Optimal. Leaf size=26 \[ \frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{p+1}}{b (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 26, 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 = {2302, 30} \[ \frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{p+1}}{b (p+1)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx &=\frac {2 \operatorname {Subst}\left (\int x^p \, dx,x,a+b \log \left (c \sqrt {x}\right )\right )}{b}\\ &=\frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{1+p}}{b (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{p+1}}{b (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 31, normalized size = 1.19 \[ \frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}}{b p + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 25, normalized size = 0.96 \[ \frac {2 \, {\left (b \log \relax (c) + \frac {1}{2} \, b \log \relax (x) + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.96 \[ \frac {2 \left (b \ln \left (c \sqrt {x}\right )+a \right )^{p +1}}{\left (p +1\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 24, normalized size = 0.92 \[ \frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 24, normalized size = 0.92 \[ \frac {2\,{\left (a+b\,\ln \left (c\,\sqrt {x}\right )\right )}^{p+1}}{b\,\left (p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.76, size = 48, normalized size = 1.85 \[ - \begin {cases} - a^{p} \log {\relax (x )} & \text {for}\: b = 0 \\- \frac {2 \left (\begin {cases} \frac {\left (a + b \log {\left (c \sqrt {x} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c \sqrt {x} \right )} \right )} & \text {otherwise} \end {cases}\right )}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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