Optimal. Leaf size=553 \[ \frac{15 d^2 2^{-2 p-1} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^4 e^6}-\frac{10 d^3 3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^3 e^6}+\frac{15 d^4 2^{-p-1} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^2 e^6}+\frac{2^{-p-1} 3^{-p} e^{-\frac{6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^6 e^6}-\frac{3 d 5^{-p} e^{-\frac{5 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^5 e^6}-\frac{3 d^5 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )}{c e^6} \]
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Rubi [A] time = 0.845662, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 21, 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} \[ \frac{15 d^2 2^{-2 p-1} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^4 e^6}-\frac{10 d^3 3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^3 e^6}+\frac{15 d^4 2^{-p-1} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^2 e^6}+\frac{2^{-p-1} 3^{-p} e^{-\frac{6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^6 e^6}-\frac{3 d 5^{-p} e^{-\frac{5 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )}{c^5 e^6}-\frac{3 d^5 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )}{c e^6} \]
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 \sqrt [3]{x}\right )\right )\right )^p \, dx &=3 \operatorname{Subst}\left (\int x^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{d^5 (a+b \log (c (d+e x)))^p}{e^5}+\frac{5 d^4 (d+e x) (a+b \log (c (d+e x)))^p}{e^5}-\frac{10 d^3 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^5}+\frac{10 d^2 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^5}-\frac{5 d (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^5}+\frac{(d+e x)^5 (a+b \log (c (d+e x)))^p}{e^5}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^5}-\frac{(15 d) \operatorname{Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^5}+\frac{\left (30 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^5}-\frac{\left (30 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^5}+\frac{\left (15 d^4\right ) \operatorname{Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^5}-\frac{\left (3 d^5\right ) \operatorname{Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt [3]{x}\right )}{e^5}\\ &=\frac{3 \operatorname{Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}-\frac{(15 d) \operatorname{Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}+\frac{\left (30 d^2\right ) \operatorname{Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}-\frac{\left (30 d^3\right ) \operatorname{Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}+\frac{\left (15 d^4\right ) \operatorname{Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}-\frac{\left (3 d^5\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}\\ &=\frac{3 \operatorname{Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^6 e^6}-\frac{(15 d) \operatorname{Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^5 e^6}+\frac{\left (30 d^2\right ) \operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^4 e^6}-\frac{\left (30 d^3\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^3 e^6}+\frac{\left (15 d^4\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c^2 e^6}-\frac{\left (3 d^5\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{c e^6}\\ &=\frac{2^{-1-p} 3^{-p} e^{-\frac{6 a}{b}} \Gamma \left (1+p,-\frac{6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^6 e^6}-\frac{3\ 5^{-p} d e^{-\frac{5 a}{b}} \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^5 e^6}+\frac{15\ 2^{-1-2 p} d^2 e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^4 e^6}-\frac{10\ 3^{-p} d^3 e^{-\frac{3 a}{b}} \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^6}+\frac{15\ 2^{-1-p} d^4 e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^2 e^6}-\frac{3 d^5 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^6}\\ \end{align*}
Mathematica [A] time = 0.910723, size = 325, normalized size = 0.59 \[ \frac{2^{-2 p-1} 15^{-p} e^{-\frac{6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p} \left (10^p \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )-c d e^{a/b} \left (2^{2 p+1} 3^{p+1} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+c d 5^p e^{a/b} \left (c d 2^p e^{a/b} \left (5\ 2^{p+2} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )+c d 3^{p+1} e^{a/b} \left (c d 2^{p+1} e^{a/b} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )-5 \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )\right )\right )-5\ 3^{p+1} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right )\right )\right )\right )}{c^6 e^6} \]
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\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{\left (b \log \left ({\left (e x^{\frac{1}{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{1}{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{1}{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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