Optimal. Leaf size=678 \[ \frac {3^{-p} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{b}\right )}{4 c^3 e^6}+\frac {15 d^2 2^{-p-2} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{b}\right )}{c^2 e^6}-\frac {3 d^5 2^{p-1} e^{-\frac {a}{2 b}} \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,\frac {-a-b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+e x^{2/3}\right )^2}}+\frac {15 d^4 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )}{4 c e^6}-\frac {5 d^3 \left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}-\frac {3 d 2^{p-1} 5^{-p} e^{-\frac {5 a}{2 b}} \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{5/2}} \]
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Rubi [A] time = 0.97, antiderivative size = 675, normalized size of antiderivative = 1.00, 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-2} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\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+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{b}\right )}{4 c^3 e^6}-\frac {5 d^3 \left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}+\frac {15 d^4 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )}{4 c e^6}-\frac {3 d^5 2^{p-1} e^{-\frac {a}{2 b}} \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+e x^{2/3}\right )^2}}-\frac {3 d 2^{p-1} 5^{-p} e^{-\frac {5 a}{2 b}} \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2401
Rule 2454
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx &=\frac {3}{2} \operatorname {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,x^{2/3}\right )\\ &=\frac {3}{2} \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,x^{2/3}\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,x^{2/3}\right )}{2 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,x^{2/3}\right )}{2 e^5}+\frac {\left (15 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,x^{2/3}\right )}{e^5}-\frac {\left (15 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,x^{2/3}\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,x^{2/3}\right )}{2 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,x^{2/3}\right )}{2 e^5}\\ &=\frac {3 \operatorname {Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 e^6}+\frac {\left (15 d^2\right ) \operatorname {Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e x^{2/3}\right )}{e^6}-\frac {\left (15 d^3\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e x^{2/3}\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 e^6}\\ &=\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 )^2\right )\right )}{4 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+e x^{2/3}\right )^2\right )\right )}{2 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+e x^{2/3}\right )^2\right )\right )}{4 c e^6}-\frac {\left (15 d \left (d+e x^{2/3}\right )^5\right ) \operatorname {Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{4 e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{5/2}}-\frac {\left (15 d^3 \left (d+e x^{2/3}\right )^3\right ) \operatorname {Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{3/2}}-\frac {\left (3 d^5 \left (d+e x^{2/3}\right )\right ) \operatorname {Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{4 e^6 \sqrt {c \left (d+e x^{2/3}\right )^2}}\\ &=\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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{4 c^3 e^6}-\frac {3\ 2^{-1+p} 5^{-p} d e^{-\frac {5 a}{2 b}} \left (d+e x^{2/3}\right )^5 \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e x^{2/3}\right )^2\right )^{5/2}}+\frac {15\ 2^{-2-p} d^2 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{c^2 e^6}-\frac {5 \left (\frac {2}{3}\right )^p d^3 e^{-\frac {3 a}{2 b}} \left (d+e x^{2/3}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e x^{2/3}\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+e x^{2/3}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{4 c e^6}-\frac {3\ 2^{-1+p} d^5 e^{-\frac {a}{2 b}} \left (d+e x^{2/3}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )}{b}\right )^{-p}}{e^6 \sqrt {c \left (d+e x^{2/3}\right )^2}}\\ \end {align*}
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Mathematica [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c e^{2} x^{\frac {4}{3}} + 2 \, c d e x^{\frac {2}{3}} + c d^{2}\right ) + a\right )}^{p} x^{3}, x\right ) \]
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 \[ \int {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int x^{3} \left (b \ln \left (\left (e \,x^{\frac {2}{3}}+d \right )^{2} c \right )+a \right )^{p}\, dx \]
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 \[ \int {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3}\,{d x} \]
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 \[ \int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^2\right )\right )}^p \,d x \]
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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