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

Optimal. Leaf size=23 \[ e^{\frac {1}{4} x \left (10-7 \left (-1+e^x\right )+2 x\right )}-x \]

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Rubi [A]  time = 0.12, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{4} \left (2 x^2-7 e^x x+17 x\right )}-x \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 27, normalized size = 1.17 \begin {gather*} e^{\frac {17 x}{4}-\frac {7 e^x x}{4}+\frac {x^2}{2}}-x \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.61, size = 19, normalized size = 0.83 \begin {gather*} -x + e^{\left (\frac {1}{2} \, x^{2} - \frac {7}{4} \, x e^{x} + \frac {17}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.22, size = 19, normalized size = 0.83 \begin {gather*} -x + e^{\left (\frac {1}{2} \, x^{2} - \frac {7}{4} \, x e^{x} + \frac {17}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.06, size = 18, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x - \frac {1}{4} \, \int {\left (7 \, {\left (x + 1\right )} e^{\left (\frac {21}{4} \, x\right )} - {\left (4 \, x + 17\right )} e^{\left (\frac {17}{4} \, x\right )}\right )} e^{\left (\frac {1}{2} \, x^{2} - \frac {7}{4} \, x e^{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.10, size = 19, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^{\frac {17\,x}{4}-\frac {7\,x\,{\mathrm {e}}^x}{4}+\frac {x^2}{2}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.21, size = 20, normalized size = 0.87 \begin {gather*} - x + e^{\frac {x^{2}}{2} - \frac {7 x e^{x}}{4} + \frac {17 x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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