Optimal. Leaf size=18 \[ e^{x+\frac {1}{3} (3-\log (x))} x \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.35, antiderivative size = 175, normalized size of antiderivative = 9.72, number of steps
used = 16, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {12, 2306,
6820, 6874, 2250, 2258, 2634, 15, 14, 6696} \begin {gather*} -\frac {9}{25} e x^{5/3} \, _2F_2\left (\frac {5}{3},\frac {5}{3};\frac {8}{3},\frac {8}{3};x\right )-\frac {3}{2} e x^{2/3} \, _2F_2\left (\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3};x\right )-\frac {e x^{2/3} \text {Gamma}\left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {e x^{5/3} \log (x) \text {Gamma}\left (\frac {5}{3},-x\right )}{(-x)^{5/3}}-\frac {2 e x^{2/3} \log (x) \text {Gamma}\left (\frac {2}{3},-x\right )}{3 (-x)^{2/3}}-\frac {3 e x^{2/3} \text {Gamma}\left (\frac {5}{3}\right ) \log \left (\sqrt [3]{x}\right )}{(-x)^{2/3}}+\frac {2 e x^{2/3} \text {Gamma}\left (\frac {2}{3}\right ) \log \left (\sqrt [3]{x}\right )}{(-x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 15
Rule 2250
Rule 2258
Rule 2306
Rule 2634
Rule 6696
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{\frac {1}{3} (3+3 x-\log (x))} (3+(2+3 x) \log (x)) \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{3} (3+3 x)} (3+(2+3 x) \log (x))}{\sqrt [3]{x}} \, dx\\ &=\frac {1}{3} \int \frac {e^{1+x} (3+(2+3 x) \log (x))}{\sqrt [3]{x}} \, dx\\ &=\text {Subst}\left (\int e^{1+x^3} x \left (3+\left (2+3 x^3\right ) \log \left (x^3\right )\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\text {Subst}\left (\int \left (3 e^{1+x^3} x+e^{1+x^3} x \left (2+3 x^3\right ) \log \left (x^3\right )\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int e^{1+x^3} x \, dx,x,\sqrt [3]{x}\right )+\text {Subst}\left (\int e^{1+x^3} x \left (2+3 x^3\right ) \log \left (x^3\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-\text {Subst}\left (\int \frac {e x \left (-2 \Gamma \left (\frac {2}{3},-x^3\right )+3 \Gamma \left (\frac {5}{3},-x^3\right )\right )}{\left (-x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-e \text {Subst}\left (\int \frac {x \left (-2 \Gamma \left (\frac {2}{3},-x^3\right )+3 \Gamma \left (\frac {5}{3},-x^3\right )\right )}{\left (-x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-\frac {\left (e x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \Gamma \left (\frac {2}{3},-x^3\right )+3 \Gamma \left (\frac {5}{3},-x^3\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}}\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}-\frac {\left (e x^{2/3}\right ) \text {Subst}\left (\int \left (-\frac {2 \Gamma \left (\frac {2}{3},-x^3\right )}{x}+\frac {3 \Gamma \left (\frac {5}{3},-x^3\right )}{x}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}}\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}+\frac {\left (2 e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {2}{3},-x^3\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}}-\frac {\left (3 e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {5}{3},-x^3\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-x)^{2/3}}\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}+\frac {\left (2 e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {2}{3},-x\right )}{x} \, dx,x,x\right )}{3 (-x)^{2/3}}-\frac {\left (e x^{2/3}\right ) \text {Subst}\left (\int \frac {\Gamma \left (\frac {5}{3},-x\right )}{x} \, dx,x,x\right )}{(-x)^{2/3}}\\ &=-\frac {e x^{2/3} \Gamma \left (\frac {2}{3},-x\right )}{(-x)^{2/3}}-\frac {3}{2} e x^{2/3} \, _2F_2\left (\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3};x\right )-\frac {9}{25} e x^{5/3} \, _2F_2\left (\frac {5}{3},\frac {5}{3};\frac {8}{3},\frac {8}{3};x\right )+\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3}\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{2/3} \Gamma \left (\frac {5}{3}\right ) \log (x)}{(-x)^{2/3}}-\frac {2 e x^{2/3} \Gamma \left (\frac {2}{3},-x\right ) \log (x)}{3 (-x)^{2/3}}-\frac {e x^{5/3} \Gamma \left (\frac {5}{3},-x\right ) \log (x)}{(-x)^{5/3}}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 13, normalized size = 0.72 \begin {gather*} e^{1+x} x^{2/3} \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 11, normalized size = 0.61
method | result | size |
default | \(x^{\frac {2}{3}} {\mathrm e}^{x +1} \ln \left (x \right )\) | \(11\) |
risch | \(x^{\frac {2}{3}} {\mathrm e}^{x +1} \ln \left (x \right )\) | \(11\) |
norman | \(x \,{\mathrm e}^{-\frac {\ln \left (x \right )}{3}+x +1} \ln \left (x \right )\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 12, normalized size = 0.67 \begin {gather*} x e^{\left (x - \frac {1}{3} \, \log \left (x\right ) + 1\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.21, size = 10, normalized size = 0.56 \begin {gather*} x^{2/3}\,\mathrm {e}\,{\mathrm {e}}^x\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________