3.19.16 \(\int \frac {-100+250 x+e^{x^4} (50 x+100 x^5)}{16 x^2-40 x^3+25 x^4+e^{2 x^4} x^4+e^{x^4} (-8 x^3+10 x^4)} \, dx\)

Optimal. Leaf size=26 \[ \frac {5}{x \left (1-x+\frac {1}{5} \left (-1-e^{x^4} x\right )\right )} \]

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Rubi [A]  time = 0.29, antiderivative size = 19, normalized size of antiderivative = 0.73, number of steps used = 3, number of rules used = 3, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 12, 6687} \begin {gather*} \frac {25}{x \left (4-\left (e^{x^4}+5\right ) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-100 + 250*x + E^x^4*(50*x + 100*x^5))/(16*x^2 - 40*x^3 + 25*x^4 + E^(2*x^4)*x^4 + E^x^4*(-8*x^3 + 10*x^4
)),x]

[Out]

25/(x*(4 - (5 + E^x^4)*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 {50 \left (-2+\left (5+e^{x^4}\right ) x+2 e^{x^4} x^5\right )}{x^2 \left (4-\left (5+e^{x^4}\right ) x\right )^2} \, dx\\ &=50 \int \frac {-2+\left (5+e^{x^4}\right ) x+2 e^{x^4} x^5}{x^2 \left (4-\left (5+e^{x^4}\right ) x\right )^2} \, dx\\ &=\frac {25}{x \left (4-\left (5+e^{x^4}\right ) x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 19, normalized size = 0.73 \begin {gather*} -\frac {25}{x \left (-4+5 x+e^{x^4} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-100 + 250*x + E^x^4*(50*x + 100*x^5))/(16*x^2 - 40*x^3 + 25*x^4 + E^(2*x^4)*x^4 + E^x^4*(-8*x^3 +
10*x^4)),x]

[Out]

-25/(x*(-4 + 5*x + E^x^4*x))

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fricas [A]  time = 0.82, size = 21, normalized size = 0.81 \begin {gather*} -\frac {25}{x^{2} e^{\left (x^{4}\right )} + 5 \, x^{2} - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^5+50*x)*exp(x^4)+250*x-100)/(x^4*exp(x^4)^2+(10*x^4-8*x^3)*exp(x^4)+25*x^4-40*x^3+16*x^2),x,
algorithm="fricas")

[Out]

-25/(x^2*e^(x^4) + 5*x^2 - 4*x)

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giac [A]  time = 0.23, size = 21, normalized size = 0.81 \begin {gather*} -\frac {25}{x^{2} e^{\left (x^{4}\right )} + 5 \, x^{2} - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^5+50*x)*exp(x^4)+250*x-100)/(x^4*exp(x^4)^2+(10*x^4-8*x^3)*exp(x^4)+25*x^4-40*x^3+16*x^2),x,
algorithm="giac")

[Out]

-25/(x^2*e^(x^4) + 5*x^2 - 4*x)

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maple [A]  time = 0.06, size = 19, normalized size = 0.73

method	result	size
norman	\(-\frac {25}{x \left (x \,{\mathrm e}^{x^{4}}+5 x -4\right )}\)	\(19\)
risch	\(-\frac {25}{x \left (x \,{\mathrm e}^{x^{4}}+5 x -4\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((100*x^5+50*x)*exp(x^4)+250*x-100)/(x^4*exp(x^4)^2+(10*x^4-8*x^3)*exp(x^4)+25*x^4-40*x^3+16*x^2),x,method
=_RETURNVERBOSE)

[Out]

-25/x/(x*exp(x^4)+5*x-4)

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maxima [A]  time = 0.51, size = 21, normalized size = 0.81 \begin {gather*} -\frac {25}{x^{2} e^{\left (x^{4}\right )} + 5 \, x^{2} - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^5+50*x)*exp(x^4)+250*x-100)/(x^4*exp(x^4)^2+(10*x^4-8*x^3)*exp(x^4)+25*x^4-40*x^3+16*x^2),x,
algorithm="maxima")

[Out]

-25/(x^2*e^(x^4) + 5*x^2 - 4*x)

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mupad [B]  time = 1.26, size = 28, normalized size = 1.08 \begin {gather*} \frac {25}{4\,x}-\frac {\frac {25\,{\mathrm {e}}^{x^4}}{4}+\frac {125}{4}}{x\,\left ({\mathrm {e}}^{x^4}+5\right )-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((250*x + exp(x^4)*(50*x + 100*x^5) - 100)/(x^4*exp(2*x^4) - exp(x^4)*(8*x^3 - 10*x^4) + 16*x^2 - 40*x^3 +
25*x^4),x)

[Out]

25/(4*x) - ((25*exp(x^4))/4 + 125/4)/(x*(exp(x^4) + 5) - 4)

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sympy [A]  time = 0.14, size = 19, normalized size = 0.73 \begin {gather*} - \frac {25}{x^{2} e^{x^{4}} + 5 x^{2} - 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x**5+50*x)*exp(x**4)+250*x-100)/(x**4*exp(x**4)**2+(10*x**4-8*x**3)*exp(x**4)+25*x**4-40*x**3+
16*x**2),x)

[Out]

-25/(x**2*exp(x**4) + 5*x**2 - 4*x)

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