Optimal. Leaf size=25 \[ \left (4-x^2\right ) \left (-7+\frac {x}{-e^2+27 x^4}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps
used = 5, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {28, 1872, 1600,
12, 30} \begin {gather*} 7 x^2-\frac {x \left (4-x^2\right )}{e^2-27 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 28
Rule 30
Rule 1600
Rule 1872
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=729 \int \frac {14 e^4 x-324 x^4+27 x^6+10206 x^9+e^2 \left (-4+3 x^2-756 x^5\right )}{\left (-27 e^2+729 x^4\right )^2} \, dx\\ &=-\frac {x \left (4-x^2\right )}{e^2-27 x^4}+\frac {\int \frac {-21695547384 e^4 x+585779779368 e^2 x^5}{-27 e^2+729 x^4} \, dx}{57395628 e^2}\\ &=-\frac {x \left (4-x^2\right )}{e^2-27 x^4}+\frac {\int 803538792 e^2 x \, dx}{57395628 e^2}\\ &=-\frac {x \left (4-x^2\right )}{e^2-27 x^4}+14 \int x \, dx\\ &=7 x^2-\frac {x \left (4-x^2\right )}{e^2-27 x^4}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.92 \begin {gather*} x \left (7 x+\frac {-4+x^2}{e^2-27 x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 25, normalized size = 1.00
method | result | size |
risch | \(7 x^{2}+\frac {x^{3}-4 x}{-27 x^{4}+{\mathrm e}^{2}}\) | \(25\) |
gosper | \(\frac {x \left (-189 x^{5}+7 \,{\mathrm e}^{2} x +x^{2}-4\right )}{-27 x^{4}+{\mathrm e}^{2}}\) | \(32\) |
norman | \(\frac {-189 x^{6}+x^{3}+7 x^{2} {\mathrm e}^{2}-4 x}{-27 x^{4}+{\mathrm e}^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 27, normalized size = 1.08 \begin {gather*} 7 \, x^{2} - \frac {x^{3} - 4 \, x}{27 \, x^{4} - e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 34, normalized size = 1.36 \begin {gather*} \frac {189 \, x^{6} - x^{3} - 7 \, x^{2} e^{2} + 4 \, x}{27 \, x^{4} - e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 19, normalized size = 0.76 \begin {gather*} 7 x^{2} + \frac {- x^{3} + 4 x}{27 x^{4} - e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 27, normalized size = 1.08 \begin {gather*} 7 \, x^{2} - \frac {x^{3} - 4 \, x}{27 \, x^{4} - e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.07, size = 23, normalized size = 0.92 \begin {gather*} 7\,x^2+\frac {x\,\left (x^2-4\right )}{{\mathrm {e}}^2-27\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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