Optimal. Leaf size=17 \[ -\frac {c^2}{2 e (d+e x)^2} \]
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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 = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32}
\begin {gather*} -\frac {c^2}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac {c^2}{(d+e x)^3} \, dx\\ &=c^2 \int \frac {1}{(d+e x)^3} \, dx\\ &=-\frac {c^2}{2 e (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {c^2}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 16, normalized size = 0.94
method | result | size |
gosper | \(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\) | \(16\) |
default | \(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\) | \(16\) |
risch | \(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\) | \(16\) |
norman | \(\frac {-\frac {c^{2} d^{4}}{2 e}-\frac {c^{2} x^{4} e^{3}}{2}-2 c^{2} d \,e^{2} x^{3}-3 c^{2} d^{2} e \,x^{2}-2 c^{2} d^{3} x}{\left (e x +d \right )^{6}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 1.53 \begin {gather*} -\frac {c^{2}}{2 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.42, size = 26, normalized size = 1.53 \begin {gather*} -\frac {c^{2}}{2 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 27, normalized size = 1.59 \begin {gather*} - \frac {c^{2}}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.59, size = 15, normalized size = 0.88 \begin {gather*} -\frac {c^{2} e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 26, normalized size = 1.53 \begin {gather*} -\frac {c^2}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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