Optimal. Leaf size=552 \[ -\frac {2^{-p} 3^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^6 e^6}+\frac {2 d 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^5 e^6}-\frac {5 d^2 4^{-p} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^4 e^6}+\frac {20 d^3 3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^3 e^6}-\frac {5 d^4 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^2 e^6}+\frac {2 d^5 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )}{c e^6} \]
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Rubi [A] time = 0.85, antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps used = 21, 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 {5 d^2 4^{-p} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^4 e^6}+\frac {20 d^3 3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^3 e^6}-\frac {5 d^4 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^2 e^6}-\frac {2^{-p} 3^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^6 e^6}+\frac {2 d 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )}{c^5 e^6}+\frac {2 d^5 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )}{c e^6} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2299
Rule 2309
Rule 2389
Rule 2390
Rule 2401
Rule 2454
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x^4} \, dx &=-\left (2 \operatorname {Subst}\left (\int x^5 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\left (2 \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,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}+\frac {(10 d) \operatorname {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}-\frac {\left (20 d^2\right ) \operatorname {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}+\frac {\left (20 d^3\right ) \operatorname {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}-\frac {\left (10 d^4\right ) \operatorname {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}+\frac {\left (2 d^5\right ) \operatorname {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^5}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {(10 d) \operatorname {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {\left (20 d^2\right ) \operatorname {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {\left (20 d^3\right ) \operatorname {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}-\frac {\left (10 d^4\right ) \operatorname {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}+\frac {\left (2 d^5\right ) \operatorname {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^6}\\ &=-\frac {2 \operatorname {Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^6 e^6}+\frac {(10 d) \operatorname {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^5 e^6}-\frac {\left (20 d^2\right ) \operatorname {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^4 e^6}+\frac {\left (20 d^3\right ) \operatorname {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^3 e^6}-\frac {\left (10 d^4\right ) \operatorname {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c^2 e^6}+\frac {\left (2 d^5\right ) \operatorname {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{c e^6}\\ &=-\frac {2^{-p} 3^{-1-p} e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^6 e^6}+\frac {2\ 5^{-p} d e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^5 e^6}-\frac {5\ 4^{-p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^4 e^6}+\frac {20\ 3^{-1-p} d^3 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^3 e^6}-\frac {5\ 2^{-p} d^4 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c^2 e^6}+\frac {2 d^5 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p}}{c e^6}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 325, normalized size = 0.59 \[ \frac {3^{-p-1} 20^{-p} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )^{-p} \left (c d e^{a/b} \left (2^{2 p+1} 3^{p+1} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d 5^p e^{a/b} \left (c d 2^p e^{a/b} \left (5\ 2^{p+2} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )+c d 3^{p+1} e^{a/b} \left (c d 2^{p+1} e^{a/b} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )}{b}\right )-5 \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )\right )-5\ 3^{p+1} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )\right )-10^p \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{b}\right )\right )}{c^6 e^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (\frac {c d x + c e \sqrt {x}}{x}\right ) + a\right )}^{p}}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (d +\frac {e}{\sqrt {x}}\right ) c \right )+a \right )^{p}}{x^{4}}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,\left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}^p}{x^4} \,d x \]
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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