3.98.9 \(\int \frac {-16+e^x (-8 x-4 x^2)}{40000 x^2-40000 e^3 x^2+10000 e^6 x^2+e^x (20000 x^3-20000 e^3 x^3+5000 e^6 x^3)+e^{2 x} (2500 x^4-2500 e^3 x^4+625 e^6 x^4)} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{625 \left (2-e^3\right )^2 x \left (1+\frac {e^x x}{4}\right )} \]

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Rubi [A]  time = 0.19, antiderivative size = 25, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6, 6688, 12, 6687} \begin {gather*} \frac {4}{625 \left (2-e^3\right )^2 x \left (e^x x+4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 + E^x*(-8*x - 4*x^2))/(40000*x^2 - 40000*E^3*x^2 + 10000*E^6*x^2 + E^x*(20000*x^3 - 20000*E^3*x^3 + 5
000*E^6*x^3) + E^(2*x)*(2500*x^4 - 2500*E^3*x^4 + 625*E^6*x^4)),x]

[Out]

4/(625*(2 - E^3)^2*x*(4 + E^x*x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16+e^x \left (-8 x-4 x^2\right )}{10000 e^6 x^2+\left (40000-40000 e^3\right ) x^2+e^x \left (20000 x^3-20000 e^3 x^3+5000 e^6 x^3\right )+e^{2 x} \left (2500 x^4-2500 e^3 x^4+625 e^6 x^4\right )} \, dx\\ &=\int \frac {-16+e^x \left (-8 x-4 x^2\right )}{\left (40000-40000 e^3+10000 e^6\right ) x^2+e^x \left (20000 x^3-20000 e^3 x^3+5000 e^6 x^3\right )+e^{2 x} \left (2500 x^4-2500 e^3 x^4+625 e^6 x^4\right )} \, dx\\ &=\int \frac {4 \left (-4-e^x x (2+x)\right )}{625 \left (2-e^3\right )^2 x^2 \left (4+e^x x\right )^2} \, dx\\ &=\frac {4 \int \frac {-4-e^x x (2+x)}{x^2 \left (4+e^x x\right )^2} \, dx}{625 \left (2-e^3\right )^2}\\ &=\frac {4}{625 \left (2-e^3\right )^2 x \left (4+e^x x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 23, normalized size = 0.82 \begin {gather*} \frac {4}{625 \left (-2+e^3\right )^2 x \left (4+e^x x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 + E^x*(-8*x - 4*x^2))/(40000*x^2 - 40000*E^3*x^2 + 10000*E^6*x^2 + E^x*(20000*x^3 - 20000*E^3*x
^3 + 5000*E^6*x^3) + E^(2*x)*(2500*x^4 - 2500*E^3*x^4 + 625*E^6*x^4)),x]

[Out]

4/(625*(-2 + E^3)^2*x*(4 + E^x*x))

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fricas [B]  time = 0.93, size = 40, normalized size = 1.43 \begin {gather*} \frac {4}{625 \, {\left (4 \, x e^{6} - 16 \, x e^{3} + {\left (x^{2} e^{6} - 4 \, x^{2} e^{3} + 4 \, x^{2}\right )} e^{x} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-8*x)*exp(x)-16)/((625*x^4*exp(3)^2-2500*x^4*exp(3)+2500*x^4)*exp(x)^2+(5000*x^3*exp(3)^2-20
000*x^3*exp(3)+20000*x^3)*exp(x)+10000*x^2*exp(3)^2-40000*x^2*exp(3)+40000*x^2),x, algorithm="fricas")

[Out]

4/625/(4*x*e^6 - 16*x*e^3 + (x^2*e^6 - 4*x^2*e^3 + 4*x^2)*e^x + 16*x)

