Optimal. Leaf size=46 \[ 2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \log \left (-\frac {b \sqrt {x}}{a}\right )+2 p \text {Li}_2\left (1+\frac {b \sqrt {x}}{a}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2504, 2441,
2352} \begin {gather*} 2 p \text {PolyLog}\left (2,\frac {b \sqrt {x}}{a}+1\right )+2 \log \left (-\frac {b \sqrt {x}}{a}\right ) \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx &=2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \log \left (-\frac {b \sqrt {x}}{a}\right )-(2 b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\sqrt {x}\right )\\ &=2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \log \left (-\frac {b \sqrt {x}}{a}\right )+2 p \text {Li}_2\left (1+\frac {b \sqrt {x}}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 47, normalized size = 1.02 \begin {gather*} 2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \log \left (-\frac {b \sqrt {x}}{a}\right )+2 p \text {Li}_2\left (\frac {a+b \sqrt {x}}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs.
\(2 (39) = 78\).
time = 0.28, size = 79, normalized size = 1.72 \begin {gather*} b p {\left (\frac {\log \left (b \sqrt {x} + a\right ) \log \left (x\right )}{b} - \frac {\log \left (x\right ) \log \left (\frac {b \sqrt {x}}{a} + 1\right ) + 2 \, {\rm Li}_2\left (-\frac {b \sqrt {x}}{a}\right )}{b}\right )} - p \log \left (b \sqrt {x} + a\right ) \log \left (x\right ) + \log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right ) \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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