3.22.18 \(\int \frac {-10 e^8-e^4 x^2+x^6 (-250-100 x-10 x^2)+x^3 (-75 x-30 x^2-3 x^3+e^4 (100+20 x))}{5 e^{12} x^2+e^8 x^3 (-50 x^2-10 x^3)+e^4 x^6 (125 x^2+50 x^3+5 x^4)} \, dx\)

Optimal. Leaf size=35 \[ \frac {2+\frac {1}{5} \left (x-\frac {x}{-x^3+\frac {e^4}{5+x}}\right )}{e^4 x} \]

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Rubi [F]  time = 0.35, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-10 e^8-e^4 x^2+x^6 \left (-250-100 x-10 x^2\right )+x^3 \left (-75 x-30 x^2-3 x^3+e^4 (100+20 x)\right )}{5 e^{12} x^2+e^8 x^3 \left (-50 x^2-10 x^3\right )+e^4 x^6 \left (125 x^2+50 x^3+5 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10*E^8 - E^4*x^2 + x^6*(-250 - 100*x - 10*x^2) + x^3*(-75*x - 30*x^2 - 3*x^3 + E^4*(100 + 20*x)))/(5*E^1
2*x^2 + E^8*x^3*(-50*x^2 - 10*x^3) + E^4*x^6*(125*x^2 + 50*x^3 + 5*x^4)),x]

[Out]

2/(E^4*x) - 3/(4*E^4*(E^4 - 5*x^3 - x^4)) - (4*Defer[Int][(E^4 - 5*x^3 - x^4)^(-2), x])/5 - (15*Defer[Int][x^2
/(E^4 - 5*x^3 - x^4)^2, x])/(4*E^4) + (3*Defer[Int][(E^4 - 5*x^3 - x^4)^(-1), x])/(5*E^4)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{e^4 x^2}+\frac {-4 e^4-75 x^2-15 x^3}{5 e^4 \left (e^4-5 x^3-x^4\right )^2}+\frac {3}{5 e^4 \left (e^4-5 x^3-x^4\right )}\right ) \, dx\\ &=\frac {2}{e^4 x}+\frac {\int \frac {-4 e^4-75 x^2-15 x^3}{\left (e^4-5 x^3-x^4\right )^2} \, dx}{5 e^4}+\frac {3 \int \frac {1}{e^4-5 x^3-x^4} \, dx}{5 e^4}\\ &=\frac {2}{e^4 x}-\frac {3}{4 e^4 \left (e^4-5 x^3-x^4\right )}-\frac {\int \frac {16 e^4+75 x^2}{\left (e^4-5 x^3-x^4\right )^2} \, dx}{20 e^4}+\frac {3 \int \frac {1}{e^4-5 x^3-x^4} \, dx}{5 e^4}\\ &=\frac {2}{e^4 x}-\frac {3}{4 e^4 \left (e^4-5 x^3-x^4\right )}-\frac {\int \left (\frac {16 e^4}{\left (e^4-5 x^3-x^4\right )^2}+\frac {75 x^2}{\left (e^4-5 x^3-x^4\right )^2}\right ) \, dx}{20 e^4}+\frac {3 \int \frac {1}{e^4-5 x^3-x^4} \, dx}{5 e^4}\\ &=\frac {2}{e^4 x}-\frac {3}{4 e^4 \left (e^4-5 x^3-x^4\right )}-\frac {4}{5} \int \frac {1}{\left (e^4-5 x^3-x^4\right )^2} \, dx+\frac {3 \int \frac {1}{e^4-5 x^3-x^4} \, dx}{5 e^4}-\frac {15 \int \frac {x^2}{\left (e^4-5 x^3-x^4\right )^2} \, dx}{4 e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 33, normalized size = 0.94 \begin {gather*} \frac {\frac {10}{x}+\frac {5+x}{-e^4+5 x^3+x^4}}{5 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*E^8 - E^4*x^2 + x^6*(-250 - 100*x - 10*x^2) + x^3*(-75*x - 30*x^2 - 3*x^3 + E^4*(100 + 20*x)))/
(5*E^12*x^2 + E^8*x^3*(-50*x^2 - 10*x^3) + E^4*x^6*(125*x^2 + 50*x^3 + 5*x^4)),x]

[Out]

(10/x + (5 + x)/(-E^4 + 5*x^3 + x^4))/(5*E^4)

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fricas [A]  time = 0.62, size = 43, normalized size = 1.23 \begin {gather*} -\frac {10 \, x^{4} + 50 \, x^{3} + x^{2} + 5 \, x - 10 \, e^{4}}{5 \, {\left (x e^{8} - {\left (x^{5} + 5 \, x^{4}\right )} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2-100*x-250)*x^6+((20*x+100)*exp(4)-3*x^3-30*x^2-75*x)*x^3-10*exp(4)^2-x^2*exp(4))/((5*x^4+5
0*x^3+125*x^2)*exp(4)*x^6+(-10*x^3-50*x^2)*exp(4)^2*x^3+5*x^2*exp(4)^3),x, algorithm="fricas")

