3.55.55 \(\int \frac {e^{4+x} (6-6 x)+2 e^{2 x} x^2+e^4 (3 x^2+2 x^3)}{e^4 x^2} \, dx\)

Optimal. Leaf size=24 \[ e^{-4+2 x}+(-1-x)^2-\frac {6 e^x}{x}+x \]

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 14, 2194, 2197} \begin {gather*} \frac {1}{4} (2 x+3)^2+e^{2 x-4}-\frac {6 e^x}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4 + x)*(6 - 6*x) + 2*E^(2*x)*x^2 + E^4*(3*x^2 + 2*x^3))/(E^4*x^2),x]

[Out]

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

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.92 \begin {gather*} e^{-4+2 x}-\frac {6 e^x}{x}+3 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 + x)*(6 - 6*x) + 2*E^(2*x)*x^2 + E^4*(3*x^2 + 2*x^3))/(E^4*x^2),x]

[Out]

E^(-4 + 2*x) - (6*E^x)/x + 3*x + x^2

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fricas [A]  time = 1.32, size = 33, normalized size = 1.38 \begin {gather*} \frac {{\left ({\left (x^{3} + 3 \, x^{2}\right )} e^{12} + x e^{\left (2 \, x + 8\right )} - 6 \, e^{\left (x + 12\right )}\right )} e^{\left (-12\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*x^2+(6-6*x)*exp(2)^2*exp(x)+(2*x^3+3*x^2)*exp(2)^2)/x^2/exp(2)^2,x, algorithm="fricas")

[Out]

((x^3 + 3*x^2)*e^12 + x*e^(2*x + 8) - 6*e^(x + 12))*e^(-12)/x

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giac [A]  time = 0.14, size = 32, normalized size = 1.33 \begin {gather*} \frac {{\left (x^{3} e^{4} + 3 \, x^{2} e^{4} + x e^{\left (2 \, x\right )} - 6 \, e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*x^2+(6-6*x)*exp(2)^2*exp(x)+(2*x^3+3*x^2)*exp(2)^2)/x^2/exp(2)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.11, size = 21, normalized size = 0.88

method	result	size
risch	\(x^{2}+3 x +{\mathrm e}^{2 x -4}-\frac {6 \,{\mathrm e}^{x}}{x}\)	\(21\)
norman	\(\frac {\left (x^{3} {\mathrm e}^{2}+x \,{\mathrm e}^{-2} {\mathrm e}^{2 x}+3 x^{2} {\mathrm e}^{2}-6 \,{\mathrm e}^{2} {\mathrm e}^{x}\right ) {\mathrm e}^{-2}}{x}\)	\(39\)
default	\({\mathrm e}^{-4} \left (3 x \,{\mathrm e}^{4}+{\mathrm e}^{2 x}+x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{x}}{x}-\expIntegralEi \left (1, -x \right )\right )+6 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )\right )\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)^2*x^2+(6-6*x)*exp(2)^2*exp(x)+(2*x^3+3*x^2)*exp(2)^2)/x^2/exp(2)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.40, size = 34, normalized size = 1.42 \begin {gather*} {\left (x^{2} e^{4} + 3 \, x e^{4} - 6 \, {\rm Ei}\relax (x) e^{4} + 6 \, e^{4} \Gamma \left (-1, -x\right ) + e^{\left (2 \, x\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*x^2+(6-6*x)*exp(2)^2*exp(x)+(2*x^3+3*x^2)*exp(2)^2)/x^2/exp(2)^2,x, algorithm="maxima")

[Out]

(x^2*e^4 + 3*x*e^4 - 6*Ei(x)*e^4 + 6*e^4*gamma(-1, -x) + e^(2*x))*e^(-4)

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mupad [B]  time = 0.08, size = 20, normalized size = 0.83 \begin {gather*} 3\,x+{\mathrm {e}}^{2\,x-4}-\frac {6\,{\mathrm {e}}^x}{x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4)*(2*x^2*exp(2*x) + exp(4)*(3*x^2 + 2*x^3) - exp(4)*exp(x)*(6*x - 6)))/x^2,x)

[Out]

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

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sympy [A]  time = 0.12, size = 26, normalized size = 1.08 \begin {gather*} x^{2} + 3 x + \frac {x e^{2 x} - 6 e^{4} e^{x}}{x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)**2*x**2+(6-6*x)*exp(2)**2*exp(x)+(2*x**3+3*x**2)*exp(2)**2)/x**2/exp(2)**2,x)

[Out]

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

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