Optimal. Leaf size=17 \[ -\frac {1}{4 c^3 e (d+e x)^4} \]
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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 = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 32}
\begin {gather*} -\frac {1}{4 c^3 e (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)^5} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^5} \, dx}{c^3}\\ &=-\frac {1}{4 c^3 e (d+e x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{4 c^3 e (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 16, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{4 c^{3} e \left (e x +d \right )^{4}}\) | \(16\) |
norman | \(\frac {-\frac {d}{4 e c}-\frac {x}{4 c}}{c^{2} \left (e x +d \right )^{5}}\) | \(28\) |
gosper | \(-\frac {1}{4 \left (e x +d \right )^{2} \left (e^{2} x^{2}+2 d x e +d^{2}\right ) c^{3} e}\) | \(34\) |
risch | \(-\frac {1}{4 \left (e x +d \right )^{2} \left (e^{2} x^{2}+2 d x e +d^{2}\right ) c^{3} e}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 29, normalized size = 1.71 \begin {gather*} -\frac {e^{\left (-1\right )}}{4 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (15) = 30\).
time = 2.64, size = 58, normalized size = 3.41 \begin {gather*} -\frac {1}{4 \, {\left (c^{3} x^{4} e^{5} + 4 \, c^{3} d x^{3} e^{4} + 6 \, c^{3} d^{2} x^{2} e^{3} + 4 \, c^{3} d^{3} x e^{2} + c^{3} d^{4} e\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. 66 vs.
\(2 (15) = 30\).
time = 0.15, size = 66, normalized size = 3.88 \begin {gather*} - \frac {1}{4 c^{3} d^{4} e + 16 c^{3} d^{3} e^{2} x + 24 c^{3} d^{2} e^{3} x^{2} + 16 c^{3} d e^{4} x^{3} + 4 c^{3} e^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.95, size = 30, normalized size = 1.76 \begin {gather*} -\frac {e^{\left (-1\right )}}{4 \, {\left (c d^{2} + {\left (x^{2} e + 2 \, d x\right )} c e\right )}^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 63, normalized size = 3.71 \begin {gather*} -\frac {1}{4\,c^3\,d^4\,e+16\,c^3\,d^3\,e^2\,x+24\,c^3\,d^2\,e^3\,x^2+16\,c^3\,d\,e^4\,x^3+4\,c^3\,e^5\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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