Optimal. Leaf size=24 \[ 5+\frac {15 \left (-1+x+x^4\right )}{3+e^{x^{x/2}}+x} \]
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Rubi [F] time = 7.32, 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 {120+360 x^3+90 x^4+e^{x^{x/2}} \left (30+120 x^3+x^{x/2} \left (15-15 x-15 x^4+\left (15-15 x-15 x^4\right ) \log (x)\right )\right )}{18+2 e^{2 x^{x/2}}+12 x+2 x^2+e^{x^{x/2}} (12+4 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {120+360 x^3+90 x^4+e^{x^{x/2}} \left (30+120 x^3+x^{x/2} \left (15-15 x-15 x^4+\left (15-15 x-15 x^4\right ) \log (x)\right )\right )}{2 \left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {120+360 x^3+90 x^4+e^{x^{x/2}} \left (30+120 x^3+x^{x/2} \left (15-15 x-15 x^4+\left (15-15 x-15 x^4\right ) \log (x)\right )\right )}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {120}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {30 e^{x^{x/2}}}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {360 x^3}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {120 e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {90 x^4}{\left (3+e^{x^{x/2}}+x\right )^2}-\frac {15 e^{x^{x/2}} x^{x/2} \left (-1+x+x^4\right ) (1+\log (x))}{\left (3+e^{x^{x/2}}+x\right )^2}\right ) \, dx\\ &=-\left (\frac {15}{2} \int \frac {e^{x^{x/2}} x^{x/2} \left (-1+x+x^4\right ) (1+\log (x))}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\right )+15 \int \frac {e^{x^{x/2}}}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {1}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=-\left (15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x \left (-1+2 x+16 x^4\right ) (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=-\left (15 \operatorname {Subst}\left (\int \left (-\frac {2^x e^{2^x x^x} x^x (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{1+x} e^{2^x x^x} x^{1+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{4+x} e^{2^x x^x} x^{4+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \left (\frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^x e^{2^x x^x} x^x \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \left (\frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{1+x} e^{2^x x^x} x^{1+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \left (\frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{4+x} e^{2^x x^x} x^{4+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx}{x} \, dx,x,\frac {x}{2}\right )+15 \operatorname {Subst}\left (\int \frac {\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx}{x} \, dx,x,\frac {x}{2}\right )+15 \operatorname {Subst}\left (\int \frac {\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx}{x} \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+(15 \log (x)) \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-(15 \log (x)) \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-(15 \log (x)) \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} \frac {15 \left (-1+x+x^4\right )}{3+e^{x^{x/2}}+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 19, normalized size = 0.79 \begin {gather*} \frac {15 \, {\left (x^{4} + x - 1\right )}}{x + e^{\left (x^{\frac {1}{2} \, x}\right )} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 0.83
method | result | size |
risch | \(\frac {15 x^{4}+15 x -15}{x +3+{\mathrm e}^{x^{\frac {x}{2}}}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 19, normalized size = 0.79 \begin {gather*} \frac {15 \, {\left (x^{4} + x - 1\right )}}{x + e^{\left (x^{\frac {1}{2} \, x}\right )} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.03, size = 170, normalized size = 7.08 \begin {gather*} \frac {15\,\left (x^2\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}-3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}-2\,x+3\,x^4\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+x^5\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}-3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)-2\,x^4+2\,x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+x^2\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+3\,x^4\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+x^5\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+2\,x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+2\right )}{\left (x+{\mathrm {e}}^{{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}}+3\right )\,\left (3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 22, normalized size = 0.92 \begin {gather*} \frac {15 x^{4} + 15 x - 15}{x + e^{e^{\frac {x \log {\relax (x )}}{2}}} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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