Optimal. Leaf size=16 \[ \frac {c \log \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {21, 260} \[ \frac {c \log \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 21
Rule 260
Rubi steps
\begin {align*} \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx &=c \int \frac {x}{a+b x^2} \, dx\\ &=\frac {c \log \left (a+b x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \frac {c \log \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 14, normalized size = 0.88 \[ \frac {c \log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 63, normalized size = 3.94 \[ -\frac {1}{2} \, c {\left (\frac {\log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x^{2} + a\right )} b}\right )} - \frac {a c}{2 \, {\left (b x^{2} + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.94 \[ \frac {c \ln \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 14, normalized size = 0.88 \[ \frac {c \log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 14, normalized size = 0.88 \[ \frac {c\,\ln \left (b\,x^2+a\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.75 \[ \frac {c \log {\left (a + b x^{2} \right )}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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