3.2.6 \(\int \frac {1}{2} e^{3+x} (-12 x^2+x^4+e^4 (40 x+20 x^2)) \, dx\)

Optimal. Leaf size=33 \[ e^3+10 e^{3+x} x^2 \left (e^4+\frac {1}{5} \left (-x+\frac {x^2}{4}\right )\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {12, 2196, 2176, 2194} \begin {gather*} \frac {1}{2} e^{x+3} x^4-2 e^{x+3} x^3+10 e^{x+7} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3 + x)*(-12*x^2 + x^4 + E^4*(40*x + 20*x^2)))/2,x]

[Out]

10*E^(7 + x)*x^2 - 2*E^(3 + x)*x^3 + (E^(3 + x)*x^4)/2

Rule 12

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{3+x} \left (-12 x^2+x^4+e^4 \left (40 x+20 x^2\right )\right ) \, dx\\ &=\frac {1}{2} \int \left (-12 e^{3+x} x^2+e^{3+x} x^4+20 e^{7+x} x (2+x)\right ) \, dx\\ &=\frac {1}{2} \int e^{3+x} x^4 \, dx-6 \int e^{3+x} x^2 \, dx+10 \int e^{7+x} x (2+x) \, dx\\ &=-6 e^{3+x} x^2+\frac {1}{2} e^{3+x} x^4-2 \int e^{3+x} x^3 \, dx+10 \int \left (2 e^{7+x} x+e^{7+x} x^2\right ) \, dx+12 \int e^{3+x} x \, dx\\ &=12 e^{3+x} x-6 e^{3+x} x^2-2 e^{3+x} x^3+\frac {1}{2} e^{3+x} x^4+6 \int e^{3+x} x^2 \, dx+10 \int e^{7+x} x^2 \, dx-12 \int e^{3+x} \, dx+20 \int e^{7+x} x \, dx\\ &=-12 e^{3+x}+12 e^{3+x} x+20 e^{7+x} x+10 e^{7+x} x^2-2 e^{3+x} x^3+\frac {1}{2} e^{3+x} x^4-12 \int e^{3+x} x \, dx-20 \int e^{7+x} \, dx-20 \int e^{7+x} x \, dx\\ &=-12 e^{3+x}-20 e^{7+x}+10 e^{7+x} x^2-2 e^{3+x} x^3+\frac {1}{2} e^{3+x} x^4+12 \int e^{3+x} \, dx+20 \int e^{7+x} \, dx\\ &=10 e^{7+x} x^2-2 e^{3+x} x^3+\frac {1}{2} e^{3+x} x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 0.70 \begin {gather*} \frac {1}{2} e^{3+x} x^2 \left (20 e^4+(-4+x) x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 + x)*(-12*x^2 + x^4 + E^4*(40*x + 20*x^2)))/2,x]

[Out]

(E^(3 + x)*x^2*(20*E^4 + (-4 + x)*x))/2

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fricas [A]  time = 0.66, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, {\left (x^{4} - 4 \, x^{3} + 20 \, x^{2} e^{4}\right )} e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((20*x^2+40*x)*exp(4)+x^4-12*x^2)*exp(3+x),x, algorithm="fricas")

[Out]

1/2*(x^4 - 4*x^3 + 20*x^2*e^4)*e^(x + 3)

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giac [A]  time = 0.28, size = 25, normalized size = 0.76 \begin {gather*} 10 \, x^{2} e^{\left (x + 7\right )} + \frac {1}{2} \, {\left (x^{4} - 4 \, x^{3}\right )} e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((20*x^2+40*x)*exp(4)+x^4-12*x^2)*exp(3+x),x, algorithm="giac")

[Out]

10*x^2*e^(x + 7) + 1/2*(x^4 - 4*x^3)*e^(x + 3)

