Optimal. Leaf size=59 \[ \frac {p}{4 x^2}-\frac {a p}{2 b x}+\frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 45}
\begin {gather*} \frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}-\frac {a p}{2 b x}+\frac {p}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^3} \, dx &=-\text {Subst}\left (\int x \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}+\frac {1}{2} (b p) \text {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}+\frac {1}{2} (b p) \text {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {p}{4 x^2}-\frac {a p}{2 b x}+\frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.00 \begin {gather*} \frac {p}{4 x^2}-\frac {a p}{2 b x}+\frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 63, normalized size = 1.07 \begin {gather*} \frac {1}{4} \, b p {\left (\frac {2 \, a^{2} \log \left (a x + b\right )}{b^{3}} - \frac {2 \, a^{2} \log \left (x\right )}{b^{3}} - \frac {2 \, a x - b}{b^{2} x^{2}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 55, normalized size = 0.93 \begin {gather*} -\frac {2 \, a b p x - b^{2} p + 2 \, b^{2} \log \left (c\right ) - 2 \, {\left (a^{2} p x^{2} - b^{2} p\right )} \log \left (\frac {a x + b}{x}\right )}{4 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.79, size = 61, normalized size = 1.03 \begin {gather*} \begin {cases} \frac {a^{2} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{2 b^{2}} - \frac {a p}{2 b x} + \frac {p}{4 x^{2}} - \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{2 x^{2}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs.
\(2 (51) = 102\).
time = 4.26, size = 150, normalized size = 2.54 \begin {gather*} \frac {\frac {4 \, {\left (a x + b\right )} a p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b x} - \frac {4 \, {\left (a x + b\right )} a p}{b x} - \frac {2 \, {\left (a x + b\right )}^{2} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b x^{2}} + \frac {4 \, {\left (a x + b\right )} a \log \left (c\right )}{b x} + \frac {{\left (a x + b\right )}^{2} p}{b x^{2}} - \frac {2 \, {\left (a x + b\right )}^{2} \log \left (c\right )}{b x^{2}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 53, normalized size = 0.90 \begin {gather*} \frac {\frac {p}{2}-\frac {a\,p\,x}{b}}{2\,x^2}-\frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{2\,x^2}+\frac {a^2\,p\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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