Optimal. Leaf size=38 \[ \frac {b p \log (x)}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2454, 2395, 36, 29, 31} \[ -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 x^2}-\frac {b p \log \left (a+b x^2\right )}{2 a}+\frac {b p \log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 x^2}+\frac {1}{2} (b p) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^2\right )\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 x^2}+\frac {(b p) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 a}-\frac {\left (b^2 p\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^2\right )}{2 a}\\ &=\frac {b p \log (x)}{a}-\frac {b p \log \left (a+b x^2\right )}{2 a}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 45, normalized size = 1.18 \[ -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 x^2}-\frac {b p \log \left (a+b x^2\right )}{2 a}+\frac {b p \log (x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 43, normalized size = 1.13 \[ \frac {2 \, b p x^{2} \log \relax (x) - {\left (b p x^{2} + a p\right )} \log \left (b x^{2} + a\right ) - a \log \relax (c)}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 58, normalized size = 1.53 \[ -\frac {\frac {b^{2} p \log \left (b x^{2} + a\right )}{a} - \frac {b^{2} p \log \left (b x^{2}\right )}{a} + \frac {b p \log \left (b x^{2} + a\right )}{x^{2}} + \frac {b \log \relax (c)}{x^{2}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 173, normalized size = 4.55 \[ -\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{2 x^{2}}-\frac {-4 b p \,x^{2} \ln \relax (x )+2 b p \,x^{2} \ln \left (b \,x^{2}+a \right )-i \pi a \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+i \pi a \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}+i \pi a \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi a \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+2 a \ln \relax (c )}{4 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 44, normalized size = 1.16 \[ -\frac {1}{2} \, b p {\left (\frac {\log \left (b x^{2} + a\right )}{a} - \frac {\log \left (x^{2}\right )}{a}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 41, normalized size = 1.08 \[ \frac {b\,p\,\ln \relax (x)}{a}-\frac {b\,p\,\ln \left (b\,x^2+a\right )}{2\,a}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.66, size = 82, normalized size = 2.16 \[ \begin {cases} - \frac {p \log {\left (a + b x^{2} \right )}}{2 x^{2}} - \frac {\log {\relax (c )}}{2 x^{2}} + \frac {b p \log {\relax (x )}}{a} - \frac {b p \log {\left (a + b x^{2} \right )}}{2 a} & \text {for}\: a \neq 0 \\- \frac {p \log {\relax (b )}}{2 x^{2}} - \frac {p \log {\relax (x )}}{x^{2}} - \frac {p}{2 x^{2}} - \frac {\log {\relax (c )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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