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giac [B]  time = 0.25, size = 42, normalized size = 1.50 \begin {gather*} \frac {4}{625 \, {\left (x^{2} e^{\left (x + 6\right )} - 4 \, x^{2} e^{\left (x + 3\right )} + 4 \, x^{2} e^{x} + 4 \, x e^{6} - 16 \, x e^{3} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-8*x)*exp(x)-16)/((625*x^4*exp(3)^2-2500*x^4*exp(3)+2500*x^4)*exp(x)^2+(5000*x^3*exp(3)^2-20
000*x^3*exp(3)+20000*x^3)*exp(x)+10000*x^2*exp(3)^2-40000*x^2*exp(3)+40000*x^2),x, algorithm="giac")

[Out]

4/625/(x^2*e^(x + 6) - 4*x^2*e^(x + 3) + 4*x^2*e^x + 4*x*e^6 - 16*x*e^3 + 16*x)

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maple [A]  time = 0.23, size = 20, normalized size = 0.71

method	result	size
norman	\(\frac {4}{625 \left ({\mathrm e}^{3}-2\right )^{2} x \left ({\mathrm e}^{x} x +4\right )}\)	\(20\)
risch	\(\frac {4}{625 x \left ({\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right ) \left ({\mathrm e}^{x} x +4\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2-8*x)*exp(x)-16)/((625*x^4*exp(3)^2-2500*x^4*exp(3)+2500*x^4)*exp(x)^2+(5000*x^3*exp(3)^2-20000*x^
3*exp(3)+20000*x^3)*exp(x)+10000*x^2*exp(3)^2-40000*x^2*exp(3)+40000*x^2),x,method=_RETURNVERBOSE)

[Out]

4/625/(exp(3)-2)^2/x/(exp(x)*x+4)

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maxima [A]  time = 0.51, size = 30, normalized size = 1.07 \begin {gather*} \frac {4}{625 \, {\left (x^{2} {\left (e^{6} - 4 \, e^{3} + 4\right )} e^{x} + 4 \, x {\left (e^{6} - 4 \, e^{3} + 4\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-8*x)*exp(x)-16)/((625*x^4*exp(3)^2-2500*x^4*exp(3)+2500*x^4)*exp(x)^2+(5000*x^3*exp(3)^2-20
000*x^3*exp(3)+20000*x^3)*exp(x)+10000*x^2*exp(3)^2-40000*x^2*exp(3)+40000*x^2),x, algorithm="maxima")

[Out]

4/625/(x^2*(e^6 - 4*e^3 + 4)*e^x + 4*x*(e^6 - 4*e^3 + 4))

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mupad [B]  time = 6.16, size = 19, normalized size = 0.68 \begin {gather*} \frac {4}{625\,x\,\left (x\,{\mathrm {e}}^x+4\right )\,{\left ({\mathrm {e}}^3-2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(8*x + 4*x^2) + 16)/(exp(x)*(5000*x^3*exp(6) - 20000*x^3*exp(3) + 20000*x^3) + exp(2*x)*(625*x^4*
exp(6) - 2500*x^4*exp(3) + 2500*x^4) - 40000*x^2*exp(3) + 10000*x^2*exp(6) + 40000*x^2),x)

[Out]

4/(625*x*(x*exp(x) + 4)*(exp(3) - 2)^2)

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sympy [A]  time = 0.16, size = 42, normalized size = 1.50 \begin {gather*} \frac {4}{- 10000 x e^{3} + 10000 x + 2500 x e^{6} + \left (- 2500 x^{2} e^{3} + 2500 x^{2} + 625 x^{2} e^{6}\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2-8*x)*exp(x)-16)/((625*x**4*exp(3)**2-2500*x**4*exp(3)+2500*x**4)*exp(x)**2+(5000*x**3*exp(
3)**2-20000*x**3*exp(3)+20000*x**3)*exp(x)+10000*x**2*exp(3)**2-40000*x**2*exp(3)+40000*x**2),x)

[Out]

4/(-10000*x*exp(3) + 10000*x + 2500*x*exp(6) + (-2500*x**2*exp(3) + 2500*x**2 + 625*x**2*exp(6))*exp(x))

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