Optimal. Leaf size=20 \[ -\frac {1}{2 c d (a e+c d x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32}
\begin {gather*} -\frac {1}{2 c d (a e+c d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3} \, dx\\ &=-\frac {1}{2 c d (a e+c d x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{2 c d (a e+c d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 19, normalized size = 0.95
method | result | size |
gosper | \(-\frac {1}{2 c d \left (c d x +a e \right )^{2}}\) | \(19\) |
default | \(-\frac {1}{2 c d \left (c d x +a e \right )^{2}}\) | \(19\) |
risch | \(-\frac {1}{2 c d \left (c d x +a e \right )^{2}}\) | \(19\) |
norman | \(\frac {-\frac {e x}{c}-\frac {d}{2 c}-\frac {e^{2} x^{2}}{2 c d}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 1.75 \begin {gather*} -\frac {1}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} x e + a^{2} c d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.59, size = 35, normalized size = 1.75 \begin {gather*} -\frac {1}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} x e + a^{2} c d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (17) = 34\).
time = 0.12, size = 39, normalized size = 1.95 \begin {gather*} - \frac {1}{2 a^{2} c d e^{2} + 4 a c^{2} d^{2} e x + 2 c^{3} d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.80, size = 19, normalized size = 0.95 \begin {gather*} -\frac {1}{2 \, {\left (c d x + a e\right )}^{2} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 37, normalized size = 1.85 \begin {gather*} -\frac {1}{2\,a^2\,c\,d\,e^2+4\,a\,c^2\,d^2\,e\,x+2\,c^3\,d^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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