Optimal. Leaf size=15 \[ -\frac {1}{c^2 e (d+e x)} \]
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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 = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ -\frac {1}{c^2 e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {1}{c^2 (d+e x)^2} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^2} \, dx}{c^2}\\ &=-\frac {1}{c^2 e (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {1}{c^2 e (d+e x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 19, normalized size = 1.27 \[ -\frac {1}{c^{2} e^{2} x + c^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 1.07 \[ -\frac {1}{\left (e x +d \right ) c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 19, normalized size = 1.27 \[ -\frac {1}{c^{2} e^{2} x + c^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 19, normalized size = 1.27 \[ -\frac {1}{x\,c^2\,e^2+d\,c^2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 17, normalized size = 1.13 \[ - \frac {1}{c^{2} d e + c^{2} e^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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