Integrand size = 44, antiderivative size = 31 \[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=\frac {4 e^{\frac {7 e^{\frac {e^6}{4}}}{3}+\frac {x}{5}} x}{1-x} \]
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=-\frac {4 e^{\frac {7 e^{\frac {e^6}{4}}}{3}+\frac {x}{5}} x}{-1+x} \]
Time = 0.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2700, 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.
\(\displaystyle \int \frac {e^{\frac {1}{15} \left (3 x+35 e^{\frac {e^6}{4}}\right )} \left (-4 x^2+4 x+20\right )}{5 x^2-10 x+5} \, dx\) |
\(\Big \downarrow \) 2700 |
\(\displaystyle \int \left (\frac {4 e^{\frac {1}{15} \left (3 x+35 e^{\frac {e^6}{4}}\right )} (6-x)}{5 x^2-10 x+5}-\frac {4}{5} e^{\frac {1}{15} \left (3 x+35 e^{\frac {e^6}{4}}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 e^{\frac {1}{15} \left (3 x+35 e^{\frac {e^6}{4}}\right )}}{1-x}-4 e^{\frac {1}{15} \left (3 x+35 e^{\frac {e^6}{4}}\right )}\) |
3.9.58.3.1 Defintions of rubi rules used
Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_ ) + (c_)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/( a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Polynomia lQ[u, x] && IntegerQ[m]
Time = 5.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {4 x \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-1+x}\) | \(21\) |
gosper | \(-\frac {4 x \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-1+x}\) | \(23\) |
norman | \(-\frac {4 x \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-1+x}\) | \(23\) |
parallelrisch | \(-\frac {4 x \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-1+x}\) | \(23\) |
derivativedivides | \(\frac {12 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-3 x +3}-\frac {4 \,{\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )}{5}+\frac {4 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}} \left (35 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}+3\right )}{5 \left (-3 x +3\right )}-180 \left (\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{675}+\frac {2}{375}\right ) {\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )-4 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}-\frac {4 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}} \left (1225 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{2}}+210 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}+9\right )}{15 \left (-3 x +3\right )}+4 \left (\frac {49 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{2}}}{9}+\frac {28 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{5}+\frac {11}{25}\right ) {\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )-420 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}} \left (\frac {{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-45 x +45}-\frac {{\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )}{225}\right )-4900 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{2}} \left (\frac {{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-45 x +45}-\frac {{\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )}{225}\right )+4200 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}} \left (\frac {{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}} \left (35 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}+3\right )}{-675 x +675}-\left (\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{675}+\frac {2}{375}\right ) {\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )\right )\) | \(476\) |
default | \(\frac {12 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-3 x +3}-\frac {4 \,{\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )}{5}+\frac {4 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}} \left (35 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}+3\right )}{5 \left (-3 x +3\right )}-180 \left (\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{675}+\frac {2}{375}\right ) {\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )-4 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}-\frac {4 \,{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}} \left (1225 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{2}}+210 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}+9\right )}{15 \left (-3 x +3\right )}+4 \left (\frac {49 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{2}}}{9}+\frac {28 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{5}+\frac {11}{25}\right ) {\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )-420 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}} \left (\frac {{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-45 x +45}-\frac {{\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )}{225}\right )-4900 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{2}} \left (\frac {{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}}}{-45 x +45}-\frac {{\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )}{225}\right )+4200 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}} \left (\frac {{\mathrm e}^{\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}+\frac {x}{5}} \left (35 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}+3\right )}{-675 x +675}-\left (\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{675}+\frac {2}{375}\right ) {\mathrm e}^{\frac {1}{5}+\frac {7 \,{\mathrm e}^{\frac {{\mathrm e}^{6}}{4}}}{3}} \operatorname {Ei}_{1}\left (-\frac {x}{5}+\frac {1}{5}\right )\right )\) | \(476\) |
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=-\frac {4 \, x e^{\left (\frac {1}{5} \, x + \frac {7}{3} \, e^{\left (\frac {1}{4} \, e^{6}\right )}\right )}}{x - 1} \]
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=- \frac {4 x e^{\frac {x}{5} + \frac {7 e^{\frac {e^{6}}{4}}}{3}}}{x - 1} \]
\[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=\int { -\frac {4 \, {\left (x^{2} - x - 5\right )} e^{\left (\frac {1}{5} \, x + \frac {7}{3} \, e^{\left (\frac {1}{4} \, e^{6}\right )}\right )}}{5 \, {\left (x^{2} - 2 \, x + 1\right )}} \,d x } \]
-4*x*e^(1/5*x + 7/3*e^(1/4*e^6))/(x - 1) - 4*e^(7/3*e^(1/4*e^6) + 1/5)*exp _integral_e(2, -1/5*x + 1/5)/(x - 1) - 4*integrate(e^(1/5*x + 7/3*e^(1/4*e ^6))/(x^2 - 2*x + 1), x)
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=-\frac {4 \, x e^{\left (\frac {1}{5} \, x + \frac {7}{3} \, e^{\left (\frac {1}{4} \, e^{6}\right )}\right )}}{x - 1} \]
Time = 9.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {1}{15} \left (35 e^{\frac {e^6}{4}}+3 x\right )} \left (20+4 x-4 x^2\right )}{5-10 x+5 x^2} \, dx=-\frac {20\,x\,{\mathrm {e}}^{x/5}\,{\mathrm {e}}^{\frac {7\,{\mathrm {e}}^{\frac {{\mathrm {e}}^6}{4}}}{3}}}{5\,x-5} \]