Optimal. Leaf size=1141 \[ \text{result too large to display} \]
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Rubi [A] time = 1.73843, antiderivative size = 1141, normalized size of antiderivative = 1., number of steps used = 33, 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} \[ \text{result too large to display} \]
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{x}}\right )^2\right )\right )^p}{x^6} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (-\frac{d^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac{9 d^8 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac{36 d^7 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac{84 d^6 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac{126 d^5 (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac{126 d^4 (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac{84 d^3 (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac{36 d^2 (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}-\frac{9 d (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}+\frac{(d+e x)^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^9}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int (d+e x)^9 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^9}+\frac{(18 d) \operatorname{Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^9}-\frac{\left (72 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^9}+\frac{\left (168 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^9}-\frac{\left (252 d^4\right ) \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{x}}\right )}{e^9}+\frac{\left (252 d^5\right ) \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{x}}\right )}{e^9}-\frac{\left (168 d^6\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{x}}\right )}{e^9}+\frac{\left (72 d^7\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{x}}\right )}{e^9}-\frac{\left (18 d^8\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{x}}\right )}{e^9}+\frac{\left (2 d^9\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^9}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^9 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}+\frac{(18 d) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}-\frac{\left (72 d^2\right ) \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}+\frac{\left (168 d^3\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}-\frac{\left (252 d^4\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}+\frac{\left (252 d^5\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}-\frac{\left (168 d^6\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}+\frac{\left (72 d^7\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}-\frac{\left (18 d^8\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}+\frac{\left (2 d^9\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^{10}}\\ &=-\frac{\operatorname{Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{c^5 e^{10}}-\frac{\left (36 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 )^2\right )\right )}{c^4 e^{10}}-\frac{\left (126 d^4\right ) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{c^3 e^{10}}-\frac{\left (84 d^6\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{c^2 e^{10}}-\frac{\left (9 d^8\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{c e^{10}}+\frac{\left (9 d \left (d+\frac{e}{\sqrt{x}}\right )^9\right ) \operatorname{Subst}\left (\int e^{9 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{9/2}}+\frac{\left (84 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^7\right ) \operatorname{Subst}\left (\int e^{7 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{7/2}}+\frac{\left (126 d^5 \left (d+\frac{e}{\sqrt{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{x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{5/2}}+\frac{\left (36 d^7 \left (d+\frac{e}{\sqrt{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{x}}\right )^2\right )\right )}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{3/2}}+\frac{\left (d^9 \left (d+\frac{e}{\sqrt{x}}\right )\right ) \operatorname{Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{e^{10} \sqrt{c \left (d+\frac{e}{\sqrt{x}}\right )^2}}\\ &=-\frac{5^{-1-p} e^{-\frac{5 a}{b}} \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{c^5 e^{10}}+\frac{2^{1+p} 9^{-p} d e^{-\frac{9 a}{2 b}} \left (d+\frac{e}{\sqrt{x}}\right )^9 \Gamma \left (1+p,-\frac{9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{9/2}}-\frac{9\ 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{c^4 e^{10}}+\frac{3\ 2^{3+p} 7^{-p} d^3 e^{-\frac{7 a}{2 b}} \left (d+\frac{e}{\sqrt{x}}\right )^7 \Gamma \left (1+p,-\frac{7 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{7/2}}-\frac{14\ 3^{1-p} d^4 e^{-\frac{3 a}{b}} \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{c^3 e^{10}}+\frac{63\ 2^{2+p} 5^{-1-p} d^5 e^{-\frac{5 a}{2 b}} \left (d+\frac{e}{\sqrt{x}}\right )^5 \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{5/2}}-\frac{21\ 2^{1-p} d^6 e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{c^2 e^{10}}+\frac{2^{3+p} 3^{1-p} d^7 e^{-\frac{3 a}{2 b}} \left (d+\frac{e}{\sqrt{x}}\right )^3 \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )^{3/2}}-\frac{9 d^8 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{c e^{10}}+\frac{2^{1+p} d^9 e^{-\frac{a}{2 b}} \left (d+\frac{e}{\sqrt{x}}\right ) \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )}{b}\right )^{-p}}{e^{10} \sqrt{c \left (d+\frac{e}{\sqrt{x}}\right )^2}}\\ \end{align*}
Mathematica [F] time = 0.132578, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p}{x^6} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.333, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{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}{\sqrt{x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}}\,{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 \sqrt{x} + c e^{2}}{x}\right ) + a\right )}^{p}}{x^{6}}, 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}{\sqrt{x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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