Optimal. Leaf size=267 \[ -\frac{3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^3 e^3}+\frac{3 d 2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^2 e^3}-\frac{3 d^2 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )}{c e^3} \]
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Rubi [A] time = 0.392388, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2454, 2401, 2389, 2299, 2181, 2390, 2309} \[ -\frac{3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^3 e^3}+\frac{3 d 2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^2 e^3}-\frac{3 d^2 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )}{c e^3} \]
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 \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^2} \, dx &=-\left (3 \operatorname{Subst}\left (\int x^2 (a+b \log (c (d+e x)))^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \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,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac{3 \operatorname{Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^2}+\frac{(6 d) \operatorname{Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^2}-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^2}\\ &=-\frac{3 \operatorname{Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}+\frac{(6 d) \operatorname{Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}\\ &=-\frac{3 \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{c^3 e^3}+\frac{(6 d) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\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+\frac{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+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{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+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{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+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c e^3}\\ \end{align*}
Mathematica [A] time = 0.23436, size = 175, normalized size = 0.66 \[ -\frac{6^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \left (2^p \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+c d 3^{p+1} e^{a/b} \left (c d 2^p e^{a/b} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )}{b}\right )-\text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )\right )\right )}{c^3 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \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 \frac{{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{2}}\,{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 (\frac{{\left (b \log \left (\frac{c d x + c e x^{\frac{2}{3}}}{x}\right ) + a\right )}^{p}}{x^{2}}, 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 \frac{{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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