Optimal. Leaf size=15 \[ -\frac {d}{a+b x+c x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643}
\begin {gather*} -\frac {d}{a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d}{a+b x+c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} -\frac {d}{a+x (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 16, normalized size = 1.07
method | result | size |
gosper | \(-\frac {d}{c \,x^{2}+b x +a}\) | \(16\) |
default | \(-\frac {d}{c \,x^{2}+b x +a}\) | \(16\) |
risch | \(-\frac {d}{c \,x^{2}+b x +a}\) | \(16\) |
norman | \(\frac {\frac {b x d}{a}+\frac {c d \,x^{2}}{a}}{c \,x^{2}+b x +a}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.54, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 12, normalized size = 0.80 \begin {gather*} - \frac {d}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 22, normalized size = 1.47 \begin {gather*} -\frac {d}{a + \frac {c d x^{2} + b d x}{d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{c\,x^2+b\,x+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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