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, 266}
\begin {gather*} \frac {c \log \left (a+b x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 266
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 \begin {gather*} \frac {c \log \left (a+b x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 15, normalized size = 0.94
method | result | size |
default | \(\frac {c \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(15\) |
norman | \(\frac {c \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(15\) |
risch | \(\frac {c \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} \frac {c \log \left (b x^{2} + a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 14, normalized size = 0.88 \begin {gather*} \frac {c \log \left (b x^{2} + a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.75 \begin {gather*} \frac {c \log {\left (a + b x^{2} \right )}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (14) = 28\).
time = 1.14, size = 63, normalized size = 3.94 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 14, normalized size = 0.88 \begin {gather*} \frac {c\,\ln \left (b\,x^2+a\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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