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} \[ -\frac {1}{2 c e (d+e x)^2} \]
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 \[ -\frac {1}{2 c e (d+e x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 27, normalized size = 1.59 \[ -\frac {1}{2 \, {\left (c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e\right )}} \]
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 = 0.94 \[ -\frac {1}{2 \left (e x +d \right )^{2} c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 27, normalized size = 1.59 \[ -\frac {1}{2 \, {\left (c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e\right )}} \]
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 \[ -\frac {1}{2\,c\,d^2\,e+4\,c\,d\,e^2\,x+2\,c\,e^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 31, normalized size = 1.82 \[ - \frac {1}{2 c d^{2} e + 4 c d e^{2} x + 2 c e^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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