Optimal. Leaf size=17 \[ \frac {(d+e x)^2}{2 c e} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 9}
\begin {gather*} \frac {(d+e x)^2}{2 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 9
Rule 27
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {d+e x}{c} \, dx\\ &=\frac {(d+e x)^2}{2 c e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {d x+\frac {e x^2}{2}}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 15, normalized size = 0.88
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right )}{2 c}\) | \(14\) |
default | \(\frac {\frac {1}{2} e \,x^{2}+d x}{c}\) | \(15\) |
risch | \(\frac {e \,x^{2}}{2 c}+\frac {x d}{c}\) | \(17\) |
norman | \(\frac {\frac {d^{2} x}{c}+\frac {e^{2} x^{3}}{2 c}+\frac {3 e d \,x^{2}}{2 c}}{e x +d}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 0.94 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.99, size = 16, normalized size = 0.94 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 12, normalized size = 0.71 \begin {gather*} \frac {d x}{c} + \frac {e x^{2}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.82, size = 16, normalized size = 0.94 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x\,\left (2\,d+e\,x\right )}{2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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