Optimal. Leaf size=673 \[ \text{result too large to display} \]
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Rubi [A] time = 0.972384, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 21, 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{15 d^2 2^{-p-1} e^{-\frac{2 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{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right )}{c^2 e^6}-\frac{3^{-p} e^{-\frac{3 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{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right )}{2 c^3 e^6}+\frac{5 d^3 2^{p+1} 3^{-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^6 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac{15 d^4 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 )}{2 c e^6}+\frac{3 d^5 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^6 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}+\frac{3 d \left (\frac{2}{5}\right )^p e^{-\frac{5 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^5 \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{5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{5/2}} \]
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^3} \, dx &=-\left (3 \operatorname{Subst}\left (\int x^5 \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^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac{5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac{10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac{10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac{5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac{(d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac{3 \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^5}+\frac{(15 d) \operatorname{Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^5}-\frac{\left (30 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt [3]{x}}\right )}{e^5}+\frac{\left (30 d^3\right ) \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^5}-\frac{\left (15 d^4\right ) \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^5}+\frac{\left (3 d^5\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^5}\\ &=-\frac{3 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^6}+\frac{(15 d) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^6}-\frac{\left (30 d^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{e^6}+\frac{\left (30 d^3\right ) \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^6}-\frac{\left (15 d^4\right ) \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^6}+\frac{\left (3 d^5\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^6}\\ &=-\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 )^2\right )\right )}{2 c^3 e^6}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{c^2 e^6}-\frac{\left (15 d^4\right ) \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 )}{2 c e^6}+\frac{\left (15 d \left (d+\frac{e}{\sqrt [3]{x}}\right )^5\right ) \operatorname{Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 e^6 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}+\frac{\left (15 d^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 )}{e^6 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac{\left (3 d^5 \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^6 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}\\ &=-\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 )^2\right )\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}}{2 c^3 e^6}+\frac{3 \left (\frac{2}{5}\right )^p d e^{-\frac{5 a}{2 b}} \left (d+\frac{e}{\sqrt [3]{x}}\right )^5 \Gamma \left (1+p,-\frac{5 \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^6 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}-\frac{15\ 2^{-1-p} d^2 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 )^2\right )\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^2 e^6}+\frac{5\ 2^{1+p} 3^{-p} d^3 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^6 \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac{15 d^4 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}}{2 c e^6}+\frac{3\ 2^p d^5 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^6 \sqrt{c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}}\\ \end{align*}
Mathematica [F] time = 0.134207, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^{3}}\,{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^{3}}, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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