3.43.27 \(\int (6-6 x+3 x^2-12 x^3+15 x^4-6 x^5+e^{-2+5 e^x} (2 x-6 x^2+4 x^3+e^x (5 x^2-10 x^3+5 x^4))) \, dx\)

Optimal. Leaf size=31 \[ (1-x)^2 \left (-3+x^2 \left (e^{-2+5 e^x}+(1-x) x\right )\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.58, number of steps used = 2, number of rules used = 1, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2288} \begin {gather*} -x^6+3 x^5-3 x^4+x^3-3 x^2+e^{5 e^x-2} \left (x^4-2 x^3+x^2\right )+6 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[6 - 6*x + 3*x^2 - 12*x^3 + 15*x^4 - 6*x^5 + E^(-2 + 5*E^x)*(2*x - 6*x^2 + 4*x^3 + E^x*(5*x^2 - 10*x^3 + 5*
x^4)),x]

[Out]

6*x - 3*x^2 + x^3 - 3*x^4 + 3*x^5 - x^6 + E^(-2 + 5*E^x)*(x^2 - 2*x^3 + 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} &=6 x-3 x^2+x^3-3 x^4+3 x^5-x^6+\int e^{-2+5 e^x} \left (2 x-6 x^2+4 x^3+e^x \left (5 x^2-10 x^3+5 x^4\right )\right ) \, dx\\ &=6 x-3 x^2+x^3-3 x^4+3 x^5-x^6+e^{-2+5 e^x} \left (x^2-2 x^3+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 49, normalized size = 1.58 \begin {gather*} -\frac {x \left (-e^{5 e^x} (-1+x)^2 x+e^2 \left (-6+3 x-x^2+3 x^3-3 x^4+x^5\right )\right )}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[6 - 6*x + 3*x^2 - 12*x^3 + 15*x^4 - 6*x^5 + E^(-2 + 5*E^x)*(2*x - 6*x^2 + 4*x^3 + E^x*(5*x^2 - 10*x^
3 + 5*x^4)),x]

[Out]

-((x*(-(E^(5*E^x)*(-1 + x)^2*x) + E^2*(-6 + 3*x - x^2 + 3*x^3 - 3*x^4 + x^5)))/E^2)

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fricas [A]  time = 0.69, size = 47, normalized size = 1.52 \begin {gather*} -x^{6} + 3 \, x^{5} - 3 \, x^{4} + x^{3} - 3 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (5 \, e^{x} - 2\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4-10*x^3+5*x^2)*exp(x)+4*x^3-6*x^2+2*x)*exp(5*exp(x)-2)-6*x^5+15*x^4-12*x^3+3*x^2-6*x+6,x, alg
orithm="fricas")

[Out]

-x^6 + 3*x^5 - 3*x^4 + x^3 - 3*x^2 + (x^4 - 2*x^3 + x^2)*e^(5*e^x - 2) + 6*x

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giac [B]  time = 0.19, size = 59, normalized size = 1.90 \begin {gather*} -x^{6} + 3 \, x^{5} - 3 \, x^{4} + x^{3} - 3 \, x^{2} + {\left (x^{4} e^{\left (5 \, e^{x}\right )} - 2 \, x^{3} e^{\left (5 \, e^{x}\right )} + x^{2} e^{\left (5 \, e^{x}\right )}\right )} e^{\left (-2\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4-10*x^3+5*x^2)*exp(x)+4*x^3-6*x^2+2*x)*exp(5*exp(x)-2)-6*x^5+15*x^4-12*x^3+3*x^2-6*x+6,x, alg
orithm="giac")

[Out]

-x^6 + 3*x^5 - 3*x^4 + x^3 - 3*x^2 + (x^4*e^(5*e^x) - 2*x^3*e^(5*e^x) + x^2*e^(5*e^x))*e^(-2) + 6*x

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maple [A]  time = 0.08, size = 47, normalized size = 1.52

method	result	size
risch	\(x^{2} \left (x^{2}-2 x +1\right ) {\mathrm e}^{5 \,{\mathrm e}^{x}-2}-x^{6}+3 x^{5}-3 x^{4}+x^{3}-3 x^{2}+6 x\)	\(47\)
default	\(6 x +{\mathrm e}^{5 \,{\mathrm e}^{x}-2} x^{2}+x^{4} {\mathrm e}^{5 \,{\mathrm e}^{x}-2}-2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}-2} x^{3}-3 x^{2}+x^{3}-3 x^{4}+3 x^{5}-x^{6}\)	\(62\)
norman	\(6 x +{\mathrm e}^{5 \,{\mathrm e}^{x}-2} x^{2}+x^{4} {\mathrm e}^{5 \,{\mathrm e}^{x}-2}-2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}-2} x^{3}-3 x^{2}+x^{3}-3 x^{4}+3 x^{5}-x^{6}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^4-10*x^3+5*x^2)*exp(x)+4*x^3-6*x^2+2*x)*exp(5*exp(x)-2)-6*x^5+15*x^4-12*x^3+3*x^2-6*x+6,x,method=_RE
TURNVERBOSE)

[Out]

x^2*(x^2-2*x+1)*exp(5*exp(x)-2)-x^6+3*x^5-3*x^4+x^3-3*x^2+6*x

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maxima [A]  time = 0.34, size = 47, normalized size = 1.52 \begin {gather*} -x^{6} + 3 \, x^{5} - 3 \, x^{4} + x^{3} - 3 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (5 \, e^{x} - 2\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4-10*x^3+5*x^2)*exp(x)+4*x^3-6*x^2+2*x)*exp(5*exp(x)-2)-6*x^5+15*x^4-12*x^3+3*x^2-6*x+6,x, alg
orithm="maxima")

[Out]

-x^6 + 3*x^5 - 3*x^4 + x^3 - 3*x^2 + (x^4 - 2*x^3 + x^2)*e^(5*e^x - 2) + 6*x

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mupad [B]  time = 0.10, size = 61, normalized size = 1.97 \begin {gather*} 6\,x+x^2\,{\mathrm {e}}^{5\,{\mathrm {e}}^x-2}-2\,x^3\,{\mathrm {e}}^{5\,{\mathrm {e}}^x-2}+x^4\,{\mathrm {e}}^{5\,{\mathrm {e}}^x-2}-3\,x^2+x^3-3\,x^4+3\,x^5-x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(5*exp(x) - 2)*(2*x + exp(x)*(5*x^2 - 10*x^3 + 5*x^4) - 6*x^2 + 4*x^3) - 6*x + 3*x^2 - 12*x^3 + 15*x^4
- 6*x^5 + 6,x)

[Out]

6*x + x^2*exp(5*exp(x) - 2) - 2*x^3*exp(5*exp(x) - 2) + x^4*exp(5*exp(x) - 2) - 3*x^2 + x^3 - 3*x^4 + 3*x^5 -
x^6

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sympy [A]  time = 0.19, size = 44, normalized size = 1.42 \begin {gather*} - x^{6} + 3 x^{5} - 3 x^{4} + x^{3} - 3 x^{2} + 6 x + \left (x^{4} - 2 x^{3} + x^{2}\right ) e^{5 e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**4-10*x**3+5*x**2)*exp(x)+4*x**3-6*x**2+2*x)*exp(5*exp(x)-2)-6*x**5+15*x**4-12*x**3+3*x**2-6*x
+6,x)

[Out]

-x**6 + 3*x**5 - 3*x**4 + x**3 - 3*x**2 + 6*x + (x**4 - 2*x**3 + x**2)*exp(5*exp(x) - 2)

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