Optimal. Leaf size=17 \[ \frac {c^2 (d+e x)^2}{2 e} \]
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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 {c^2 (d+e x)^2}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 9
Rule 27
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)^3} \, dx &=\int c^2 (d+e x) \, dx\\ &=\frac {c^2 (d+e x)^2}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} c^2 \left (d x+\frac {e x^2}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 15, normalized size = 0.88
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) c^{2}}{2}\) | \(14\) |
default | \(c^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )\) | \(15\) |
risch | \(\frac {1}{2} c^{2} e \,x^{2}+c^{2} d x\) | \(17\) |
norman | \(\frac {\frac {c^{2} x^{4} e^{3}}{2}-\frac {5 c^{2} d^{4}}{2 e}+2 c^{2} d \,e^{2} x^{3}-4 c^{2} d^{3} x}{\left (e x +d \right )^{2}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} e + c^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.15, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} e + c^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 15, normalized size = 0.88 \begin {gather*} c^{2} d x + \frac {c^{2} e x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.16, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} e + c^{2} d x \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 {c^2\,x\,\left (2\,d+e\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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