Optimal. Leaf size=87 \[ \frac {a^4 p \log \left (a+\frac {b}{x}\right )}{4 b^4}-\frac {a^3 p}{4 b^3 x}+\frac {a^2 p}{8 b^2 x^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 x^4}-\frac {a p}{12 b x^3}+\frac {p}{16 x^4} \]
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Rubi [A] time = 0.06, antiderivative size = 87, 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} \[ \frac {a^2 p}{8 b^2 x^2}-\frac {a^3 p}{4 b^3 x}+\frac {a^4 p \log \left (a+\frac {b}{x}\right )}{4 b^4}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 x^4}-\frac {a p}{12 b x^3}+\frac {p}{16 x^4} \]
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^5} \, dx &=-\operatorname {Subst}\left (\int x^3 \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 )}{4 x^4}+\frac {1}{4} (b p) \operatorname {Subst}\left (\int \frac {x^4}{a+b x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 x^4}+\frac {1}{4} (b p) \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{b^2}+\frac {x^3}{b}+\frac {a^4}{b^4 (a+b x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {p}{16 x^4}-\frac {a p}{12 b x^3}+\frac {a^2 p}{8 b^2 x^2}-\frac {a^3 p}{4 b^3 x}+\frac {a^4 p \log \left (a+\frac {b}{x}\right )}{4 b^4}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 87, normalized size = 1.00 \[ \frac {a^4 p \log \left (a+\frac {b}{x}\right )}{4 b^4}-\frac {a^3 p}{4 b^3 x}+\frac {a^2 p}{8 b^2 x^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{4 x^4}-\frac {a p}{12 b x^3}+\frac {p}{16 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 79, normalized size = 0.91 \[ -\frac {12 \, a^{3} b p x^{3} - 6 \, a^{2} b^{2} p x^{2} + 4 \, a b^{3} p x - 3 \, b^{4} p + 12 \, b^{4} \log \relax (c) - 12 \, {\left (a^{4} p x^{4} - b^{4} p\right )} \log \left (\frac {a x + b}{x}\right )}{48 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 317, normalized size = 3.64 \[ \frac {\frac {48 \, {\left (a x + b\right )} a^{3} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{3} x} - \frac {48 \, {\left (a x + b\right )} a^{3} p}{b^{3} x} - \frac {72 \, {\left (a x + b\right )}^{2} a^{2} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{3} x^{2}} + \frac {48 \, {\left (a x + b\right )} a^{3} \log \relax (c)}{b^{3} x} + \frac {36 \, {\left (a x + b\right )}^{2} a^{2} p}{b^{3} x^{2}} + \frac {48 \, {\left (a x + b\right )}^{3} a p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{3} x^{3}} - \frac {72 \, {\left (a x + b\right )}^{2} a^{2} \log \relax (c)}{b^{3} x^{2}} - \frac {16 \, {\left (a x + b\right )}^{3} a p}{b^{3} x^{3}} - \frac {12 \, {\left (a x + b\right )}^{4} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{3} x^{4}} + \frac {48 \, {\left (a x + b\right )}^{3} a \log \relax (c)}{b^{3} x^{3}} + \frac {3 \, {\left (a x + b\right )}^{4} p}{b^{3} x^{4}} - \frac {12 \, {\left (a x + b\right )}^{4} \log \relax (c)}{b^{3} x^{4}}}{48 \, b} \]
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 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 85, normalized size = 0.98 \[ \frac {1}{48} \, b p {\left (\frac {12 \, a^{4} \log \left (a x + b\right )}{b^{5}} - \frac {12 \, a^{4} \log \relax (x)}{b^{5}} - \frac {12 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 4 \, a b^{2} x - 3 \, b^{3}}{b^{4} x^{4}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 78, normalized size = 0.90 \[ \frac {\frac {p}{4}+\frac {a^2\,p\,x^2}{2\,b^2}-\frac {a^3\,p\,x^3}{b^3}-\frac {a\,p\,x}{3\,b}}{4\,x^4}-\frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{4\,x^4}+\frac {a^4\,p\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{2\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.47, size = 94, normalized size = 1.08 \[ \begin {cases} \frac {a^{4} p \log {\left (a + \frac {b}{x} \right )}}{4 b^{4}} - \frac {a^{3} p}{4 b^{3} x} + \frac {a^{2} p}{8 b^{2} x^{2}} - \frac {a p}{12 b x^{3}} - \frac {p \log {\left (a + \frac {b}{x} \right )}}{4 x^{4}} + \frac {p}{16 x^{4}} - \frac {\log {\relax (c )}}{4 x^{4}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{4 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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