Optimal. Leaf size=273 \[ \frac{3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{2 c^3 e^3}-\frac{3 d 2^{-p-1} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^2 e^3}+\frac{3 d^2 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )}{2 c e^3} \]
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Rubi [A] time = 0.38078, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2454, 2401, 2389, 2299, 2181, 2390, 2309} \[ \frac{3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{2 c^3 e^3}-\frac{3 d 2^{-p-1} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^2 e^3}+\frac{3 d^2 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )}{2 c e^3} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2401
Rule 2389
Rule 2299
Rule 2181
Rule 2390
Rule 2309
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int x^2 (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )\\ &=\frac{3}{2} \operatorname{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,x^{2/3}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )}{2 e^2}-\frac{(3 d) \operatorname{Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )}{e^2}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,x^{2/3}\right )}{2 e^2}\\ &=\frac{3 \operatorname{Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e x^{2/3}\right )}{2 e^3}-\frac{(3 d) \operatorname{Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e x^{2/3}\right )}{e^3}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e x^{2/3}\right )}{2 e^3}\\ &=\frac{3 \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )\right )\right )}{2 c^3 e^3}-\frac{(3 d) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )\right )\right )}{c^2 e^3}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )\right )\right )}{2 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 x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c^3 e^3}-\frac{3\ 2^{-1-p} d e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\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 x^{2/3}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p}}{2 c e^3}\\ \end{align*}
Mathematica [A] time = 0.236596, size = 181, normalized size = 0.66 \[ \frac{2^{-p-1} 3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \left (2^p \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )+c d 3^{p+1} e^{a/b} \left (c d 2^p e^{a/b} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )-\text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )\right )\right )}{c^3 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )} c\right ) + a\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c e x^{\frac{2}{3}} + c d\right ) + a\right )}^{p} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )} c\right ) + a\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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