Optimal. Leaf size=13 \[ \frac {\log (d+e x)}{c e} \]
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Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 31} \[ \frac {\log (d+e x)}{c e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rubi steps
\begin {align*} \int \frac {d+e x}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {1}{c (d+e x)} \, dx\\ &=\frac {\int \frac {1}{d+e x} \, dx}{c}\\ &=\frac {\log (d+e x)}{c e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.23 \[ \frac {\log (c d+c e x)}{c e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 13, normalized size = 1.00 \[ \frac {\log \left (e x + d\right )}{c e} \]
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 = 0.04, size = 14, normalized size = 1.08 \[ \frac {\ln \left (e x +d \right )}{c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 29, normalized size = 2.23 \[ \frac {\log \left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}{2 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 13, normalized size = 1.00 \[ \frac {\ln \left (d+e\,x\right )}{c\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 12, normalized size = 0.92 \[ \frac {\log {\left (c d + c e x \right )}}{c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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