3.87.31 \(\int e^{x-25 x^3+10 x^4-x^5} (1+x-75 x^3+40 x^4-5 x^5) \, dx\)

Optimal. Leaf size=18 \[ e^{x-(5-x)^2 x^3} x \]

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Rubi [B]  time = 0.10, antiderivative size = 56, normalized size of antiderivative = 3.11, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2288} \begin {gather*} \frac {e^{-x^5+10 x^4-25 x^3+x} \left (-5 x^5+40 x^4-75 x^3+x\right )}{-5 x^4+40 x^3-75 x^2+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(x - 25*x^3 + 10*x^4 - x^5)*(1 + x - 75*x^3 + 40*x^4 - 5*x^5),x]

[Out]

(E^(x - 25*x^3 + 10*x^4 - x^5)*(x - 75*x^3 + 40*x^4 - 5*x^5))/(1 - 75*x^2 + 40*x^3 - 5*x^4)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{x-25 x^3+10 x^4-x^5} \left (x-75 x^3+40 x^4-5 x^5\right )}{1-75 x^2+40 x^3-5 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 21, normalized size = 1.17 \begin {gather*} e^{x-25 x^3+10 x^4-x^5} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(x - 25*x^3 + 10*x^4 - x^5)*(1 + x - 75*x^3 + 40*x^4 - 5*x^5),x]

[Out]

E^(x - 25*x^3 + 10*x^4 - x^5)*x

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fricas [A]  time = 0.57, size = 20, normalized size = 1.11 \begin {gather*} e^{\left (-x^{5} + 10 \, x^{4} - 25 \, x^{3} + x + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^5+40*x^4-75*x^3+x+1)*exp(log(x)-x^5+10*x^4-25*x^3+x)/x,x, algorithm="fricas")

[Out]

e^(-x^5 + 10*x^4 - 25*x^3 + x + log(x))

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giac [A]  time = 0.13, size = 20, normalized size = 1.11 \begin {gather*} e^{\left (-x^{5} + 10 \, x^{4} - 25 \, x^{3} + x + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^5+40*x^4-75*x^3+x+1)*exp(log(x)-x^5+10*x^4-25*x^3+x)/x,x, algorithm="giac")

[Out]

e^(-x^5 + 10*x^4 - 25*x^3 + x + log(x))

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maple [A]  time = 0.05, size = 21, normalized size = 1.17

method	result	size
gosper	\({\mathrm e}^{\ln \relax (x )-x^{5}+10 x^{4}-25 x^{3}+x}\)	\(21\)
norman	\({\mathrm e}^{\ln \relax (x )-x^{5}+10 x^{4}-25 x^{3}+x}\)	\(21\)
risch	\(x \,{\mathrm e}^{-x \left (x^{2}-5 x +1\right ) \left (x^{2}-5 x -1\right )}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*x^5+40*x^4-75*x^3+x+1)*exp(ln(x)-x^5+10*x^4-25*x^3+x)/x,x,method=_RETURNVERBOSE)

[Out]

exp(ln(x)-x^5+10*x^4-25*x^3+x)

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maxima [A]  time = 0.42, size = 20, normalized size = 1.11 \begin {gather*} x e^{\left (-x^{5} + 10 \, x^{4} - 25 \, x^{3} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^5+40*x^4-75*x^3+x+1)*exp(log(x)-x^5+10*x^4-25*x^3+x)/x,x, algorithm="maxima")

[Out]

x*e^(-x^5 + 10*x^4 - 25*x^3 + x)

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mupad [B]  time = 5.25, size = 22, normalized size = 1.22 \begin {gather*} x\,{\mathrm {e}}^{-x^5}\,{\mathrm {e}}^{10\,x^4}\,{\mathrm {e}}^{-25\,x^3}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + log(x) - 25*x^3 + 10*x^4 - x^5)*(x - 75*x^3 + 40*x^4 - 5*x^5 + 1))/x,x)

[Out]

x*exp(-x^5)*exp(10*x^4)*exp(-25*x^3)*exp(x)

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sympy [A]  time = 0.12, size = 17, normalized size = 0.94 \begin {gather*} x e^{- x^{5} + 10 x^{4} - 25 x^{3} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x**5+40*x**4-75*x**3+x+1)*exp(ln(x)-x**5+10*x**4-25*x**3+x)/x,x)

[Out]

x*exp(-x**5 + 10*x**4 - 25*x**3 + x)

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