3.5.62 \(\int \frac {2 x+2 x^2+2 x^3+e (2+2 x)}{e+x^2} \, dx\)

Optimal. Leaf size=14 \[ 3+2 x+x^2+\log \left (e+x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1810, 260} \begin {gather*} x^2+\log \left (x^2+e\right )+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*x + x^2 + Log[E + x^2]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.50 \begin {gather*} 2 \left (x+\frac {x^2}{2}+\frac {1}{2} \log \left (e+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(x + x^2/2 + Log[E + x^2]/2)

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fricas [A]  time = 0.63, size = 14, normalized size = 1.00 \begin {gather*} x^{2} + 2 \, x + \log \left (x^{2} + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + 2*x + log(x^2 + e)

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giac [A]  time = 0.37, size = 14, normalized size = 1.00 \begin {gather*} x^{2} + 2 \, x + \log \left (x^{2} + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + 2*x + log(x^2 + e)

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maple [A]  time = 0.30, size = 15, normalized size = 1.07

method	result	size
default	\(x^{2}+2 x +\ln \left ({\mathrm e}+x^{2}\right )\)	\(15\)
norman	\(x^{2}+2 x +\ln \left ({\mathrm e}+x^{2}\right )\)	\(15\)
risch	\(x^{2}+2 x +\ln \left ({\mathrm e}+x^{2}\right )\)	\(15\)
meijerg	\(2 \,{\mathrm e}^{\frac {1}{2}} \arctan \left (x \,{\mathrm e}^{-\frac {1}{2}}\right )+\frac {\left (2 \,{\mathrm e}+2\right ) \ln \left (1+x^{2} {\mathrm e}^{-1}\right )}{2}+{\mathrm e} \left (x^{2} {\mathrm e}^{-1}-\ln \left (1+x^{2} {\mathrm e}^{-1}\right )\right )+{\mathrm e}^{\frac {1}{2}} \left (2 x \,{\mathrm e}^{-\frac {1}{2}}-2 \arctan \left (x \,{\mathrm e}^{-\frac {1}{2}}\right )\right )\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2+2*x+ln(exp(1)+x^2)

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maxima [A]  time = 0.35, size = 14, normalized size = 1.00 \begin {gather*} x^{2} + 2 \, x + \log \left (x^{2} + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + 2*x + log(x^2 + e)

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mupad [B]  time = 0.06, size = 14, normalized size = 1.00 \begin {gather*} 2\,x+\ln \left (x^2+\mathrm {e}\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 2*x^2 + 2*x^3 + exp(1)*(2*x + 2))/(exp(1) + x^2),x)

[Out]

2*x + log(exp(1) + x^2) + x^2

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sympy [A]  time = 0.10, size = 14, normalized size = 1.00 \begin {gather*} x^{2} + 2 x + \log {\left (x^{2} + e \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2 + 2*x + log(x**2 + E)

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