Optimal. Leaf size=41 \[ -\frac {c \log \left (a+b x^2\right )}{2 a^2}+\frac {c \log (x)}{a^2}+\frac {c}{2 a \left (a+b x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 266, 44} \[ -\frac {c \log \left (a+b x^2\right )}{2 a^2}+\frac {c \log (x)}{a^2}+\frac {c}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 21
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^3} \, dx &=c \int \frac {1}{x \left (a+b x^2\right )^2} \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {c}{2 a \left (a+b x^2\right )}+\frac {c \log (x)}{a^2}-\frac {c \log \left (a+b x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.83 \[ \frac {c \left (\frac {a}{a+b x^2}-\log \left (a+b x^2\right )+2 \log (x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 54, normalized size = 1.32 \[ \frac {a c - {\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (b c x^{2} + a c\right )} \log \relax (x)}{2 \, {\left (a^{2} b x^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 51, normalized size = 1.24 \[ \frac {c \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac {b c x^{2} + 2 \, a c}{2 \, {\left (b x^{2} + a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.93 \[ \frac {c}{2 \left (b \,x^{2}+a \right ) a}+\frac {c \ln \relax (x )}{a^{2}}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 40, normalized size = 0.98 \[ \frac {c}{2 \, {\left (a b x^{2} + a^{2}\right )}} - \frac {c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {c \log \left (x^{2}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 37, normalized size = 0.90 \[ \frac {c}{2\,a\,\left (b\,x^2+a\right )}-\frac {c\,\ln \left (b\,x^2+a\right )}{2\,a^2}+\frac {c\,\ln \relax (x)}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 36, normalized size = 0.88 \[ c \left (\frac {1}{2 a^{2} + 2 a b x^{2}} + \frac {\log {\relax (x )}}{a^{2}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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