Optimal. Leaf size=75 \[ c^4 \left (-2^{-2 p-1}\right ) e^{\frac {4 a}{b}} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p} \Gamma \left (p+1,\frac {4 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 2181
Rule 2310
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*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 1.00 \[ c^4 \left (-2^{-2 p-1}\right ) e^{\frac {4 a}{b}} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p} \Gamma \left (p+1,\frac {4 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \sqrt {x}\right )+a \right )^{p}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 48, normalized size = 0.64 \[ -\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p + 1} c^{4} e^{\left (\frac {4 \, a}{b}\right )} E_{-p}\left (\frac {4 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,\sqrt {x}\right )\right )}^p}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \sqrt {x} \right )}\right )^{p}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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