Optimal. Leaf size=24 \[ \frac {c \log (x)}{a}-\frac {c \log \left (a+b x^2\right )}{2 a} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {21, 266, 36, 29, 31} \begin {gather*} \frac {c \log (x)}{a}-\frac {c \log \left (a+b x^2\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 29
Rule 31
Rule 36
Rule 266
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx &=c \int \frac {1}{x \left (a+b x^2\right )} \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {c \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 a}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c \log (x)}{a}-\frac {c \log \left (a+b x^2\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} c \left (\frac {\log (x)}{a}-\frac {\log \left (a+b x^2\right )}{2 a}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.71, size = 21, normalized size = 0.88 \begin {gather*} -\frac {c \log \left (b x^{2} + a\right ) - 2 \, c \log \relax (x)}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 26, normalized size = 1.08 \begin {gather*} \frac {c \log \left (x^{2}\right )}{2 \, a} - \frac {c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.96 \begin {gather*} \frac {c \ln \relax (x )}{a}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 25, normalized size = 1.04 \begin {gather*} -\frac {c \log \left (b x^{2} + a\right )}{2 \, a} + \frac {c \log \left (x^{2}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 19, normalized size = 0.79 \begin {gather*} -\frac {c\,\left (\ln \left (b\,x^2+a\right )-2\,\ln \relax (x)\right )}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 17, normalized size = 0.71 \begin {gather*} c \left (\frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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