Optimal. Leaf size=14 \[ a e x+\frac {1}{2} c d x^2 \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {24} \begin {gather*} a e x+\frac {1}{2} c d x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{d+e x} \, dx &=\frac {\int \left (a e^3+c d e^2 x\right ) \, dx}{e^2}\\ &=a e x+\frac {1}{2} c d x^2\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} a e x+\frac {1}{2} c d x^2 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 20, normalized size = 1.43 \begin {gather*} \frac {(a e+c d x)^2}{2 c d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, c d x^{2} + a e x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 19, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, {\left (c d x^{2} e^{2} + 2 \, a x e^{3}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{2} c d \,x^{2}+a e x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, c d x^{2} + a e x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} \frac {c\,d\,x^2}{2}+a\,e\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 12, normalized size = 0.86 \begin {gather*} a e x + \frac {c d x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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