Optimal. Leaf size=175 \[ \frac{2 d e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )}{c e^2}-\frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )}{b}\right )}{c^2 e^2} \]
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Rubi [A] time = 0.246925, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 d e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )}{c e^2}-\frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )}{b}\right )}{c^2 e^2} \]
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 \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p}{x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int x (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 (a+b \log (c (d+e x)))^p}{e}+\frac{(d+e x) (a+b \log (c (d+e x)))^p}{e}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e}+\frac{(2 d) \operatorname{Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\frac{1}{\sqrt{x}}\right )}{e}\\ &=-\frac{2 \operatorname{Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}+\frac{(2 d) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}\\ &=-\frac{2 \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^2}+\frac{(2 d) \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^2}\\ &=-\frac{2^{-p} 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^2}+\frac{2 d 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^2}\\ \end{align*}
Mathematica [A] time = 0.171015, size = 131, normalized size = 0.75 \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )^{-p} \left (c d 2^{p+1} e^{a/b} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )}{b}\right )-\text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )}{b}\right )\right )}{c^2 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.361, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{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 \frac{{\left (b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}\right ) + a\right )}^{p}}{x^{2}}\,{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 x + c e \sqrt{x}}{x}\right ) + a\right )}^{p}}{x^{2}}, 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 )}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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