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maple [A]  time = 0.04, size = 21, normalized size = 0.64

method	result	size
gosper	\(\frac {{\mathrm e}^{3+x} \left (x^{2}+20 \,{\mathrm e}^{4}-4 x \right ) x^{2}}{2}\)	\(21\)
risch	\(\frac {\left (x^{4}-4 x^{3}+20 x^{2} {\mathrm e}^{4}\right ) {\mathrm e}^{3+x}}{2}\)	\(23\)
norman	\(-2 x^{3} {\mathrm e}^{3+x}+\frac {x^{4} {\mathrm e}^{3+x}}{2}+10 x^{2} {\mathrm e}^{4} {\mathrm e}^{3+x}\)	\(31\)
meijerg	\(-\left (10 \,{\mathrm e}^{4}-6\right ) {\mathrm e}^{3} \left (2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\right )+20 \,{\mathrm e}^{7} \left (1-\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}\right )-\frac {{\mathrm e}^{3} \left (24-\frac {\left (5 x^{4}-20 x^{3}+60 x^{2}-120 x +120\right ) {\mathrm e}^{x}}{5}\right )}{2}\)	\(73\)
derivativedivides	\(\frac {{\mathrm e}^{3+x} \left (3+x \right )^{4}}{2}-8 \,{\mathrm e}^{3+x} \left (3+x \right )^{3}+45 \,{\mathrm e}^{3+x} \left (3+x \right )^{2}-108 \,{\mathrm e}^{3+x} \left (3+x \right )+\frac {189 \,{\mathrm e}^{3+x}}{2}+30 \,{\mathrm e}^{3+x} {\mathrm e}^{4}-40 \,{\mathrm e}^{4} \left ({\mathrm e}^{3+x} \left (3+x \right )-{\mathrm e}^{3+x}\right )+10 \,{\mathrm e}^{4} \left ({\mathrm e}^{3+x} \left (3+x \right )^{2}-2 \,{\mathrm e}^{3+x} \left (3+x \right )+2 \,{\mathrm e}^{3+x}\right )\)	\(107\)
default	\(\frac {{\mathrm e}^{3+x} \left (3+x \right )^{4}}{2}-8 \,{\mathrm e}^{3+x} \left (3+x \right )^{3}+45 \,{\mathrm e}^{3+x} \left (3+x \right )^{2}-108 \,{\mathrm e}^{3+x} \left (3+x \right )+\frac {189 \,{\mathrm e}^{3+x}}{2}+30 \,{\mathrm e}^{3+x} {\mathrm e}^{4}-40 \,{\mathrm e}^{4} \left ({\mathrm e}^{3+x} \left (3+x \right )-{\mathrm e}^{3+x}\right )+10 \,{\mathrm e}^{4} \left ({\mathrm e}^{3+x} \left (3+x \right )^{2}-2 \,{\mathrm e}^{3+x} \left (3+x \right )+2 \,{\mathrm e}^{3+x}\right )\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((20*x^2+40*x)*exp(4)+x^4-12*x^2)*exp(3+x),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(3+x)*(x^2+20*exp(4)-4*x)*x^2

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maxima [B]  time = 0.45, size = 88, normalized size = 2.67 \begin {gather*} \frac {1}{2} \, {\left (x^{4} e^{3} - 4 \, x^{3} e^{3} + 12 \, x^{2} e^{3} - 24 \, x e^{3} + 24 \, e^{3}\right )} e^{x} + 10 \, {\left (x^{2} e^{7} - 2 \, x e^{7} + 2 \, e^{7}\right )} e^{x} - 6 \, {\left (x^{2} e^{3} - 2 \, x e^{3} + 2 \, e^{3}\right )} e^{x} + 20 \, {\left (x e^{7} - e^{7}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((20*x^2+40*x)*exp(4)+x^4-12*x^2)*exp(3+x),x, algorithm="maxima")

[Out]

1/2*(x^4*e^3 - 4*x^3*e^3 + 12*x^2*e^3 - 24*x*e^3 + 24*e^3)*e^x + 10*(x^2*e^7 - 2*x*e^7 + 2*e^7)*e^x - 6*(x^2*e
^3 - 2*x*e^3 + 2*e^3)*e^x + 20*(x*e^7 - e^7)*e^x

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mupad [B]  time = 0.06, size = 20, normalized size = 0.61 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{x+3}\,\left (x^2-4\,x+20\,{\mathrm {e}}^4\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 3)*(exp(4)*(40*x + 20*x^2) - 12*x^2 + x^4))/2,x)

[Out]

(x^2*exp(x + 3)*(20*exp(4) - 4*x + x^2))/2

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sympy [A]  time = 0.11, size = 22, normalized size = 0.67 \begin {gather*} \frac {\left (x^{4} - 4 x^{3} + 20 x^{2} e^{4}\right ) e^{x + 3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((20*x**2+40*x)*exp(4)+x**4-12*x**2)*exp(3+x),x)

[Out]

(x**4 - 4*x**3 + 20*x**2*exp(4))*exp(x + 3)/2

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