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 = {626, 32} \[ -\frac {1}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 626
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 \[ -\frac {1}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 18, normalized size = 1.00 \[ -\frac {1}{c^{2} d^{2} x + a c d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 109, normalized size = 6.06 \[ -\frac {c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}}{{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 1.06 \[ -\frac {1}{\left (c d x +a e \right ) c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 18, normalized size = 1.00 \[ -\frac {1}{c^{2} d^{2} x + a c d e} \]
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 \[ -\frac {1}{c\,d\,\left (a\,e+c\,d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 17, normalized size = 0.94 \[ - \frac {1}{a c d e + c^{2} d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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