Integrand size = 47, antiderivative size = 19 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\frac {1}{2}+\frac {3}{25} \left (1+e^5\right ) x (1+x)} \] Output:
exp(1/2+3/25*(exp(5)+1)*(1+x)*x)
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\frac {1}{50} \left (25+6 \left (1+e^5\right ) x+6 \left (1+e^5\right ) x^2\right )} \] Input:
Integrate[(E^((25 + 6*x + 6*x^2 + E^5*(6*x + 6*x^2))/50)*(3 + 6*x + E^5*(3 + 6*x)))/25,x]
Output:
E^((25 + 6*(1 + E^5)*x + 6*(1 + E^5)*x^2)/50)
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {27, 27, 2674, 2666}
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.
\(\displaystyle \int \frac {1}{25} \left (6 x+e^5 (6 x+3)+3\right ) \exp \left (\frac {1}{50} \left (6 x^2+e^5 \left (6 x^2+6 x\right )+6 x+25\right )\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{25} \int 3 e^{\frac {1}{50} \left (6 x^2+6 x+6 e^5 \left (x^2+x\right )+25\right )} \left (2 x+e^5 (2 x+1)+1\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{25} \int e^{\frac {1}{50} \left (6 x^2+6 x+6 e^5 \left (x^2+x\right )+25\right )} \left (2 x+e^5 (2 x+1)+1\right )dx\) |
\(\Big \downarrow \) 2674 |
\(\displaystyle \frac {3}{25} \int e^{\frac {3}{25} \left (1+e^5\right ) x^2+\frac {3}{25} \left (1+e^5\right ) x+\frac {1}{2}} \left (2 \left (1+e^5\right ) x+e^5+1\right )dx\) |
\(\Big \downarrow \) 2666 |
\(\displaystyle e^{\frac {3}{25} \left (1+e^5\right ) x^2+\frac {3}{25} \left (1+e^5\right ) x+\frac {1}{2}}\) |
Input:
Int[(E^((25 + 6*x + 6*x^2 + E^5*(6*x + 6*x^2))/50)*(3 + 6*x + E^5*(3 + 6*x )))/25,x]
Output:
E^(1/2 + (3*(1 + E^5)*x)/25 + (3*(1 + E^5)*x^2)/25)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSu m[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !( LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26
method | result | size |
gosper | \({\mathrm e}^{\frac {3 x^{2} {\mathrm e}^{5}}{25}+\frac {3 x \,{\mathrm e}^{5}}{25}+\frac {3 x^{2}}{25}+\frac {3 x}{25}+\frac {1}{2}}\) | \(24\) |
risch | \({\mathrm e}^{\frac {3 x^{2} {\mathrm e}^{5}}{25}+\frac {3 x \,{\mathrm e}^{5}}{25}+\frac {3 x^{2}}{25}+\frac {3 x}{25}+\frac {1}{2}}\) | \(24\) |
norman | \({\mathrm e}^{\frac {\left (6 x^{2}+6 x \right ) {\mathrm e}^{5}}{50}+\frac {3 x^{2}}{25}+\frac {3 x}{25}+\frac {1}{2}}\) | \(25\) |
parallelrisch | \({\mathrm e}^{\frac {\left (6 x^{2}+6 x \right ) {\mathrm e}^{5}}{50}+\frac {3 x^{2}}{25}+\frac {3 x}{25}+\frac {1}{2}}\) | \(25\) |
orering | \(\frac {\left (\left (6 x +3\right ) {\mathrm e}^{5}+6 x +3\right ) {\mathrm e}^{\frac {\left (6 x^{2}+6 x \right ) {\mathrm e}^{5}}{50}+\frac {3 x^{2}}{25}+\frac {3 x}{25}+\frac {1}{2}}}{3 \left ({\mathrm e}^{5}+1\right ) \left (1+2 x \right )}\) | \(53\) |
default | \(-\frac {3 i {\mathrm e}^{\frac {1}{2}} \sqrt {\pi }\, {\mathrm e}^{-\frac {3}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{10 \sqrt {3 \,{\mathrm e}^{5}+3}}+\frac {6 \,{\mathrm e}^{\frac {1}{2}} \left (\frac {{\mathrm e}^{\left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right ) x^{2}+\left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right ) x}}{\frac {6 \,{\mathrm e}^{5}}{25}+\frac {6}{25}}+\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {3}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{4 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{25}-\frac {3 i {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{5} \sqrt {\pi }\, {\mathrm e}^{-\frac {3}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{10 \sqrt {3 \,{\mathrm e}^{5}+3}}+\frac {6 \,{\mathrm e}^{\frac {1}{2}} {\mathrm e}^{5} \left (\frac {{\mathrm e}^{\left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right ) x^{2}+\left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right ) x}}{\frac {6 \,{\mathrm e}^{5}}{25}+\frac {6}{25}}+\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {3}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{4 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{25}\) | \(288\) |
