Optimal. Leaf size=18 \[ -\frac {1}{c d (a e+c d x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32}
\begin {gather*} -\frac {1}{c d (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2} \, dx\\ &=-\frac {1}{c d (a e+c d x)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{c d (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 19, normalized size = 1.06
method | result | size |
gosper | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
default | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
risch | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
norman | \(\frac {-\frac {1}{c}-\frac {e x}{c d}}{\left (c d x +a e \right ) \left (e x +d \right )}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 19, normalized size = 1.06 \begin {gather*} -\frac {1}{c^{2} d^{2} x + a c d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 19, normalized size = 1.06 \begin {gather*} -\frac {1}{c^{2} d^{2} x + a c d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 17, normalized size = 0.94 \begin {gather*} - \frac {1}{a c d e + c^{2} d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 19, normalized size = 1.06 \begin {gather*} -\frac {1}{{\left (c d x + a e\right )} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{c\,d\,\left (a\,e+c\,d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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