Optimal. Leaf size=339 \[ -\frac{3 d^2 2^p e^{-\frac{a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right )}{e^3 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}-\frac{\left (\frac{2}{3}\right )^p e^{-\frac{3 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^3 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac{3 d e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )}{c e^3} \]
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Rubi [A] time = 0.478085, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2454, 2401, 2389, 2300, 2181, 2390, 2310} \[ -\frac{3 d^2 2^p e^{-\frac{a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right )}{e^3 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}-\frac{\left (\frac{2}{3}\right )^p e^{-\frac{3 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^3 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac{3 d e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )}{c e^3} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2401
Rule 2389
Rule 2300
Rule 2181
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx &=-\left (3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{d^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^2}-\frac{2 d (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^2}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^2}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac{3 \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^2}+\frac{(6 d) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^2}-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^2}\\ &=-\frac{3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}+\frac{(6 d) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}\\ &=\frac{(3 d) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{c e^3}-\frac{\left (3 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3\right ) \operatorname{Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 e^3 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac{\left (3 d^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 e^3 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}\\ &=-\frac{\left (\frac{2}{3}\right )^p e^{-\frac{3 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^3 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac{3 d e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^3}-\frac{3\ 2^p d^2 e^{-\frac{a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^3 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}\\ \end{align*}
Mathematica [F] time = 0.139524, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.335, 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 ) ^{2} \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 )}^{2}\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^{2} x + 2 \, c d e x^{\frac{2}{3}} + c e^{2} x^{\frac{1}{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 )}^{2}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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