3.96.67 \(\int \frac {-96+48 x-6 x^2+e^2 (-160+16 x-8 x^2)}{e^2 (16000-3200 x-920 x^2+76 x^3+22 x^4+x^5)} \, dx\)

Optimal. Leaf size=20 \[ \frac {\frac {3}{e^2}+\frac {4 x}{-4+x}}{(10+x)^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 40, normalized size of antiderivative = 2.00, number of steps used = 3, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 2074} \begin {gather*} -\frac {4}{49 (x+10)}+\frac {21+20 e^2}{7 e^2 (x+10)^2}-\frac {4}{49 (4-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-96 + 48*x - 6*x^2 + E^2*(-160 + 16*x - 8*x^2))/(E^2*(16000 - 3200*x - 920*x^2 + 76*x^3 + 22*x^4 + x^5)),
x]

[Out]

-4/(49*(4 - x)) + (21 + 20*E^2)/(7*E^2*(10 + x)^2) - 4/(49*(10 + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-96+48 x-6 x^2+e^2 \left (-160+16 x-8 x^2\right )}{16000-3200 x-920 x^2+76 x^3+22 x^4+x^5} \, dx}{e^2}\\ &=\frac {\int \left (-\frac {4 e^2}{49 (-4+x)^2}-\frac {2 \left (21+20 e^2\right )}{7 (10+x)^3}+\frac {4 e^2}{49 (10+x)^2}\right ) \, dx}{e^2}\\ &=-\frac {4}{49 (4-x)}+\frac {21+20 e^2}{7 e^2 (10+x)^2}-\frac {4}{49 (10+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 1.25 \begin {gather*} \frac {-12+\left (3+4 e^2\right ) x}{e^2 (-4+x) (10+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96 + 48*x - 6*x^2 + E^2*(-160 + 16*x - 8*x^2))/(E^2*(16000 - 3200*x - 920*x^2 + 76*x^3 + 22*x^4 +
x^5)),x]

[Out]

(-12 + (3 + 4*E^2)*x)/(E^2*(-4 + x)*(10 + x)^2)

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fricas [A]  time = 0.46, size = 28, normalized size = 1.40 \begin {gather*} \frac {{\left (4 \, x e^{2} + 3 \, x - 12\right )} e^{\left (-2\right )}}{x^{3} + 16 \, x^{2} + 20 \, x - 400} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+16*x-160)*exp(2)-6*x^2+48*x-96)/(x^5+22*x^4+76*x^3-920*x^2-3200*x+16000)/exp(2),x, algorith
m="fricas")

[Out]

(4*x*e^2 + 3*x - 12)*e^(-2)/(x^3 + 16*x^2 + 20*x - 400)

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giac [A]  time = 0.15, size = 32, normalized size = 1.60 \begin {gather*} \frac {1}{49} \, {\left (\frac {4 \, e^{2}}{x - 4} - \frac {4 \, x e^{2} - 100 \, e^{2} - 147}{{\left (x + 10\right )}^{2}}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+16*x-160)*exp(2)-6*x^2+48*x-96)/(x^5+22*x^4+76*x^3-920*x^2-3200*x+16000)/exp(2),x, algorith
m="giac")

[Out]

1/49*(4*e^2/(x - 4) - (4*x*e^2 - 100*e^2 - 147)/(x + 10)^2)*e^(-2)

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maple [A]  time = 0.08, size = 29, normalized size = 1.45

method	result	size
risch	\(\frac {{\mathrm e}^{-2} \left (-12+\left (4 \,{\mathrm e}^{2}+3\right ) x \right )}{x^{3}+16 x^{2}+20 x -400}\)	\(29\)
gosper	\(\frac {\left (4 \,{\mathrm e}^{2} x +3 x -12\right ) {\mathrm e}^{-2}}{x^{3}+16 x^{2}+20 x -400}\)	\(31\)
norman	\(\frac {\left (4 \,{\mathrm e}^{2}+3\right ) {\mathrm e}^{-2} x -12 \,{\mathrm e}^{-2}}{\left (x -4\right ) \left (x +10\right )^{2}}\)	\(31\)
default	\({\mathrm e}^{-2} \left (\frac {4 \,{\mathrm e}^{2}}{49 \left (x -4\right )}-\frac {-\frac {20 \,{\mathrm e}^{2}}{7}-3}{\left (x +10\right )^{2}}-\frac {4 \,{\mathrm e}^{2}}{49 \left (x +10\right )}\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2+16*x-160)*exp(2)-6*x^2+48*x-96)/(x^5+22*x^4+76*x^3-920*x^2-3200*x+16000)/exp(2),x,method=_RETURNV
ERBOSE)

[Out]

exp(-2)*(-12+(4*exp(2)+3)*x)/(x^3+16*x^2+20*x-400)

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maxima [A]  time = 0.37, size = 28, normalized size = 1.40 \begin {gather*} \frac {{\left (x {\left (4 \, e^{2} + 3\right )} - 12\right )} e^{\left (-2\right )}}{x^{3} + 16 \, x^{2} + 20 \, x - 400} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+16*x-160)*exp(2)-6*x^2+48*x-96)/(x^5+22*x^4+76*x^3-920*x^2-3200*x+16000)/exp(2),x, algorith
m="maxima")

[Out]

(x*(4*e^2 + 3) - 12)*e^(-2)/(x^3 + 16*x^2 + 20*x - 400)

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mupad [B]  time = 0.26, size = 34, normalized size = 1.70 \begin {gather*} \frac {4}{49\,\left (x-4\right )}-\frac {4}{49\,\left (x+10\right )}+\frac {{\mathrm {e}}^{-2}\,\left (20\,{\mathrm {e}}^2+21\right )}{7\,{\left (x+10\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2)*(exp(2)*(8*x^2 - 16*x + 160) - 48*x + 6*x^2 + 96))/(76*x^3 - 920*x^2 - 3200*x + 22*x^4 + x^5 + 1
6000),x)

[Out]

4/(49*(x - 4)) - 4/(49*(x + 10)) + (exp(-2)*(20*exp(2) + 21))/(7*(x + 10)^2)

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sympy [B]  time = 0.57, size = 39, normalized size = 1.95 \begin {gather*} - \frac {x \left (- 4 e^{2} - 3\right ) + 12}{x^{3} e^{2} + 16 x^{2} e^{2} + 20 x e^{2} - 400 e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2+16*x-160)*exp(2)-6*x**2+48*x-96)/(x**5+22*x**4+76*x**3-920*x**2-3200*x+16000)/exp(2),x)

[Out]

-(x*(-4*exp(2) - 3) + 12)/(x**3*exp(2) + 16*x**2*exp(2) + 20*x*exp(2) - 400*exp(2))

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