Optimal. Leaf size=26 \[ \frac {e^5 x}{3 \left (5-\frac {5}{2 x}\right ) (3+3 x)} \]
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Rubi [A]
time = 0.10, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps
used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {12, 1607, 1694,
790} \begin {gather*} -\frac {8 e^5 (1-x)}{45 \left (9-(4 x+1)^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 790
Rule 1607
Rule 1694
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^5 \int \frac {-4 x+2 x^2}{45-90 x-135 x^2+180 x^3+180 x^4} \, dx\\ &=e^5 \int \frac {x (-4+2 x)}{45-90 x-135 x^2+180 x^3+180 x^4} \, dx\\ &=e^5 \text {Subst}\left (\int \frac {8 (1-4 x) (9-4 x)}{45 \left (9-16 x^2\right )^2} \, dx,x,\frac {1}{4}+x\right )\\ &=\frac {1}{45} \left (8 e^5\right ) \text {Subst}\left (\int \frac {(1-4 x) (9-4 x)}{\left (9-16 x^2\right )^2} \, dx,x,\frac {1}{4}+x\right )\\ &=-\frac {8 e^5 (1-x)}{45 \left (9-(1+4 x)^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.85 \begin {gather*} \frac {e^5 (1-x)}{45 \left (-1+x+2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 22, normalized size = 0.85
method | result | size |
gosper | \(-\frac {\left (x -1\right ) {\mathrm e}^{5}}{45 \left (2 x^{2}+x -1\right )}\) | \(18\) |
norman | \(\frac {2 \,{\mathrm e}^{5} x^{2}}{45 \left (2 x^{2}+x -1\right )}\) | \(18\) |
risch | \(\frac {{\mathrm e}^{5} \left (-\frac {x}{90}+\frac {1}{90}\right )}{x^{2}+\frac {1}{2} x -\frac {1}{2}}\) | \(19\) |
default | \(\frac {2 \,{\mathrm e}^{5} \left (-\frac {1}{3 \left (x +1\right )}+\frac {1}{12 x -6}\right )}{45}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 17, normalized size = 0.65 \begin {gather*} -\frac {{\left (x - 1\right )} e^{5}}{45 \, {\left (2 \, x^{2} + x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 17, normalized size = 0.65 \begin {gather*} -\frac {{\left (x - 1\right )} e^{5}}{45 \, {\left (2 \, x^{2} + x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 17, normalized size = 0.65 \begin {gather*} \frac {- x e^{5} + e^{5}}{90 x^{2} + 45 x - 45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 17, normalized size = 0.65 \begin {gather*} -\frac {{\left (x - 1\right )} e^{5}}{45 \, {\left (2 \, x^{2} + x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 22, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {e}}^5}{135\,\left (2\,x-1\right )}-\frac {2\,{\mathrm {e}}^5}{135\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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