Optimal. Leaf size=19 \[ \frac {6 \left (e^{e^x}-x\right ) x}{-5+\frac {1}{x}} \]
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Rubi [F]
time = 0.86, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-18 x^2+60 x^3+e^{e^x} \left (12 x-30 x^2+e^x \left (6 x^2-30 x^3\right )\right )}{1-10 x+25 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-18 x^2+60 x^3+e^{e^x} \left (12 x-30 x^2+e^x \left (6 x^2-30 x^3\right )\right )}{(-1+5 x)^2} \, dx\\ &=\int \left (\frac {12 e^{e^x} x}{(-1+5 x)^2}-\frac {18 x^2}{(-1+5 x)^2}-\frac {30 e^{e^x} x^2}{(-1+5 x)^2}+\frac {60 x^3}{(-1+5 x)^2}-\frac {6 e^{e^x+x} x^2}{-1+5 x}\right ) \, dx\\ &=-\left (6 \int \frac {e^{e^x+x} x^2}{-1+5 x} \, dx\right )+12 \int \frac {e^{e^x} x}{(-1+5 x)^2} \, dx-18 \int \frac {x^2}{(-1+5 x)^2} \, dx-30 \int \frac {e^{e^x} x^2}{(-1+5 x)^2} \, dx+60 \int \frac {x^3}{(-1+5 x)^2} \, dx\\ &=-\left (6 \int \left (\frac {e^{e^x+x}}{25}+\frac {1}{5} e^{e^x+x} x+\frac {e^{e^x+x}}{25 (-1+5 x)}\right ) \, dx\right )+12 \int \left (\frac {e^{e^x}}{5 (-1+5 x)^2}+\frac {e^{e^x}}{5 (-1+5 x)}\right ) \, dx-18 \int \left (\frac {1}{25}+\frac {1}{25 (-1+5 x)^2}+\frac {2}{25 (-1+5 x)}\right ) \, dx-30 \int \left (\frac {e^{e^x}}{25}+\frac {e^{e^x}}{25 (-1+5 x)^2}+\frac {2 e^{e^x}}{25 (-1+5 x)}\right ) \, dx+60 \int \left (\frac {2}{125}+\frac {x}{25}+\frac {1}{125 (-1+5 x)^2}+\frac {3}{125 (-1+5 x)}\right ) \, dx\\ &=-\frac {6}{125 (1-5 x)}+\frac {6 x}{25}+\frac {6 x^2}{5}-\frac {6}{25} \int e^{e^x+x} \, dx-\frac {6}{25} \int \frac {e^{e^x+x}}{-1+5 x} \, dx-\frac {6}{5} \int e^{e^x} \, dx-\frac {6}{5} \int e^{e^x+x} x \, dx-\frac {6}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx+\frac {12}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx\\ &=-\frac {6}{125 (1-5 x)}+\frac {6 x}{25}+\frac {6 x^2}{5}-\frac {6}{25} \int \frac {e^{e^x+x}}{-1+5 x} \, dx-\frac {6}{25} \text {Subst}\left (\int e^x \, dx,x,e^x\right )-\frac {6}{5} \int e^{e^x+x} x \, dx-\frac {6}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx-\frac {6}{5} \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )+\frac {12}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx\\ &=-\frac {6 e^{e^x}}{25}-\frac {6}{125 (1-5 x)}+\frac {6 x}{25}+\frac {6 x^2}{5}-\frac {6 \text {Ei}\left (e^x\right )}{5}-\frac {6}{25} \int \frac {e^{e^x+x}}{-1+5 x} \, dx-\frac {6}{5} \int e^{e^x+x} x \, dx-\frac {6}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx+\frac {12}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 29, normalized size = 1.53 \begin {gather*} -\frac {6 \left (3-15 x-125 e^{e^x} x^2+125 x^3\right )}{125-625 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 23, normalized size = 1.21
method | result | size |
norman | \(\frac {6 x^{3}-6 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}}{5 x -1}\) | \(23\) |
risch | \(\frac {6 x^{2}}{5}+\frac {6 x}{25}+\frac {6}{625 \left (x -\frac {1}{5}\right )}-\frac {6 x^{2} {\mathrm e}^{{\mathrm e}^{x}}}{5 x -1}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 33, normalized size = 1.74 \begin {gather*} \frac {6}{5} \, x^{2} - \frac {6 \, x^{2} e^{\left (e^{x}\right )}}{5 \, x - 1} + \frac {6}{25} \, x + \frac {6}{125 \, {\left (5 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 27, normalized size = 1.42 \begin {gather*} \frac {6 \, {\left (125 \, x^{3} - 125 \, x^{2} e^{\left (e^{x}\right )} - 5 \, x + 1\right )}}{125 \, {\left (5 \, x - 1\right )}} \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. 32 vs.
\(2 (15) = 30\).
time = 0.08, size = 32, normalized size = 1.68 \begin {gather*} \frac {6 x^{2}}{5} - \frac {6 x^{2} e^{e^{x}}}{5 x - 1} + \frac {6 x}{25} + \frac {6}{625 x - 125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (17) = 34\).
time = 0.43, size = 39, normalized size = 2.05 \begin {gather*} \frac {6 \, {\left (125 \, x^{3} e^{x} - 125 \, x^{2} e^{\left (x + e^{x}\right )} - 5 \, x e^{x} + e^{x}\right )}}{125 \, {\left (5 \, x e^{x} - e^{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.83, size = 19, normalized size = 1.00 \begin {gather*} \frac {6\,x^2\,\left (x-{\mathrm {e}}^{{\mathrm {e}}^x}\right )}{5\,x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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