Optimal. Leaf size=17 \[ -\frac {c}{2 b \left (a+b x^2\right )} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, 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, 267}
\begin {gather*} -\frac {c}{2 b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 267
Rubi steps
\begin {align*} \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx &=c \int \frac {x}{\left (a+b x^2\right )^2} \, dx\\ &=-\frac {c}{2 b \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {c}{2 b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 16, normalized size = 0.94
method | result | size |
gosper | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
default | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
risch | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
norman | \(\frac {-\frac {a c}{2 b}-\frac {c \,x^{2}}{2}}{\left (b \,x^{2}+a \right )^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 16, normalized size = 0.94 \begin {gather*} -\frac {c}{2 \, {\left (b^{2} x^{2} + a b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 16, normalized size = 0.94 \begin {gather*} -\frac {c}{2 \, {\left (b^{2} x^{2} + a b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 15, normalized size = 0.88 \begin {gather*} - \frac {c}{2 a b + 2 b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.90, size = 15, normalized size = 0.88 \begin {gather*} -\frac {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 = 15, normalized size = 0.88 \begin {gather*} -\frac {c}{2\,b\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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