Integrand size = 16, antiderivative size = 64 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {b p}{4 a x^2}-\frac {b^2 p \log (x)}{2 a^2}+\frac {b^2 p \log \left (a+b x^2\right )}{4 a^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]
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Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 46} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {b^2 p \log \left (a+b x^2\right )}{4 a^2}-\frac {b^2 p \log (x)}{2 a^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {b p}{4 a x^2} \]
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Rule 46
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^3} \, dx,x,x^2\right ) \\ & = -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{4} (b p) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,x^2\right ) \\ & = -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{4} (b p) \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b p}{4 a x^2}-\frac {b^2 p \log (x)}{2 a^2}+\frac {b^2 p \log \left (a+b x^2\right )}{4 a^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{4 x^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {1}{4} b p \left (-\frac {1}{a x^2}-\frac {2 b \log (x)}{a^2}+\frac {b \log \left (a+b x^2\right )}{a^2}\right )-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]
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Time = 0.42 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{4 x^{4}}+\frac {p b \left (-\frac {1}{2 a \,x^{2}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\right )}{2}\) | \(54\) |
parallelrisch | \(-\frac {2 b^{2} p^{2} \ln \left (x \right ) x^{4}-x^{4} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) b^{2} p -b^{2} p^{2} x^{4}+a b \,p^{2} x^{2}+\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a^{2} p}{4 x^{4} a^{2} p}\) | \(84\) |
risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{4 x^{4}}-\frac {4 b^{2} p \ln \left (x \right ) x^{4}-2 b^{2} p \ln \left (-b \,x^{2}-a \right ) x^{4}+i \pi \,a^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,a^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi \,a^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi \,a^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 a b p \,x^{2}+2 \ln \left (c \right ) a^{2}}{8 a^{2} x^{4}}\) | \(198\) |
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Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {2 \, b^{2} p x^{4} \log \left (x\right ) + a b p x^{2} + a^{2} \log \left (c\right ) - {\left (b^{2} p x^{4} - a^{2} p\right )} \log \left (b x^{2} + a\right )}{4 \, a^{2} x^{4}} \]
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Time = 2.63 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.30 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\begin {cases} - \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{4 x^{4}} - \frac {b p}{4 a x^{2}} - \frac {b^{2} p \log {\left (x \right )}}{2 a^{2}} + \frac {b^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\- \frac {p}{8 x^{4}} - \frac {\log {\left (c \left (b x^{2}\right )^{p} \right )}}{4 x^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {1}{4} \, b p {\left (\frac {b \log \left (b x^{2} + a\right )}{a^{2}} - \frac {b \log \left (x^{2}\right )}{a^{2}} - \frac {1}{a x^{2}}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (56) = 112\).
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.06 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {\frac {b^{3} p \log \left (b x^{2} + a\right )}{{\left (b x^{2} + a\right )}^{2} - 2 \, {\left (b x^{2} + a\right )} a + a^{2}} - \frac {b^{3} p \log \left (b x^{2} + a\right )}{a^{2}} + \frac {b^{3} p \log \left (b x^{2}\right )}{a^{2}} + \frac {{\left (b x^{2} + a\right )} b^{3} p - a b^{3} p + a b^{3} \log \left (c\right )}{{\left (b x^{2} + a\right )}^{2} a - 2 \, {\left (b x^{2} + a\right )} a^{2} + a^{3}}}{4 \, b} \]
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Time = 1.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {b^2\,p\,\ln \left (b\,x^2+a\right )}{4\,a^2}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{4\,x^4}-\frac {b^2\,p\,\ln \left (x\right )}{2\,a^2}-\frac {b\,p}{4\,a\,x^2} \]
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