Integrand size = 77, antiderivative size = 28 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=3+\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \]
Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {e^{\frac {4+x}{5+3 x}} \left (3+\log ^2(\log (4))\right )}{3+x} \]
Integrate[(E^((4 + x)/(5 + 3*x))*(-138 - 111*x - 27*x^2) + E^((4 + x)/(5 + 3*x))*(-46 - 37*x - 9*x^2)*Log[Log[4]]^2)/(225 + 420*x + 286*x^2 + 84*x^3 + 9*x^4),x]
Time = 1.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2463, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{\frac {x+4}{3 x+5}} \left (-27 x^2-111 x-138\right )+e^{\frac {x+4}{3 x+5}} \left (-9 x^2-37 x-46\right ) \log ^2(\log (4))}{9 x^4+84 x^3+286 x^2+420 x+225} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {3 \left (e^{\frac {x+4}{3 x+5}} \left (-27 x^2-111 x-138\right )+e^{\frac {x+4}{3 x+5}} \left (-9 x^2-37 x-46\right ) \log ^2(\log (4))\right )}{32 (x+3)}-\frac {9 \left (e^{\frac {x+4}{3 x+5}} \left (-27 x^2-111 x-138\right )+e^{\frac {x+4}{3 x+5}} \left (-9 x^2-37 x-46\right ) \log ^2(\log (4))\right )}{32 (3 x+5)}+\frac {e^{\frac {x+4}{3 x+5}} \left (-27 x^2-111 x-138\right )+e^{\frac {x+4}{3 x+5}} \left (-9 x^2-37 x-46\right ) \log ^2(\log (4))}{16 (x+3)^2}+\frac {9 \left (e^{\frac {x+4}{3 x+5}} \left (-27 x^2-111 x-138\right )+e^{\frac {x+4}{3 x+5}} \left (-9 x^2-37 x-46\right ) \log ^2(\log (4))\right )}{16 (3 x+5)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^{\frac {x+4}{3 x+5}} \left (3+\log ^2(\log (4))\right )}{x+3}\) |
Int[(E^((4 + x)/(5 + 3*x))*(-138 - 111*x - 27*x^2) + E^((4 + x)/(5 + 3*x)) *(-46 - 37*x - 9*x^2)*Log[Log[4]]^2)/(225 + 420*x + 286*x^2 + 84*x^3 + 9*x ^4),x]
3.23.5.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
| method | result | size |
| gosper | \(\frac {\left (\ln \left (2 \ln \left (2\right )\right )^{2}+3\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}}{3+x}\) | \(28\) |
| risch | \(\frac {\left (\ln \left (2\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}+3\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}}{3+x}\) | \(37\) |
| parallelrisch | \(\frac {81 \,{\mathrm e}^{\frac {4+x}{3 x +5}} \ln \left (2 \ln \left (2\right )\right )^{2}+243 \,{\mathrm e}^{\frac {4+x}{3 x +5}}}{243+81 x}\) | \(44\) |
| norman | \(\frac {\left (15+10 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+5 \ln \left (2\right )^{2}+5 \ln \left (\ln \left (2\right )\right )^{2}\right ) {\mathrm e}^{\frac {4+x}{3 x +5}}+\left (9+6 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+3 \ln \left (2\right )^{2}+3 \ln \left (\ln \left (2\right )\right )^{2}\right ) x \,{\mathrm e}^{\frac {4+x}{3 x +5}}}{3 x^{2}+14 x +15}\) | \(86\) |
| 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 )}+588 \ln \left (2\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+588 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )-\frac {35 \ln \left (2\right )^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{192 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {119 \ln \left (2\right )^{2} {\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{192}+3087 \ln \left (\ln \left (2\right )\right )^{2} \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+1029 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )+1176 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+6174 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+2058 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )\) | \(500\) |
