Optimal. Leaf size=17 \[ \frac {(d+e x)^3}{3 c e} \]
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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 {(d+e x)^3}{3 c e} \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)^4}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {(d+e x)^2}{c} \, dx\\ &=\frac {\int (d+e x)^2 \, dx}{c}\\ &=\frac {(d+e x)^3}{3 c e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(d+e x)^3}{3 c e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {\left (e x +d \right )^{3}}{3 c e}\) | \(16\) |
gosper | \(\frac {x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right )}{3 c}\) | \(25\) |
risch | \(\frac {e^{2} x^{3}}{3 c}+\frac {e d \,x^{2}}{c}+\frac {d^{2} x}{c}+\frac {d^{3}}{3 c e}\) | \(41\) |
norman | \(\frac {-\frac {d^{4}}{c e}+\frac {e^{3} x^{4}}{3 c}+\frac {2 e \,d^{2} x^{2}}{c}+\frac {4 e^{2} d \,x^{3}}{3 c}}{e x +d}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 1.53 \begin {gather*} \frac {x^{3} e^{2} + 3 \, d x^{2} e + 3 \, d^{2} x}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.12, size = 26, normalized size = 1.53 \begin {gather*} \frac {x^{3} e^{2} + 3 \, d x^{2} e + 3 \, d^{2} x}{3 \, c} \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. 24 vs.
\(2 (10) = 20\).
time = 0.02, size = 24, normalized size = 1.41 \begin {gather*} \frac {d^{2} x}{c} + \frac {d e x^{2}}{c} + \frac {e^{2} x^{3}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.35, size = 26, normalized size = 1.53 \begin {gather*} \frac {x^{3} e^{2} + 3 \, d x^{2} e + 3 \, d^{2} x}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 1.41 \begin {gather*} \frac {x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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