3.77.82 \(\int \frac {2500-10000 x+6250 x^2-1250 x^3+e (-200+200 x-50 x^2)+e^2 (-4+4 x-x^2)}{7500-7500 x+1875 x^2} \, dx\)

Optimal. Leaf size=32 \[ x-\frac {1}{3} x \left (\left (1+\frac {e}{25}\right )^2+\frac {2}{2-x}+\frac {2}{x}+x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {27, 12, 1850} \begin {gather*} -\frac {x^2}{3}+\frac {(1250-e (50+e)) x}{1875}-\frac {4}{3 (2-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2500 - 10000*x + 6250*x^2 - 1250*x^3 + E*(-200 + 200*x - 50*x^2) + E^2*(-4 + 4*x - x^2))/(7500 - 7500*x +
1875*x^2),x]

[Out]

-4/(3*(2 - x)) + ((1250 - E*(50 + E))*x)/1875 - x^2/3

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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{1875 (-2+x)^2} \, dx\\ &=\frac {\int \frac {2500-10000 x+6250 x^2-1250 x^3+e \left (-200+200 x-50 x^2\right )+e^2 \left (-4+4 x-x^2\right )}{(-2+x)^2} \, dx}{1875}\\ &=\frac {\int \left (1250 \left (1-\frac {e (50+e)}{1250}\right )-\frac {2500}{(-2+x)^2}-1250 x\right ) \, dx}{1875}\\ &=-\frac {4}{3 (2-x)}+\frac {(1250-e (50+e)) x}{1875}-\frac {x^2}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.19 \begin {gather*} \frac {4}{3 (-2+x)}-\frac {1}{3} (-2+x)^2-\frac {2 x}{3}-\frac {2 e x}{75}-\frac {e^2 x}{1875} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2500 - 10000*x + 6250*x^2 - 1250*x^3 + E*(-200 + 200*x - 50*x^2) + E^2*(-4 + 4*x - x^2))/(7500 - 75
00*x + 1875*x^2),x]

[Out]

4/(3*(-2 + x)) - (-2 + x)^2/3 - (2*x)/3 - (2*E*x)/75 - (E^2*x)/1875

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fricas [A]  time = 1.15, size = 43, normalized size = 1.34 \begin {gather*} -\frac {625 \, x^{3} - 2500 \, x^{2} + {\left (x^{2} - 2 \, x\right )} e^{2} + 50 \, {\left (x^{2} - 2 \, x\right )} e + 2500 \, x - 2500}{1875 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(1)^2+(-50*x^2+200*x-200)*exp(1)-1250*x^3+6250*x^2-10000*x+2500)/(1875*x^2-7500*x+7
500),x, algorithm="fricas")

[Out]

-1/1875*(625*x^3 - 2500*x^2 + (x^2 - 2*x)*e^2 + 50*(x^2 - 2*x)*e + 2500*x - 2500)/(x - 2)

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giac [A]  time = 0.13, size = 26, normalized size = 0.81 \begin {gather*} -\frac {1}{3} \, x^{2} - \frac {1}{1875} \, x e^{2} - \frac {2}{75} \, x e + \frac {2}{3} \, x + \frac {4}{3 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(1)^2+(-50*x^2+200*x-200)*exp(1)-1250*x^3+6250*x^2-10000*x+2500)/(1875*x^2-7500*x+7
500),x, algorithm="giac")

[Out]

-1/3*x^2 - 1/1875*x*e^2 - 2/75*x*e + 2/3*x + 4/3/(x - 2)

