d^5 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^2 e^9}-\frac {7\ 2^{p+2} 3^{-p} d^6 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac {12 d^7 e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^9}-\frac {3\ 2^p d^8 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}} \]
[In]
[Out]
Rule 2212
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\left (3 \text {Subst}\left (\int \left (\frac {d^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {8 d^7 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {28 d^6 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {56 d^5 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {70 d^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {56 d^3 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {28 d^2 (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}-\frac {8 d (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}+\frac {(d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^8}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\frac {3 \text {Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}+\frac {(24 d) \text {Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}-\frac {\left (84 d^2\right ) \text {Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}+\frac {\left (168 d^3\right ) \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}-\frac {\left (210 d^4\right ) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}+\frac {\left (168 d^5\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}-\frac {\left (84 d^6\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}+\frac {\left (24 d^7\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8}-\frac {\left (3 d^8\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^8} \\ & = -\frac {3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}+\frac {(24 d) \text {Subst}\left (\int x^7 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}-\frac {\left (84 d^2\right ) \text {Subst}\left (\int x^6 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}+\frac {\left (168 d^3\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}-\frac {\left (210 d^4\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}+\frac {\left (168 d^5\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}-\frac {\left (84 d^6\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}+\frac {\left (24 d^7\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9}-\frac {\left (3 d^8\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^9} \\ & = \frac {(12 d) \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^4 e^9}+\frac {\left (84 d^3\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^3 e^9}+\frac {\left (84 d^5\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^2 e^9}+\frac {\left (12 d^7\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{c e^9}-\frac {\left (3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^9\right ) \text {Subst}\left (\int e^{9 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{9/2}}-\frac {\left (42 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^7\right ) \text {Subst}\left (\int e^{7 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{7/2}}-\frac {\left (105 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5\right ) \text {Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}-\frac {\left (42 d^6 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3\right ) \text {Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac {\left (3 d^8 \left (d+\frac {e}{\sqrt [3]{x}}\right )\right ) \text {Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 e^9 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}} \\ & = \text {Too large to display} \\ \end{align*}
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx \]
[In]
[Out]
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{2}\right )\right )}^{p}}{x^{4}}d x\]
[In]
[Out]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^2\right )\right )}^p}{x^4} \,d x \]
[In]
[Out]