3.103.36 \(\int \frac {20-20 e^2+20 e x-5 x^2}{4 e^2-4 e x+x^2} \, dx\)

Optimal. Leaf size=22 \[ -4071+4 \left (\frac {5}{2 e-x}-x\right )-x \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {27, 683} \begin {gather*} \frac {20}{2 e-x}-5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 - 20*E^2 + 20*E*x - 5*x^2)/(4*E^2 - 4*E*x + x^2),x]

[Out]

20/(2*E - x) - 5*x

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20-20 e^2+20 e x-5 x^2}{(-2 e+x)^2} \, dx\\ &=\int \left (-5+\frac {20}{(2 e-x)^2}\right ) \, dx\\ &=\frac {20}{2 e-x}-5 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.59 \begin {gather*} -5 \left (x+\frac {4}{-2 e+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 - 20*E^2 + 20*E*x - 5*x^2)/(4*E^2 - 4*E*x + x^2),x]

[Out]

-5*(x + 4/(-2*E + x))

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fricas [A]  time = 0.62, size = 20, normalized size = 0.91 \begin {gather*} -\frac {5 \, {\left (x^{2} - 2 \, x e + 4\right )}}{x - 2 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(1)^2+20*x*exp(1)-5*x^2+20)/(4*exp(1)^2-4*x*exp(1)+x^2),x, algorithm="fricas")

[Out]

-5*(x^2 - 2*x*e + 4)/(x - 2*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(1)^2+20*x*exp(1)-5*x^2+20)/(4*exp(1)^2-4*x*exp(1)+x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -5*(sageVARx-8*1/4/sqrt(-exp(1)^2+exp(2)
)*atan((sageVARx-2*exp(1))*1/2/sqrt(-exp(1)^2+exp(2))))

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maple [A]  time = 0.14, size = 15, normalized size = 0.68

method	result	size
risch	\(-5 x +\frac {10}{{\mathrm e}-\frac {x}{2}}\)	\(15\)
norman	\(\frac {5 x^{2}+20-20 \,{\mathrm e}^{2}}{2 \,{\mathrm e}-x}\)	\(25\)
gosper	\(-\frac {5 \left (-x^{2}-4+4 \,{\mathrm e}^{2}\right )}{2 \,{\mathrm e}-x}\)	\(26\)
meijerg	\(-\frac {5 x}{1-\frac {{\mathrm e}^{-1} x}{2}}+\frac {5 \,{\mathrm e}^{-2} x}{1-\frac {{\mathrm e}^{-1} x}{2}}+20 \,{\mathrm e} \left (\frac {{\mathrm e}^{-1} x}{2-{\mathrm e}^{-1} x}+\ln \left (1-\frac {{\mathrm e}^{-1} x}{2}\right )\right )+10 \,{\mathrm e} \left (-\frac {x \,{\mathrm e}^{-1} \left (-\frac {3 \,{\mathrm e}^{-1} x}{2}+6\right )}{6 \left (1-\frac {{\mathrm e}^{-1} x}{2}\right )}-2 \ln \left (1-\frac {{\mathrm e}^{-1} x}{2}\right )\right )\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-20*exp(1)^2+20*x*exp(1)-5*x^2+20)/(4*exp(1)^2-4*x*exp(1)+x^2),x,method=_RETURNVERBOSE)

[Out]

-5*x+10/(exp(1)-1/2*x)

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maxima [A]  time = 0.36, size = 14, normalized size = 0.64 \begin {gather*} -5 \, x - \frac {20}{x - 2 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(1)^2+20*x*exp(1)-5*x^2+20)/(4*exp(1)^2-4*x*exp(1)+x^2),x, algorithm="maxima")

[Out]

-5*x - 20/(x - 2*e)

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mupad [B]  time = 0.08, size = 14, normalized size = 0.64 \begin {gather*} -5\,x-\frac {20}{x-2\,\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*exp(2) - 20*x*exp(1) + 5*x^2 - 20)/(4*exp(2) - 4*x*exp(1) + x^2),x)

[Out]

- 5*x - 20/(x - 2*exp(1))

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sympy [A]  time = 0.18, size = 12, normalized size = 0.55 \begin {gather*} - 5 x - \frac {20}{x - 2 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*exp(1)**2+20*x*exp(1)-5*x**2+20)/(4*exp(1)**2-4*x*exp(1)+x**2),x)

[Out]

-5*x - 20/(x - 2*E)

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