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maple [A]  time = 0.27, size = 27, normalized size = 0.84

method	result	size
default	\(-\frac {x^{2}}{3}+\frac {2 x}{3}-\frac {{\mathrm e}^{2} x}{1875}-\frac {2 x \,{\mathrm e}}{75}+\frac {4}{3 \left (x -2\right )}\)	\(27\)
risch	\(-\frac {x^{2}}{3}+\frac {2 x}{3}-\frac {{\mathrm e}^{2} x}{1875}-\frac {2 x \,{\mathrm e}}{75}+\frac {4}{3 \left (x -2\right )}\)	\(27\)
norman	\(\frac {\left (-\frac {{\mathrm e}^{2}}{1875}-\frac {2 \,{\mathrm e}}{75}+\frac {4}{3}\right ) x^{2}-\frac {x^{3}}{3}-\frac {4}{3}+\frac {4 \,{\mathrm e}^{2}}{1875}+\frac {8 \,{\mathrm e}}{75}}{x -2}\)	\(40\)
gosper	\(-\frac {x^{2} {\mathrm e}^{2}+50 x^{2} {\mathrm e}+625 x^{3}-4 \,{\mathrm e}^{2}-2500 x^{2}-200 \,{\mathrm e}+2500}{1875 \left (x -2\right )}\)	\(45\)
meijerg	\(\frac {x}{3-\frac {3 x}{2}}-2 \left (-\frac {{\mathrm e}^{2}}{1875}-\frac {2 \,{\mathrm e}}{75}+\frac {10}{3}\right ) \left (-\frac {x \left (-\frac {3 x}{2}+6\right )}{6 \left (1-\frac {x}{2}\right )}-2 \ln \left (1-\frac {x}{2}\right )\right )-2 \left (-\frac {2 \,{\mathrm e}^{2}}{1875}-\frac {4 \,{\mathrm e}}{75}+\frac {8}{3}\right ) \left (\frac {x}{2-x}+\ln \left (1-\frac {x}{2}\right )\right )-\frac {x \left (-\frac {1}{2} x^{2}-3 x +12\right )}{3 \left (1-\frac {x}{2}\right )}-8 \ln \left (1-\frac {x}{2}\right )-\frac {{\mathrm e}^{2} x}{1875 \left (1-\frac {x}{2}\right )}-\frac {2 \,{\mathrm e} x}{75 \left (1-\frac {x}{2}\right )}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+4*x-4)*exp(1)^2+(-50*x^2+200*x-200)*exp(1)-1250*x^3+6250*x^2-10000*x+2500)/(1875*x^2-7500*x+7500),x
,method=_RETURNVERBOSE)

[Out]

-1/3*x^2+2/3*x-1/1875*exp(2)*x-2/75*x*exp(1)+4/3/(x-2)

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maxima [A]  time = 0.35, size = 24, normalized size = 0.75 \begin {gather*} -\frac {1}{3} \, x^{2} - \frac {1}{1875} \, x {\left (e^{2} + 50 \, e - 1250\right )} + \frac {4}{3 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(1)^2+(-50*x^2+200*x-200)*exp(1)-1250*x^3+6250*x^2-10000*x+2500)/(1875*x^2-7500*x+7
500),x, algorithm="maxima")

[Out]

-1/3*x^2 - 1/1875*x*(e^2 + 50*e - 1250) + 4/3/(x - 2)

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mupad [B]  time = 5.30, size = 28, normalized size = 0.88 \begin {gather*} \frac {4}{3\,\left (x-2\right )}-x\,\left (\frac {2\,\mathrm {e}}{75}+\frac {{\mathrm {e}}^2}{1875}-\frac {2}{3}\right )-\frac {x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10000*x + exp(1)*(50*x^2 - 200*x + 200) + exp(2)*(x^2 - 4*x + 4) - 6250*x^2 + 1250*x^3 - 2500)/(1875*x^2
- 7500*x + 7500),x)

[Out]

4/(3*(x - 2)) - x*((2*exp(1))/75 + exp(2)/1875 - 2/3) - x^2/3

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sympy [A]  time = 0.14, size = 27, normalized size = 0.84 \begin {gather*} - \frac {x^{2}}{3} - x \left (- \frac {2}{3} + \frac {e^{2}}{1875} + \frac {2 e}{75}\right ) + \frac {4}{3 x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+4*x-4)*exp(1)**2+(-50*x**2+200*x-200)*exp(1)-1250*x**3+6250*x**2-10000*x+2500)/(1875*x**2-75
00*x+7500),x)

[Out]

-x**2/3 - x*(-2/3 + exp(2)/1875 + 2*E/75) + 4/(3*x - 6)

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