3.43.53 \(\int \frac {4 x^2+24 x^4+e^{2 x} (-1+2 x)}{x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac {e^{2 x}}{x}+2 x+x \left (2+8 x^2\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14, 2197} \begin {gather*} 8 x^3+4 x+\frac {e^{2 x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^2 + 24*x^4 + E^(2*x)*(-1 + 2*x))/x^2,x]

[Out]

E^(2*x)/x + 4*x + 8*x^3

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 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} &=\int \left (\frac {e^{2 x} (-1+2 x)}{x^2}+4 \left (1+6 x^2\right )\right ) \, dx\\ &=4 \int \left (1+6 x^2\right ) \, dx+\int \frac {e^{2 x} (-1+2 x)}{x^2} \, dx\\ &=\frac {e^{2 x}}{x}+4 x+8 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.82 \begin {gather*} \frac {e^{2 x}}{x}+4 x+8 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 + 24*x^4 + E^(2*x)*(-1 + 2*x))/x^2,x]

[Out]

E^(2*x)/x + 4*x + 8*x^3

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fricas [A]  time = 0.55, size = 19, normalized size = 0.86 \begin {gather*} \frac {8 \, x^{4} + 4 \, x^{2} + e^{\left (2 \, x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(x)^2+24*x^4+4*x^2)/x^2,x, algorithm="fricas")

[Out]

(8*x^4 + 4*x^2 + e^(2*x))/x

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giac [A]  time = 0.14, size = 19, normalized size = 0.86 \begin {gather*} \frac {8 \, x^{4} + 4 \, x^{2} + e^{\left (2 \, x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(x)^2+24*x^4+4*x^2)/x^2,x, algorithm="giac")

[Out]

(8*x^4 + 4*x^2 + e^(2*x))/x

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maple [A]  time = 0.04, size = 18, normalized size = 0.82

method	result	size
default	\(8 x^{3}+4 x +\frac {{\mathrm e}^{2 x}}{x}\)	\(18\)
risch	\(8 x^{3}+4 x +\frac {{\mathrm e}^{2 x}}{x}\)	\(18\)
norman	\(\frac {{\mathrm e}^{2 x}+4 x^{2}+8 x^{4}}{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-1)*exp(x)^2+24*x^4+4*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

8*x^3+4*x+exp(x)^2/x

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maxima [C]  time = 0.36, size = 22, normalized size = 1.00 \begin {gather*} 8 \, x^{3} + 4 \, x + 2 \, {\rm Ei}\left (2 \, x\right ) - 2 \, \Gamma \left (-1, -2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(x)^2+24*x^4+4*x^2)/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} 4\,x\,\left (2\,x^2+1\right )+\frac {{\mathrm {e}}^{2\,x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(2*x - 1) + 4*x^2 + 24*x^4)/x^2,x)

[Out]

4*x*(2*x^2 + 1) + exp(2*x)/x

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sympy [A]  time = 0.08, size = 14, normalized size = 0.64 \begin {gather*} 8 x^{3} + 4 x + \frac {e^{2 x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp(x)**2+24*x**4+4*x**2)/x**2,x)

[Out]

8*x**3 + 4*x + exp(2*x)/x

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