3.101.52 \(\int \frac {e^{-x} (e^x (4-36 x^2)+e^{4+2 x} (x^2+x^3))}{4 x^2} \, dx\)

Optimal. Leaf size=29 \[ 1+\left (\frac {e^{4+x}}{4}-\left (4+\frac {1-x}{x}\right )^2\right ) x \]

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Rubi [A]  time = 0.13, antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 6688, 2176, 2194} \begin {gather*} -9 x-\frac {e^{x+4}}{4}+\frac {1}{4} e^{x+4} (x+1)-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(4 - 36*x^2) + E^(4 + 2*x)*(x^2 + x^3))/(4*E^x*x^2),x]

[Out]

-1/4*E^(4 + x) - x^(-1) - 9*x + (E^(4 + x)*(1 + x))/4

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{-x} \left (e^x \left (4-36 x^2\right )+e^{4+2 x} \left (x^2+x^3\right )\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (4 \left (-9+\frac {1}{x^2}\right )+e^{4+x} (1+x)\right ) \, dx\\ &=\frac {1}{4} \int e^{4+x} (1+x) \, dx+\int \left (-9+\frac {1}{x^2}\right ) \, dx\\ &=-\frac {1}{x}-9 x+\frac {1}{4} e^{4+x} (1+x)-\frac {1}{4} \int e^{4+x} \, dx\\ &=-\frac {e^{4+x}}{4}-\frac {1}{x}-9 x+\frac {1}{4} e^{4+x} (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.66 \begin {gather*} -\frac {1}{x}-9 x+\frac {1}{4} e^{4+x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(4 - 36*x^2) + E^(4 + 2*x)*(x^2 + x^3))/(4*E^x*x^2),x]

[Out]

-x^(-1) - 9*x + (E^(4 + x)*x)/4

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fricas [A]  time = 0.53, size = 20, normalized size = 0.69 \begin {gather*} \frac {x^{2} e^{\left (x + 4\right )} - 36 \, x^{2} - 4}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^3+x^2)*exp(2+x)^2+(-36*x^2+4)*exp(x))/exp(x)/x^2,x, algorithm="fricas")

[Out]

1/4*(x^2*e^(x + 4) - 36*x^2 - 4)/x

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giac [A]  time = 0.21, size = 20, normalized size = 0.69 \begin {gather*} \frac {x^{2} e^{\left (x + 4\right )} - 36 \, x^{2} - 4}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^3+x^2)*exp(2+x)^2+(-36*x^2+4)*exp(x))/exp(x)/x^2,x, algorithm="giac")

[Out]

1/4*(x^2*e^(x + 4) - 36*x^2 - 4)/x

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maple [A]  time = 0.04, size = 17, normalized size = 0.59

method	result	size
risch	\(-9 x -\frac {1}{x}+\frac {x \,{\mathrm e}^{4+x}}{4}\)	\(17\)
default	\(-9 x +\frac {{\mathrm e}^{4} {\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{4} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )}{4}-\frac {1}{x}\)	\(33\)
norman	\(\frac {\left (-9 \,{\mathrm e}^{x} x^{2}+\frac {{\mathrm e}^{4} {\mathrm e}^{2 x} x^{2}}{4}-{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((x^3+x^2)*exp(2+x)^2+(-36*x^2+4)*exp(x))/exp(x)/x^2,x,method=_RETURNVERBOSE)

[Out]

-9*x-1/x+1/4*x*exp(4+x)

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maxima [A]  time = 0.50, size = 28, normalized size = 0.97 \begin {gather*} \frac {1}{4} \, {\left (x e^{4} - e^{4}\right )} e^{x} - 9 \, x - \frac {1}{x} + \frac {1}{4} \, e^{\left (x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^3+x^2)*exp(2+x)^2+(-36*x^2+4)*exp(x))/exp(x)/x^2,x, algorithm="maxima")

[Out]

1/4*(x*e^4 - e^4)*e^x - 9*x - 1/x + 1/4*e^(x + 4)

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mupad [B]  time = 6.43, size = 16, normalized size = 0.55 \begin {gather*} x\,\left (\frac {{\mathrm {e}}^{x+4}}{4}-9\right )-\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*((exp(2*x + 4)*(x^2 + x^3))/4 - (exp(x)*(36*x^2 - 4))/4))/x^2,x)

[Out]

x*(exp(x + 4)/4 - 9) - 1/x

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sympy [A]  time = 0.15, size = 15, normalized size = 0.52 \begin {gather*} \frac {x e^{4} e^{x}}{4} - 9 x - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x**3+x**2)*exp(2+x)**2+(-36*x**2+4)*exp(x))/exp(x)/x**2,x)

[Out]

x*exp(4)*exp(x)/4 - 9*x - 1/x

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