parts | \(-\frac {3 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right ) {\mathrm e}^{5} x}{5 \sqrt {3 \,{\mathrm e}^{5}+3}}-\frac {3 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right ) {\mathrm e}^{5}}{10 \sqrt {3 \,{\mathrm e}^{5}+3}}-\frac {3 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right ) x}{5 \sqrt {3 \,{\mathrm e}^{5}+3}}-\frac {3 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, x}{5}+\frac {5 i \left (\frac {3 \,{\mathrm e}^{5}}{25}+\frac {3}{25}\right )}{2 \sqrt {3 \,{\mathrm e}^{5}+3}}\right )}{10 \sqrt {3 \,{\mathrm e}^{5}+3}}+\frac {6 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} x \,\operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, \left (1+2 x \right )}{10}\right ) {\mathrm e}^{5}+3 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, \left (1+2 x \right )}{10}\right ) {\mathrm e}^{5}+6 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} x \,\operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, \left (1+2 x \right )}{10}\right )+3 i \sqrt {\pi }\, {\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \operatorname {erf}\left (\frac {i \sqrt {3 \,{\mathrm e}^{5}+3}\, \left (1+2 x \right )}{10}\right )+10 \,{\mathrm e}^{\frac {47}{100}-\frac {3 \,{\mathrm e}^{5}}{100}} \sqrt {3 \,{\mathrm e}^{5}+3}\, {\mathrm e}^{\frac {\left (3 \,{\mathrm e}^{5}+3\right ) \left (1+2 x \right )^{2}}{100}}}{10 \sqrt {3 \,{\mathrm e}^{5}+3}}\) | \(386\) |
Input:
int(1/25*((6*x+3)*exp(5)+6*x+3)*exp(1/50*(6*x^2+6*x)*exp(5)+3/25*x^2+3/25* x+1/2),x,method=_RETURNVERBOSE)
Output:
exp(3/25*x^2*exp(5)+3/25*x*exp(5)+3/25*x^2+3/25*x+1/2)
Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\left (\frac {3}{25} \, x^{2} + \frac {3}{25} \, {\left (x^{2} + x\right )} e^{5} + \frac {3}{25} \, x + \frac {1}{2}\right )} \] Input:
integrate(1/25*((6*x+3)*exp(5)+6*x+3)*exp(1/50*(6*x^2+6*x)*exp(5)+3/25*x^2 +3/25*x+1/2),x, algorithm="fricas")
Output:
e^(3/25*x^2 + 3/25*(x^2 + x)*e^5 + 3/25*x + 1/2)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\frac {3 x^{2}}{25} + \frac {3 x}{25} + \left (\frac {3 x^{2}}{25} + \frac {3 x}{25}\right ) e^{5} + \frac {1}{2}} \] Input:
integrate(1/25*((6*x+3)*exp(5)+6*x+3)*exp(1/50*(6*x**2+6*x)*exp(5)+3/25*x* *2+3/25*x+1/2),x)
Output:
exp(3*x**2/25 + 3*x/25 + (3*x**2/25 + 3*x/25)*exp(5) + 1/2)
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\left (\frac {3}{25} \, x^{2} + \frac {3}{25} \, {\left (x^{2} + x\right )} e^{5} + \frac {3}{25} \, x + \frac {1}{2}\right )} \] Input:
integrate(1/25*((6*x+3)*exp(5)+6*x+3)*exp(1/50*(6*x^2+6*x)*exp(5)+3/25*x^2 +3/25*x+1/2),x, algorithm="maxima")
Output:
e^(3/25*x^2 + 3/25*(x^2 + x)*e^5 + 3/25*x + 1/2)
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\left (\frac {3}{25} \, x^{2} e^{5} + \frac {3}{25} \, x^{2} + \frac {3}{25} \, x e^{5} + \frac {3}{25} \, x + \frac {1}{2}\right )} \] Input:
integrate(1/25*((6*x+3)*exp(5)+6*x+3)*exp(1/50*(6*x^2+6*x)*exp(5)+3/25*x^2 +3/25*x+1/2),x, algorithm="giac")
Output:
e^(3/25*x^2*e^5 + 3/25*x^2 + 3/25*x*e^5 + 3/25*x + 1/2)
Time = 3.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx={\mathrm {e}}^{\frac {3\,x^2\,{\mathrm {e}}^5}{25}}\,{\mathrm {e}}^{\frac {3\,x}{25}}\,\sqrt {\mathrm {e}}\,{\mathrm {e}}^{\frac {3\,x^2}{25}}\,{\mathrm {e}}^{\frac {3\,x\,{\mathrm {e}}^5}{25}} \] Input:
int((exp((3*x)/25 + (exp(5)*(6*x + 6*x^2))/50 + (3*x^2)/25 + 1/2)*(6*x + e xp(5)*(6*x + 3) + 3))/25,x)
Output:
exp((3*x^2*exp(5))/25)*exp((3*x)/25)*exp(1/2)*exp((3*x^2)/25)*exp((3*x*exp (5))/25)
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {1}{25} e^{\frac {1}{50} \left (25+6 x+6 x^2+e^5 \left (6 x+6 x^2\right )\right )} \left (3+6 x+e^5 (3+6 x)\right ) \, dx=e^{\frac {3}{25} e^{5} x^{2}+\frac {3}{25} e^{5} x +\frac {3}{25} x^{2}+\frac {3}{25} x +\frac {1}{2}} \] Input:
int(1/25*((6*x+3)*exp(5)+6*x+3)*exp(1/50*(6*x^2+6*x)*exp(5)+3/25*x^2+3/25* x+1/2),x)
Output:
e**((6*e**5*x**2 + 6*e**5*x + 6*x**2 + 6*x + 25)/50)