3.101.17 \(\int \frac {e^4+2 x^2+20 e^{20 x} x^2}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -1+e^{20 x}-\frac {e^4}{x}+2 x \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {14, 2194} \begin {gather*} 2 x+e^{20 x}-\frac {e^4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4 + 2*x^2 + 20*E^(20*x)*x^2)/x^2,x]

[Out]

E^(20*x) - E^4/x + 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 (20 e^{20 x}+\frac {e^4+2 x^2}{x^2}\right ) \, dx\\ &=20 \int e^{20 x} \, dx+\int \frac {e^4+2 x^2}{x^2} \, dx\\ &=e^{20 x}+\int \left (2+\frac {e^4}{x^2}\right ) \, dx\\ &=e^{20 x}-\frac {e^4}{x}+2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} e^{20 x}-\frac {e^4}{x}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4 + 2*x^2 + 20*E^(20*x)*x^2)/x^2,x]

[Out]

E^(20*x) - E^4/x + 2*x

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fricas [A]  time = 1.38, size = 20, normalized size = 1.11 \begin {gather*} \frac {2 \, x^{2} + x e^{\left (20 \, x\right )} - e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^2*exp(20*x)+exp(4)+2*x^2)/x^2,x, algorithm="fricas")

[Out]

(2*x^2 + x*e^(20*x) - e^4)/x

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giac [A]  time = 0.22, size = 20, normalized size = 1.11 \begin {gather*} \frac {2 \, x^{2} + x e^{\left (20 \, x\right )} - e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^2*exp(20*x)+exp(4)+2*x^2)/x^2,x, algorithm="giac")

[Out]

(2*x^2 + x*e^(20*x) - e^4)/x

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maple [A]  time = 0.04, size = 16, normalized size = 0.89

method	result	size
derivativedivides	\(2 x -\frac {{\mathrm e}^{4}}{x}+{\mathrm e}^{20 x}\)	\(16\)
default	\(2 x -\frac {{\mathrm e}^{4}}{x}+{\mathrm e}^{20 x}\)	\(16\)
risch	\(2 x -\frac {{\mathrm e}^{4}}{x}+{\mathrm e}^{20 x}\)	\(16\)
norman	\(\frac {x \,{\mathrm e}^{20 x}+2 x^{2}-{\mathrm e}^{4}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x^2*exp(20*x)+exp(4)+2*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 15, normalized size = 0.83 \begin {gather*} 2 \, x - \frac {e^{4}}{x} + e^{\left (20 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^2*exp(20*x)+exp(4)+2*x^2)/x^2,x, algorithm="maxima")

[Out]

2*x - e^4/x + e^(20*x)

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mupad [B]  time = 7.57, size = 15, normalized size = 0.83 \begin {gather*} 2\,x+{\mathrm {e}}^{20\,x}-\frac {{\mathrm {e}}^4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4) + 20*x^2*exp(20*x) + 2*x^2)/x^2,x)

[Out]

2*x + exp(20*x) - exp(4)/x

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sympy [A]  time = 0.11, size = 12, normalized size = 0.67 \begin {gather*} 2 x + e^{20 x} - \frac {e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x**2*exp(20*x)+exp(4)+2*x**2)/x**2,x)

[Out]

2*x + exp(20*x) - exp(4)/x

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