Optimal. Leaf size=17 \[ -\frac {1}{2 c e (d+e x)^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32}
\begin {gather*} -\frac {1}{2 c e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx &=\int \frac {1}{c (d+e x)^3} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^3} \, dx}{c}\\ &=-\frac {1}{2 c e (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{2 c e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 16, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{2 c e \left (e x +d \right )^{2}}\) | \(16\) |
norman | \(-\frac {1}{2 c e \left (e x +d \right )^{2}}\) | \(16\) |
risch | \(-\frac {1}{2 c e \left (e x +d \right )^{2}}\) | \(16\) |
gosper | \(-\frac {1}{2 e c \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 1.53 \begin {gather*} -\frac {1}{2 \, {\left (c x^{2} e^{3} + 2 \, c d x e^{2} + c d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.73, size = 26, normalized size = 1.53 \begin {gather*} -\frac {1}{2 \, {\left (c x^{2} e^{3} + 2 \, c d x e^{2} + c d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (14) = 28\).
time = 0.08, size = 31, normalized size = 1.82 \begin {gather*} - \frac {1}{2 c d^{2} e + 4 c d e^{2} x + 2 c e^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 15, normalized size = 0.88 \begin {gather*} -\frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 29, normalized size = 1.71 \begin {gather*} -\frac {1}{2\,c\,d^2\,e+4\,c\,d\,e^2\,x+2\,c\,e^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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