Optimal. Leaf size=17 \[ \frac {(d+e x)^4}{4 c^2 e} \]
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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} \begin {gather*} \frac {(d+e x)^4}{4 c^2 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)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^3}{c^2} \, dx\\ &=\frac {\int (d+e x)^3 \, dx}{c^2}\\ &=\frac {(d+e x)^4}{4 c^2 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(d+e x)^4}{4 c^2 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.37, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (e x +d \right )^{4}}{4 c^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 43, normalized size = 2.53 \begin {gather*} \frac {d^3\,x}{c^2}+\frac {e^3\,x^4}{4\,c^2}+\frac {3\,d^2\,e\,x^2}{2\,c^2}+\frac {d\,e^2\,x^3}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 46, normalized size = 2.71 \begin {gather*} \frac {d^{3} x}{c^{2}} + \frac {3 d^{2} e x^{2}}{2 c^{2}} + \frac {d e^{2} x^{3}}{c^{2}} + \frac {e^{3} x^{4}}{4 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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