Optimal. Leaf size=59 \[ \frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}-\frac {a p}{2 b x}+\frac {p}{4 x^2} \]
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Rubi [A] time = 0.04, 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 = {2454, 2395, 43} \[ \frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}-\frac {a p}{2 b x}+\frac {p}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^3} \, dx &=-\operatorname {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) \operatorname {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) \operatorname {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 \[ \frac {a^2 p \log \left (a+\frac {b}{x}\right )}{2 b^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 x^2}-\frac {a p}{2 b x}+\frac {p}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 55, normalized size = 0.93 \[ -\frac {2 \, a b p x - b^{2} p + 2 \, b^{2} \log \relax (c) - 2 \, {\left (a^{2} p x^{2} - b^{2} p\right )} \log \left (\frac {a x + b}{x}\right )}{4 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 150, normalized size = 2.54 \[ \frac {\frac {4 \, {\left (a x + b\right )} a p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b x} - \frac {4 \, {\left (a x + b\right )} a p}{b x} - \frac {2 \, {\left (a x + b\right )}^{2} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b x^{2}} + \frac {4 \, {\left (a x + b\right )} a \log \relax (c)}{b x} + \frac {{\left (a x + b\right )}^{2} p}{b x^{2}} - \frac {2 \, {\left (a x + b\right )}^{2} \log \relax (c)}{b x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, 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 = 0.65, size = 63, normalized size = 1.07 \[ \frac {1}{4} \, b p {\left (\frac {2 \, a^{2} \log \left (a x + b\right )}{b^{3}} - \frac {2 \, a^{2} \log \relax (x)}{b^{3}} - \frac {2 \, a x - b}{b^{2} x^{2}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{2 \, x^{2}} \]
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 \[ \frac {\frac {p}{2}-\frac {a\,p\,x}{b}}{2\,x^2}-\frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{2\,x^2}+\frac {a^2\,p\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.93, size = 66, normalized size = 1.12 \[ \begin {cases} \frac {a^{2} p \log {\left (a + \frac {b}{x} \right )}}{2 b^{2}} - \frac {a p}{2 b x} - \frac {p \log {\left (a + \frac {b}{x} \right )}}{2 x^{2}} + \frac {p}{4 x^{2}} - \frac {\log {\relax (c )}}{2 x^{2}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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