3.10.52 \(\int \frac {-100+e^4+320 x-61 x^2+3 x^3}{100-20 x+x^2} \, dx\)

Optimal. Leaf size=21 \[ -\frac {e^4}{-10+x}-x+\frac {3 x^2}{2} \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.05, 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 = {27, 1850} \begin {gather*} \frac {3 x^2}{2}-x+\frac {e^4}{10-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-100 + E^4 + 320*x - 61*x^2 + 3*x^3)/(100 - 20*x + x^2),x]

[Out]

E^4/(10 - x) - x + (3*x^2)/2

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 {-100+e^4+320 x-61 x^2+3 x^3}{(-10+x)^2} \, dx\\ &=\int \left (-1+\frac {e^4}{(-10+x)^2}+3 x\right ) \, dx\\ &=\frac {e^4}{10-x}-x+\frac {3 x^2}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {e^4}{-10+x}-x+\frac {3 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-100 + E^4 + 320*x - 61*x^2 + 3*x^3)/(100 - 20*x + x^2),x]

[Out]

-(E^4/(-10 + x)) - x + (3*x^2)/2

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fricas [A]  time = 0.81, size = 25, normalized size = 1.19 \begin {gather*} \frac {3 \, x^{3} - 32 \, x^{2} + 20 \, x - 2 \, e^{4}}{2 \, {\left (x - 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)+3*x^3-61*x^2+320*x-100)/(x^2-20*x+100),x, algorithm="fricas")

[Out]

1/2*(3*x^3 - 32*x^2 + 20*x - 2*e^4)/(x - 10)

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giac [A]  time = 0.30, size = 18, normalized size = 0.86 \begin {gather*} \frac {3}{2} \, x^{2} - x - \frac {e^{4}}{x - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)+3*x^3-61*x^2+320*x-100)/(x^2-20*x+100),x, algorithm="giac")

[Out]

3/2*x^2 - x - e^4/(x - 10)

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maple [A]  time = 0.19, size = 19, normalized size = 0.90

method	result	size
default	\(\frac {3 x^{2}}{2}-\frac {{\mathrm e}^{4}}{x -10}-x\)	\(19\)
risch	\(\frac {3 x^{2}}{2}-\frac {{\mathrm e}^{4}}{x -10}-x\)	\(19\)
norman	\(\frac {\frac {3 x^{3}}{2}-16 x^{2}+100-{\mathrm e}^{4}}{x -10}\)	\(23\)
gosper	\(-\frac {-3 x^{3}+32 x^{2}+2 \,{\mathrm e}^{4}-200}{2 \left (x -10\right )}\)	\(24\)
meijerg	\(\frac {31 x}{1-\frac {x}{10}}+\frac {x \,{\mathrm e}^{4}}{100-10 x}+\frac {15 x \left (-\frac {1}{50} x^{2}-\frac {3}{5} x +12\right )}{2 \left (1-\frac {x}{10}\right )}-\frac {61 x \left (-\frac {3 x}{10}+6\right )}{3 \left (1-\frac {x}{10}\right )}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)+3*x^3-61*x^2+320*x-100)/(x^2-20*x+100),x,method=_RETURNVERBOSE)

[Out]

3/2*x^2-exp(4)/(x-10)-x

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maxima [A]  time = 0.49, size = 18, normalized size = 0.86 \begin {gather*} \frac {3}{2} \, x^{2} - x - \frac {e^{4}}{x - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)+3*x^3-61*x^2+320*x-100)/(x^2-20*x+100),x, algorithm="maxima")

[Out]

3/2*x^2 - x - e^4/(x - 10)

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mupad [B]  time = 0.06, size = 18, normalized size = 0.86 \begin {gather*} \frac {3\,x^2}{2}-x-\frac {{\mathrm {e}}^4}{x-10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((320*x + exp(4) - 61*x^2 + 3*x^3 - 100)/(x^2 - 20*x + 100),x)

[Out]

(3*x^2)/2 - x - exp(4)/(x - 10)

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sympy [A]  time = 0.09, size = 14, normalized size = 0.67 \begin {gather*} \frac {3 x^{2}}{2} - x - \frac {e^{4}}{x - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(4)+3*x**3-61*x**2+320*x-100)/(x**2-20*x+100),x)

[Out]

3*x**2/2 - x - exp(4)/(x - 10)

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