Integrand size = 18, antiderivative size = 266 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^3}-\frac {3\ 2^{-p} d e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^3} \]
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Time = 0.27 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2501, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^3 e^3}-\frac {3 d 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )}{c e^3} \]
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Rule 2212
Rule 2336
Rule 2346
Rule 2436
Rule 2437
Rule 2448
Rule 2501
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {d^2 (a+b \log (c (d+e x)))^p}{e^2}-\frac {2 d (d+e x) (a+b \log (c (d+e x)))^p}{e^2}+\frac {(d+e x)^2 (a+b \log (c (d+e x)))^p}{e^2}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^2}-\frac {(6 d) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^2} \\ & = \frac {3 \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}-\frac {(6 d) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^3} \\ & = \frac {3 \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^3}-\frac {(6 d) \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^3}+\frac {\left (3 d^2\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^3} \\ & = \frac {3^{-p} 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^3}-\frac {3\ 2^{-p} d 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^3}+\frac {3 d^2 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^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.65 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\frac {6^{-p} e^{-\frac {3 a}{b}} \left (2^p \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+3^{1+p} c d e^{a/b} \left (-\Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^3} \]
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\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )\right )\right )}^{p}d x\]
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\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} \,d x } \]
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\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right ) \right )}\right )^{p}\, dx \]
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\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} \,d x } \]
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\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )} c\right ) + a\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx=\int {\left (a+b\,\ln \left (c\,\left (d+e\,x^{1/3}\right )\right )\right )}^p \,d x \]
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