[Out]

-1/5*(10*x^4 + 50*x^3 + x^2 + 5*x - 10*e^4)/(x*e^8 - (x^5 + 5*x^4)*e^4)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2-100*x-250)*x^6+((20*x+100)*exp(4)-3*x^3-30*x^2-75*x)*x^3-10*exp(4)^2-x^2*exp(4))/((5*x^4+5
0*x^3+125*x^2)*exp(4)*x^6+(-10*x^3-50*x^2)*exp(4)^2*x^3+5*x^2*exp(4)^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.16, size = 45, normalized size = 1.29

method	result	size
risch	\(\frac {\left (-2 x^{4}-10 x^{3}-\frac {x^{2}}{5}+2 \,{\mathrm e}^{4}-x \right ) {\mathrm e}^{-4}}{x \left (-x^{4}-5 x^{3}+{\mathrm e}^{4}\right )}\)	\(45\)
gosper	\(\frac {\left (-10 x^{4}-50 x^{3}-x^{2}+10 \,{\mathrm e}^{4}-5 x \right ) {\mathrm e}^{-4}}{5 x \left (-x^{4}-5 x^{3}+{\mathrm e}^{4}\right )}\)	\(48\)
norman	\(\frac {2-2 \,{\mathrm e}^{-4} x^{4}-x \,{\mathrm e}^{-4}-\frac {x^{2} {\mathrm e}^{-4}}{5}-10 \,{\mathrm e}^{-4} x^{3}}{x \left (-x^{4}-5 x^{3}+{\mathrm e}^{4}\right )}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^2-100*x-250)*x^6+((20*x+100)*exp(4)-3*x^3-30*x^2-75*x)*x^3-10*exp(4)^2-x^2*exp(4))/((5*x^4+50*x^3+
125*x^2)*exp(4)*x^6+(-10*x^3-50*x^2)*exp(4)^2*x^3+5*x^2*exp(4)^3),x,method=_RETURNVERBOSE)

[Out]

(-2*x^4-10*x^3-1/5*x^2+2*exp(4)-x)/x/(-x^4-5*x^3+exp(4))*exp(-4)

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maxima [A]  time = 0.64, size = 44, normalized size = 1.26 \begin {gather*} \frac {10 \, x^{4} + 50 \, x^{3} + x^{2} + 5 \, x - 10 \, e^{4}}{5 \, {\left (x^{5} e^{4} + 5 \, x^{4} e^{4} - x e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2-100*x-250)*x^6+((20*x+100)*exp(4)-3*x^3-30*x^2-75*x)*x^3-10*exp(4)^2-x^2*exp(4))/((5*x^4+5
0*x^3+125*x^2)*exp(4)*x^6+(-10*x^3-50*x^2)*exp(4)^2*x^3+5*x^2*exp(4)^3),x, algorithm="maxima")

[Out]

1/5*(10*x^4 + 50*x^3 + x^2 + 5*x - 10*e^4)/(x^5*e^4 + 5*x^4*e^4 - x*e^8)

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mupad [B]  time = 1.40, size = 34, normalized size = 0.97 \begin {gather*} \frac {\frac {x}{5}+1}{{\mathrm {e}}^4\,x^4+5\,{\mathrm {e}}^4\,x^3-{\mathrm {e}}^8}+\frac {2\,{\mathrm {e}}^{-4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*exp(8) + x^2*exp(4) + x^6*(100*x + 10*x^2 + 250) + x^3*(75*x + 30*x^2 + 3*x^3 - exp(4)*(20*x + 100)))
/(5*x^2*exp(12) - x^3*exp(8)*(50*x^2 + 10*x^3) + x^6*exp(4)*(125*x^2 + 50*x^3 + 5*x^4)),x)

[Out]

(x/5 + 1)/(5*x^3*exp(4) - exp(8) + x^4*exp(4)) + (2*exp(-4))/x

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sympy [A]  time = 1.49, size = 46, normalized size = 1.31 \begin {gather*} - \frac {- 10 x^{4} - 50 x^{3} - x^{2} - 5 x + 10 e^{4}}{5 x^{5} e^{4} + 25 x^{4} e^{4} - 5 x e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**2-100*x-250)*x**6+((20*x+100)*exp(4)-3*x**3-30*x**2-75*x)*x**3-10*exp(4)**2-x**2*exp(4))/((
5*x**4+50*x**3+125*x**2)*exp(4)*x**6+(-10*x**3-50*x**2)*exp(4)**2*x**3+5*x**2*exp(4)**3),x)

[Out]

-(-10*x**4 - 50*x**3 - x**2 - 5*x + 10*exp(4))/(5*x**5*exp(4) + 25*x**4*exp(4) - 5*x*exp(8))

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