Integrand size = 22, antiderivative size = 1121 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} 4^{-1-p} e^{-\frac {12 a}{b}} \Gamma \left (1+p,-\frac {12 \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^{12} e^{12}}-\frac {3\ 11^{-p} d e^{-\frac {11 a}{b}} \Gamma \left (1+p,-\frac {11 \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^{11} e^{12}}+\frac {33\ 2^{-1-p} 5^{-p} d^2 e^{-\frac {10 a}{b}} \Gamma \left (1+p,-\frac {10 \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^{10} e^{12}}-\frac {55\ 9^{-p} d^3 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^{12}}+\frac {495\ 2^{-2-3 p} d^4 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^{12}}-\frac {198\ 7^{-p} d^5 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^{12}}+\frac {77\ 2^{-p} 3^{1-p} d^6 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^{12}}-\frac {198\ 5^{-p} d^7 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^{12}}+\frac {495\ 4^{-1-p} d^8 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^{12}}-\frac {55\ 3^{-p} d^9 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^{12}}+\frac {33\ 2^{-1-p} d^{10} 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^{12}}-\frac {3 d^{11} 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^{12}} \]
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Time = 1.26 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2504, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} 4^{-p-1} e^{-\frac {12 a}{b}} \Gamma \left (p+1,-\frac {12 \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^{12} e^{12}}-\frac {3\ 11^{-p} d e^{-\frac {11 a}{b}} \Gamma \left (p+1,-\frac {11 \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^{11} e^{12}}+\frac {33\ 2^{-p-1} 5^{-p} d^2 e^{-\frac {10 a}{b}} \Gamma \left (p+1,-\frac {10 \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^{10} e^{12}}-\frac {55\ 9^{-p} d^3 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^{12}}+\frac {495\ 2^{-3 p-2} d^4 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^{12}}-\frac {198\ 7^{-p} d^5 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^{12}}+\frac {77\ 2^{-p} 3^{1-p} d^6 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^{12}}-\frac {198\ 5^{-p} d^7 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^{12}}+\frac {495\ 4^{-p-1} d^8 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^{12}}-\frac {55\ 3^{-p} d^9 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^{12}}+\frac {33\ 2^{-p-1} d^{10} 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^{12}}-\frac {3 d^{11} 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^{12}} \]
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Rule 2212
Rule 2336
Rule 2346
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^{11} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (-\frac {d^{11} (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {11 d^{10} (d+e x) (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {55 d^9 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {165 d^8 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {330 d^7 (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {462 d^6 (d+e x)^5 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {462 d^5 (d+e x)^6 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {330 d^4 (d+e x)^7 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {165 d^3 (d+e x)^8 (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {55 d^2 (d+e x)^9 (a+b \log (c (d+e x)))^p}{e^{11}}-\frac {11 d (d+e x)^{10} (a+b \log (c (d+e x)))^p}{e^{11}}+\frac {(d+e x)^{11} (a+b \log (c (d+e x)))^p}{e^{11}}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^{11} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {(33 d) \text {Subst}\left (\int (d+e x)^{10} (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int (d+e x)^9 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int (d+e x)^8 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int (d+e x)^7 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int (d+e x)^6 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^{11}} \\ & = \frac {3 \text {Subst}\left (\int x^{11} (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {(33 d) \text {Subst}\left (\int x^{10} (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int x^9 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int x^8 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int x^7 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int x^6 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}} \\ & = \frac {3 \text {Subst}\left (\int e^{12 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^{12} e^{12}}-\frac {(33 d) \text {Subst}\left (\int e^{11 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^{11} e^{12}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int e^{10 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^{10} e^{12}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int e^{9 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^9 e^{12}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int e^{8 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^8 e^{12}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int e^{7 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^7 e^{12}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^6 e^{12}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^5 e^{12}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^4 e^{12}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^3 e^{12}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^2 e^{12}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c e^{12}} \\ & = \text {Too large to display} \\ \end{align*}
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx \]
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\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )\right )\right )}^{p}d x\]
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\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\text {Timed out} \]
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\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{1/3}\right )\right )\right )}^p \,d x \]
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