Optimal. Leaf size=273 \[ \frac {3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{2 c^3 e^3}-\frac {3 d 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )}{c^2 e^3}+\frac {3 d^2 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )}{2 c e^3} \]
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Rubi [A] time = 0.38, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, 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 2181
Rule 2299
Rule 2309
Rule 2389
Rule 2390
Rule 2401
Rule 2454
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*}
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Mathematica [A] time = 0.22, size = 181, normalized size = 0.66 \[ \frac {2^{-p-1} 3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )^{-p} \left (2^p \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )+c d 3^{p+1} e^{a/b} \left (c d 2^p e^{a/b} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )\right )}{b}\right )-\Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )}{b}\right )\right )\right )}{c^3 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c e x^{\frac {2}{3}} + c d\right ) + a\right )}^{p} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x \left (b \ln \left (\left (e \,x^{\frac {2}{3}}+d \right ) c \right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )} c\right ) + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\ln \left (c\,\left (d+e\,x^{2/3}\right )\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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