\(\int \frac {1}{4} (-4+e^{\frac {1}{4} (17 x-7 e^x x+2 x^2)} (17+e^x (-7-7 x)+4 x)) \, dx\) [4929]

Optimal result

Integrand size = 42, antiderivative size = 23 \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=e^{\frac {1}{4} x \left (10-7 \left (-1+e^x\right )+2 x\right )}-x \]

[Out]

exp(1/4*(17-7*exp(x)+2*x)*x)-x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 6838} \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=e^{\frac {1}{4} \left (2 x^2-7 e^x x+17 x\right )}-x \]

[In]

Int[(-4 + E^((17*x - 7*E^x*x + 2*x^2)/4)*(17 + E^x*(-7 - 7*x) + 4*x))/4,x]

[Out]

E^((17*x - 7*E^x*x + 2*x^2)/4) - x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx \\ & = -x+\frac {1}{4} \int e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right ) \, dx \\ & = e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )}-x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=e^{\frac {17 x}{4}-\frac {7 e^x x}{4}+\frac {x^2}{2}}-x \]

[In]

Integrate[(-4 + E^((17*x - 7*E^x*x + 2*x^2)/4)*(17 + E^x*(-7 - 7*x) + 4*x))/4,x]

[Out]

E^((17*x)/4 - (7*E^x*x)/4 + x^2/2) - x

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
risch	\(-x +{\mathrm e}^{-\frac {x \left (7 \,{\mathrm e}^{x}-2 x -17\right )}{4}}\)	\(18\)
parallelrisch	\(-x +{\mathrm e}^{-\frac {x \left (7 \,{\mathrm e}^{x}-2 x -17\right )}{4}}\)	\(18\)
default	\(-x +{\mathrm e}^{-\frac {7 \,{\mathrm e}^{x} x}{4}+\frac {x^{2}}{2}+\frac {17 x}{4}}\)	\(20\)
norman	\(-x +{\mathrm e}^{-\frac {7 \,{\mathrm e}^{x} x}{4}+\frac {x^{2}}{2}+\frac {17 x}{4}}\)	\(20\)

[In]

int(1/4*((-7*x-7)*exp(x)+4*x+17)*exp(-7/4*exp(x)*x+1/2*x^2+17/4*x)-1,x,method=_RETURNVERBOSE)

[Out]

-x+exp(-1/4*x*(7*exp(x)-2*x-17))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=-x + e^{\left (\frac {1}{2} \, x^{2} - \frac {7}{4} \, x e^{x} + \frac {17}{4} \, x\right )} \]

[In]

integrate(1/4*((-7*x-7)*exp(x)+4*x+17)*exp(-7/4*exp(x)*x+1/2*x^2+17/4*x)-1,x, algorithm="fricas")

[Out]

-x + e^(1/2*x^2 - 7/4*x*e^x + 17/4*x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=- x + e^{\frac {x^{2}}{2} - \frac {7 x e^{x}}{4} + \frac {17 x}{4}} \]

[In]

integrate(1/4*((-7*x-7)*exp(x)+4*x+17)*exp(-7/4*exp(x)*x+1/2*x**2+17/4*x)-1,x)

[Out]

-x + exp(x**2/2 - 7*x*exp(x)/4 + 17*x/4)

Maxima [F]

\[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=\int { -\frac {1}{4} \, {\left (7 \, {\left (x + 1\right )} e^{x} - 4 \, x - 17\right )} e^{\left (\frac {1}{2} \, x^{2} - \frac {7}{4} \, x e^{x} + \frac {17}{4} \, x\right )} - 1 \,d x } \]

[In]

integrate(1/4*((-7*x-7)*exp(x)+4*x+17)*exp(-7/4*exp(x)*x+1/2*x^2+17/4*x)-1,x, algorithm="maxima")

[Out]

-x - 1/4*integrate((7*(x + 1)*e^(21/4*x) - (4*x + 17)*e^(17/4*x))*e^(1/2*x^2 - 7/4*x*e^x), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx=-x + e^{\left (\frac {1}{2} \, x^{2} - \frac {7}{4} \, x e^{x} + \frac {17}{4} \, x\right )} \]

[In]

integrate(1/4*((-7*x-7)*exp(x)+4*x+17)*exp(-7/4*exp(x)*x+1/2*x^2+17/4*x)-1,x, algorithm="giac")

[Out]

-x + e^(1/2*x^2 - 7/4*x*e^x + 17/4*x)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{4} \left (-4+e^{\frac {1}{4} \left (17 x-7 e^x x+2 x^2\right )} \left (17+e^x (-7-7 x)+4 x\right )\right ) \, dx={\mathrm {e}}^{\frac {17\,x}{4}-\frac {7\,x\,{\mathrm {e}}^x}{4}+\frac {x^2}{2}}-x \]

[In]

int((exp((17*x)/4 - (7*x*exp(x))/4 + x^2/2)*(4*x - exp(x)*(7*x + 7) + 17))/4 - 1,x)

[Out]

exp((17*x)/4 - (7*x*exp(x))/4 + x^2/2) - x