| 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 )}+588 \ln \left (2\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+588 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )-\frac {35 \ln \left (2\right )^{2} {\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{192 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {119 \ln \left (2\right )^{2} {\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{192}+3087 \ln \left (\ln \left (2\right )\right )^{2} \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+1029 \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )+1176 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{784}-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{2304 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}+\frac {17 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{16128}\right )+6174 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (-\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{7056 \left (\frac {7}{3 \left (3 x +5\right )}+\frac {7}{12}\right )}-\frac {{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{7056}\right )+2058 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) \left (\frac {{\mathrm e}^{\frac {1}{3}+\frac {7}{3 \left (3 x +5\right )}}}{\frac {9408}{3 x +5}+2352}-\frac {5 \,{\mathrm e}^{-\frac {1}{4}} \operatorname {Ei}_{1}\left (-\frac {7}{3 \left (3 x +5\right )}-\frac {7}{12}\right )}{28224}\right )\) | \(500\) |
int(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*ln(2*ln(2))^2+(-27*x^2-111*x-138) *exp((4+x)/(3*x+5)))/(9*x^4+84*x^3+286*x^2+420*x+225),x,method=_RETURNVERB OSE)
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2 \, \log \left (2\right )\right )^{2} + 3 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{x + 3} \]
integrate(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*log(2*log(2))^2+(-27*x^2-11 1*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+84*x^3+286*x^2+420*x+225),x, algorithm =\
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {\left (2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \log {\left (\log {\left (2 \right )} \right )}^{2} + \log {\left (2 \right )}^{2} + 3\right ) e^{\frac {x + 4}{3 x + 5}}}{x + 3} \]
integrate(((-9*x**2-37*x-46)*exp((4+x)/(3*x+5))*ln(2*ln(2))**2+(-27*x**2-1 11*x-138)*exp((4+x)/(3*x+5)))/(9*x**4+84*x**3+286*x**2+420*x+225),x)
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {{\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2} + 3\right )} e^{\left (\frac {7}{3 \, {\left (3 \, x + 5\right )}} + \frac {1}{3}\right )}}{x + 3} \]
integrate(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*log(2*log(2))^2+(-27*x^2-11 1*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+84*x^3+286*x^2+420*x+225),x, algorithm =\
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (29) = 58\).
Time = 0.56 (sec) , antiderivative size = 198, normalized size of antiderivative = 7.07 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {\frac {3 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right )^{2}}{3 \, x + 5} - e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right )^{2} + \frac {6 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right ) \log \left (\log \left (2\right )\right )}{3 \, x + 5} - 2 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \frac {3 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (\log \left (2\right )\right )^{2}}{3 \, x + 5} - e^{\left (\frac {x + 4}{3 \, x + 5}\right )} \log \left (\log \left (2\right )\right )^{2} + \frac {9 \, {\left (x + 4\right )} e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{3 \, x + 5} - 3 \, e^{\left (\frac {x + 4}{3 \, x + 5}\right )}}{\frac {4 \, {\left (x + 4\right )}}{3 \, x + 5} + 1} \]
integrate(((-9*x^2-37*x-46)*exp((4+x)/(3*x+5))*log(2*log(2))^2+(-27*x^2-11 1*x-138)*exp((4+x)/(3*x+5)))/(9*x^4+84*x^3+286*x^2+420*x+225),x, algorithm =\
(3*(x + 4)*e^((x + 4)/(3*x + 5))*log(2)^2/(3*x + 5) - e^((x + 4)/(3*x + 5) )*log(2)^2 + 6*(x + 4)*e^((x + 4)/(3*x + 5))*log(2)*log(log(2))/(3*x + 5) - 2*e^((x + 4)/(3*x + 5))*log(2)*log(log(2)) + 3*(x + 4)*e^((x + 4)/(3*x + 5))*log(log(2))^2/(3*x + 5) - e^((x + 4)/(3*x + 5))*log(log(2))^2 + 9*(x + 4)*e^((x + 4)/(3*x + 5))/(3*x + 5) - 3*e^((x + 4)/(3*x + 5)))/(4*(x + 4) /(3*x + 5) + 1)
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {4+x}{5+3 x}} \left (-138-111 x-27 x^2\right )+e^{\frac {4+x}{5+3 x}} \left (-46-37 x-9 x^2\right ) \log ^2(\log (4))}{225+420 x+286 x^2+84 x^3+9 x^4} \, dx=\frac {{\mathrm {e}}^{\frac {x+4}{3\,x+5}}\,\left ({\ln \left (\ln \left (4\right )\right )}^2+3\right )}{x+3} \]