3.57.21 \(\int \frac {20+34 e^2-68 e x+34 x^2}{5 e^2-10 e x+5 x^2} \, dx\)

Optimal. Leaf size=16 \[ -1+\frac {4}{e-x}+\frac {34 x}{5} \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {27, 12, 683} \begin {gather*} \frac {34 x}{5}+\frac {4}{e-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 + 34*E^2 - 68*E*x + 34*x^2)/(5*E^2 - 10*E*x + 5*x^2),x]

[Out]

4/(E - x) + (34*x)/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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+34 e^2-68 e x+34 x^2}{5 (e-x)^2} \, dx\\ &=\frac {1}{5} \int \frac {20+34 e^2-68 e x+34 x^2}{(e-x)^2} \, dx\\ &=\frac {1}{5} \int \left (34+\frac {20}{(e-x)^2}\right ) \, dx\\ &=\frac {4}{e-x}+\frac {34 x}{5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.31 \begin {gather*} \frac {2}{5} \left (\frac {10}{e-x}-17 (e-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 + 34*E^2 - 68*E*x + 34*x^2)/(5*E^2 - 10*E*x + 5*x^2),x]

[Out]

(2*(10/(E - x) - 17*(E - x)))/5

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fricas [A]  time = 1.10, size = 22, normalized size = 1.38 \begin {gather*} \frac {2 \, {\left (17 \, x^{2} - 17 \, x e - 10\right )}}{5 \, {\left (x - e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((34*exp(1)^2-68*x*exp(1)+34*x^2+20)/(5*exp(1)^2-10*x*exp(1)+5*x^2),x, algorithm="fricas")

[Out]

2/5*(17*x^2 - 17*x*e - 10)/(x - 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((34*exp(1)^2-68*x*exp(1)+34*x^2+20)/(5*exp(1)^2-10*x*exp(1)+5*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2/5*(17*sageVARx+20*1/2/sqrt(-exp(1)^2+e
xp(2))*atan((sageVARx-exp(1))/sqrt(-exp(1)^2+exp(2))))

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maple [A]  time = 0.35, size = 15, normalized size = 0.94

method	result	size
risch	\(\frac {34 x}{5}+\frac {4}{{\mathrm e}-x}\)	\(15\)
norman	\(\frac {-\frac {34 x^{2}}{5}+4+\frac {34 \,{\mathrm e}^{2}}{5}}{{\mathrm e}-x}\)	\(23\)
gosper	\(\frac {-\frac {34 x^{2}}{5}+4+\frac {34 \,{\mathrm e}^{2}}{5}}{{\mathrm e}-x}\)	\(24\)
meijerg	\(\frac {34 x}{5 \left (1-{\mathrm e}^{-1} x \right )}+\frac {4 \,{\mathrm e}^{-2} x}{1-{\mathrm e}^{-1} x}-\frac {68 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{-1}}{1-{\mathrm e}^{-1} x}+\ln \left (1-{\mathrm e}^{-1} x \right )\right )}{5}-\frac {34 \,{\mathrm e} \left (-\frac {x \,{\mathrm e}^{-1} \left (-3 \,{\mathrm e}^{-1} x +6\right )}{3 \left (1-{\mathrm e}^{-1} x \right )}-2 \ln \left (1-{\mathrm e}^{-1} x \right )\right )}{5}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((34*exp(1)^2-68*x*exp(1)+34*x^2+20)/(5*exp(1)^2-10*x*exp(1)+5*x^2),x,method=_RETURNVERBOSE)

[Out]

34/5*x+4/(exp(1)-x)

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maxima [A]  time = 0.38, size = 14, normalized size = 0.88 \begin {gather*} \frac {34}{5} \, x - \frac {4}{x - e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((34*exp(1)^2-68*x*exp(1)+34*x^2+20)/(5*exp(1)^2-10*x*exp(1)+5*x^2),x, algorithm="maxima")

[Out]

34/5*x - 4/(x - e)

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mupad [B]  time = 0.08, size = 14, normalized size = 0.88 \begin {gather*} \frac {34\,x}{5}-\frac {4}{x-\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((34*exp(2) - 68*x*exp(1) + 34*x^2 + 20)/(5*exp(2) - 10*x*exp(1) + 5*x^2),x)

[Out]

(34*x)/5 - 4/(x - exp(1))

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sympy [A]  time = 0.10, size = 10, normalized size = 0.62 \begin {gather*} \frac {34 x}{5} - \frac {4}{x - e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((34*exp(1)**2-68*x*exp(1)+34*x**2+20)/(5*exp(1)**2-10*x*exp(1)+5*x**2),x)

[Out]

34*x/5 - 4/(x - E)

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