Optimal. Leaf size=28 \[ 3+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \]
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Rubi [A] time = 1.18, antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 25, number of rules used = 10, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6688, 12, 6742, 2232, 2231, 2230, 2210, 2233, 2178, 2209} \begin {gather*} \frac {e^{\frac {x+4}{3 x+5}} \left (3+\log ^2(\log (4))\right )}{x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2209
Rule 2210
Rule 2230
Rule 2231
Rule 2232
Rule 2233
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {4+x}{5+3 x}} \left (46+37 x+9 x^2\right ) \left (-3-\log ^2(\log (4))\right )}{\left (15+14 x+3 x^2\right )^2} \, dx\\ &=\left (-3-\log ^2(\log (4))\right ) \int \frac {e^{\frac {4+x}{5+3 x}} \left (46+37 x+9 x^2\right )}{\left (15+14 x+3 x^2\right )^2} \, dx\\ &=\left (-3-\log ^2(\log (4))\right ) \int \left (\frac {e^{\frac {4+x}{5+3 x}}}{(3+x)^2}+\frac {7 e^{\frac {4+x}{5+3 x}}}{16 (3+x)}+\frac {21 e^{\frac {4+x}{5+3 x}}}{4 (5+3 x)^2}-\frac {21 e^{\frac {4+x}{5+3 x}}}{16 (5+3 x)}\right ) \, dx\\ &=\left (-3-\log ^2(\log (4))\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x)^2} \, dx-\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{3+x} \, dx+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx-\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(5+3 x)^2} \, dx\\ &=\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx+\frac {1}{4} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x) (5+3 x)} \, dx-\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{(5+3 x)^2} \, dx+\left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x) (5+3 x)^2} \, dx\\ &=\frac {3}{4} e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}} \left (3+\log ^2(\log (4))\right )+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {7}{16} \sqrt [3]{e} \text {Ei}\left (\frac {7}{3 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )-\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \operatorname {Subst}\left (\int \frac {e^{-\frac {1}{4}+\frac {7 x}{4}}}{x} \, dx,x,\frac {3+x}{5+3 x}\right )-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx+\left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \left (\frac {e^{\frac {4+x}{5+3 x}}}{16 (3+x)}+\frac {3 e^{\frac {4+x}{5+3 x}}}{4 (5+3 x)^2}-\frac {3 e^{\frac {4+x}{5+3 x}}}{16 (5+3 x)}\right ) \, dx\\ &=\frac {3}{4} e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}} \left (3+\log ^2(\log (4))\right )+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {7 \text {Ei}\left (\frac {7 (3+x)}{4 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )}{16 \sqrt [4]{e}}+\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{3+x} \, dx-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx+\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(5+3 x)^2} \, dx\\ &=\frac {3}{4} e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}} \left (3+\log ^2(\log (4))\right )+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}-\frac {7 \text {Ei}\left (\frac {7 (3+x)}{4 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )}{16 \sqrt [4]{e}}+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{5+3 x} \, dx-\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx-\frac {1}{4} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {4+x}{5+3 x}}}{(3+x) (5+3 x)} \, dx+\frac {1}{4} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{(5+3 x)^2} \, dx\\ &=\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}+\frac {7}{16} \sqrt [3]{e} \text {Ei}\left (\frac {7}{3 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )-\frac {7 \text {Ei}\left (\frac {7 (3+x)}{4 (5+3 x)}\right ) \left (3+\log ^2(\log (4))\right )}{16 \sqrt [4]{e}}+\frac {1}{16} \left (7 \left (3+\log ^2(\log (4))\right )\right ) \operatorname {Subst}\left (\int \frac {e^{-\frac {1}{4}+\frac {7 x}{4}}}{x} \, dx,x,\frac {3+x}{5+3 x}\right )+\frac {1}{16} \left (21 \left (3+\log ^2(\log (4))\right )\right ) \int \frac {e^{\frac {1}{3}+\frac {7}{3 (5+3 x)}}}{5+3 x} \, dx\\ &=\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 26, normalized size = 0.93 \begin {gather*} \frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 41, normalized size = 1.46 \begin {gather*} \frac {e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2 \, \log \relax (2)\right )^{2} + 3 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.22, size = 198, normalized size = 7.07 \begin {gather*} \frac {\frac {3 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \relax (2)^{2}}{3 \, x + 5} - e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \relax (2)^{2} + \frac {6 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \relax (2) \log \left (\log \relax (2)\right )}{3 \, x + 5} - 2 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \relax (2) \log \left (\log \relax (2)\right ) + \frac {3 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (\log \relax (2)\right )^{2}}{3 \, x + 5} - e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (\log \relax (2)\right )^{2} + \frac {9 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{3 \, x + 5} - 3 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{\frac {4 \, {\left (x + 4\right )}}{3 \, x + 5} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 28, normalized size = 1.00
method | result | size |
gosper | \(\frac {\left (\ln \left (2 \ln \relax (2)\right )^{2}+3\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}}{3+x}\) | \(28\) |
risch | \(\frac {\left (\ln \relax (2)^{2}+2 \ln \relax (2) \ln \left (\ln \relax (2)\right )+\ln \left (\ln \relax (2)\right )^{2}+3\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}}{3+x}\) | \(37\) |
norman | \(\frac {\left (15+10 \ln \relax (2) \ln \left (\ln \relax (2)\right )+5 \ln \relax (2)^{2}+5 \ln \left (\ln \relax (2)\right )^{2}\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}+\left (9+6 \ln \relax (2) \ln \left (\ln \relax (2)\right )+3 \ln \relax (2)^{2}+3 \ln \left (\ln \relax (2)\right )^{2}\right ) x \,{\mathrm e}^{\frac {4+x}{3 x +5}}}{3 x^{2}+14 x +15}\) | \(86\) |
derivativedivides | \(\frac {9 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{4}-\frac {21 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {3 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}} \ln \relax (2)^{2}}{4}-\frac {7 \ln \relax (2)^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {3 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}} \ln \left (\ln \relax (2)\right )^{2}}{4}-\frac {7 \ln \left (\ln \relax (2)\right )^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {3 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}} \ln \relax (2) \ln \left (\ln \relax (2)\right )}{2}-\frac {7 \ln \relax (2) \ln \left (\ln \relax (2)\right ) {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{8 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}\) | \(194\) |
default | \(\frac {9 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{4}-\frac {21 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {3 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}} \ln \relax (2)^{2}}{4}-\frac {7 \ln \relax (2)^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {3 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}} \ln \left (\ln \relax (2)\right )^{2}}{4}-\frac {7 \ln \left (\ln \relax (2)\right )^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{16 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {3 \,{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}} \ln \relax (2) \ln \left (\ln \relax (2)\right )}{2}-\frac {7 \ln \relax (2) \ln \left (\ln \relax (2)\right ) {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{8 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 36, normalized size = 1.29 \begin {gather*} \frac {{\left (\log \relax (2)^{2} + 2 \, \log \relax (2) \log \left (\log \relax (2)\right ) + \log \left (\log \relax (2)\right )^{2} + 3\right )} e^{\left (\frac {7}{3 \, {\left (3 \, x + 5\right )}} + \frac {1}{3}\right )}}{x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x+4}{3\,x+5}}\,\left ({\ln \left (\ln \relax (4)\right )}^2+3\right )}{x+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 36, normalized size = 1.29 \begin {gather*} \frac {\left (2 \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )} + \log {\left (\log {\relax (2 )} \right )}^{2} + \log {\relax (2 )}^{2} + 3\right ) e^{\frac {x + 4}{3 x + 5}}}{x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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