Optimal. Leaf size=17 \[ \frac {(d+e x)^3}{3 c^2 e} \]
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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} \[ \frac {(d+e x)^3}{3 c^2 e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^2}{c^2} \, dx\\ &=\frac {\int (d+e x)^2 \, dx}{c^2}\\ &=\frac {(d+e x)^3}{3 c^2 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ \frac {(d+e x)^3}{3 c^2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 26, normalized size = 1.53 \[ \frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 0.94 \[ \frac {\left (e x +d \right )^{3}}{3 c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 26, normalized size = 1.53 \[ \frac {e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c^{2}} \]
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 \[ \frac {x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 29, normalized size = 1.71 \[ \frac {d^{2} x}{c^{2}} + \frac {d e x^{2}}{c^{2}} + \frac {e^{2} x^{3}}{3 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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