3.47.52 \(\int \frac {12 e^2+4 e^{6+x} x^2}{x^2} \, dx\)

Optimal. Leaf size=20 \[ 3-4 e^2 \left (-e^{4+x}+\frac {3}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14, 2194} \begin {gather*} 4 e^{x+6}-\frac {12 e^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12*E^2 + 4*E^(6 + x)*x^2)/x^2,x]

[Out]

4*E^(6 + x) - (12*E^2)/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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} 4 e^2 \left (e^{4+x}-\frac {3}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*E^2 + 4*E^(6 + x)*x^2)/x^2,x]

[Out]

4*E^2*(E^(4 + x) - 3/x)

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fricas [A]  time = 0.85, size = 16, normalized size = 0.80 \begin {gather*} \frac {4 \, {\left (x e^{\left (x + 6\right )} - 3 \, e^{2}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(2)*exp(4+x)+12*exp(2))/x^2,x, algorithm="fricas")

[Out]

4*(x*e^(x + 6) - 3*e^2)/x

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giac [A]  time = 0.11, size = 16, normalized size = 0.80 \begin {gather*} \frac {4 \, {\left (x e^{\left (x + 6\right )} - 3 \, e^{2}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(2)*exp(4+x)+12*exp(2))/x^2,x, algorithm="giac")

[Out]

4*(x*e^(x + 6) - 3*e^2)/x

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maple [A]  time = 0.04, size = 15, normalized size = 0.75

method	result	size
risch	\(-\frac {12 \,{\mathrm e}^{2}}{x}+4 \,{\mathrm e}^{x +6}\)	\(15\)
norman	\(\frac {4 x \,{\mathrm e}^{2} {\mathrm e}^{4+x}-12 \,{\mathrm e}^{2}}{x}\)	\(19\)
derivativedivides	\(4 \,{\mathrm e}^{2} \left ({\mathrm e}^{4+x}-24 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )-\frac {16 \,{\mathrm e}^{4+x}}{x}\right )-\frac {12 \,{\mathrm e}^{2}}{x}+64 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{4+x}}{x}-{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )\right )-32 \,{\mathrm e}^{2} \left (-5 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )-\frac {4 \,{\mathrm e}^{4+x}}{x}\right )\)	\(82\)
default	\(4 \,{\mathrm e}^{2} \left ({\mathrm e}^{4+x}-24 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )-\frac {16 \,{\mathrm e}^{4+x}}{x}\right )-\frac {12 \,{\mathrm e}^{2}}{x}+64 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{4+x}}{x}-{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )\right )-32 \,{\mathrm e}^{2} \left (-5 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x \right )-\frac {4 \,{\mathrm e}^{4+x}}{x}\right )\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*exp(2)*exp(4+x)+12*exp(2))/x^2,x,method=_RETURNVERBOSE)

[Out]

-12*exp(2)/x+4*exp(x+6)

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maxima [A]  time = 0.38, size = 14, normalized size = 0.70 \begin {gather*} -\frac {12 \, e^{2}}{x} + 4 \, e^{\left (x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(2)*exp(4+x)+12*exp(2))/x^2,x, algorithm="maxima")

[Out]

-12*e^2/x + 4*e^(x + 6)

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mupad [B]  time = 0.06, size = 14, normalized size = 0.70 \begin {gather*} 4\,{\mathrm {e}}^6\,{\mathrm {e}}^x-\frac {12\,{\mathrm {e}}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(2) + 4*x^2*exp(x + 4)*exp(2))/x^2,x)

[Out]

4*exp(6)*exp(x) - (12*exp(2))/x

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sympy [A]  time = 0.10, size = 15, normalized size = 0.75 \begin {gather*} 4 e^{2} e^{x + 4} - \frac {12 e^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2*exp(2)*exp(4+x)+12*exp(2))/x**2,x)

[Out]

4*exp(2)*exp(x + 4) - 12*exp(2)/x

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