Optimal. Leaf size=75 \[ c^4 \left (-2^{-2 p-1}\right ) e^{\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,\frac{4 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right ) \]
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Rubi [A] time = 0.0583778, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2310, 2181} \[ c^4 \left (-2^{-2 p-1}\right ) e^{\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,\frac{4 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right ) \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \sqrt{x}\right )\right )^p}{x^3} \, dx &=\left (2 c^4\right ) \operatorname{Subst}\left (\int e^{-4 x} (a+b x)^p \, dx,x,\log \left (c \sqrt{x}\right )\right )\\ &=-2^{-1-2 p} c^4 e^{\frac{4 a}{b}} \Gamma \left (1+p,\frac{4 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p}\\ \end{align*}
Mathematica [A] time = 0.0449017, size = 75, normalized size = 1. \[ c^4 \left (-2^{-2 p-1}\right ) e^{\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,\frac{4 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c\sqrt{x} \right ) \right ) ^{p}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24732, size = 65, normalized size = 0.87 \begin{align*} -\frac{2 \,{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p + 1} c^{4} e^{\left (\frac{4 \, a}{b}\right )} E_{-p}\left (\frac{4 \,{\left (b \log \left (c \sqrt{x}\right ) + a\right )}}{b}\right )}{b} \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 (c \sqrt{x}\right ) + a\right )}^{p}}{x^{3}}, 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 \sqrt{x}\right ) + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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