Optimal. Leaf size=15 \[ -\frac {1}{c e (d+e x)} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {27, 12, 32}
\begin {gather*} -\frac {1}{c e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {1}{c (d+e x)^2} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^2} \, dx}{c}\\ &=-\frac {1}{c e (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{c e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 16, normalized size = 1.07
method | result | size |
gosper | \(-\frac {1}{c e \left (e x +d \right )}\) | \(16\) |
default | \(-\frac {1}{c e \left (e x +d \right )}\) | \(16\) |
norman | \(\frac {x}{c d \left (e x +d \right )}\) | \(16\) |
risch | \(-\frac {1}{c e \left (e x +d \right )}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{c x e^{2} + c d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.16, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{c x e^{2} + c d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.93 \begin {gather*} - \frac {1}{c d e + c e^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 15, normalized size = 1.00 \begin {gather*} -\frac {e^{\left (-1\right )}}{{\left (x e + d\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{c\,x\,e^2+c\,d\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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