3.51.57 \(\int \frac {-400 e^{-3+x}-4 e^{-3+3 x} x^2+40 \log (6)+e^x (80 e^{-3+x} x+(-4-8 x) \log (6))}{100 e^{-3+x}-20 e^{-3+2 x} x+e^{-3+3 x} x^2} \, dx\)

Optimal. Leaf size=29 \[ -4 x+\frac {4 e^{3-x} \log (6)}{\left (e^x-\frac {10}{x}\right ) x} \]

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Rubi [F]  time = 1.95, 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 {-400 e^{-3+x}-4 e^{-3+3 x} x^2+40 \log (6)+e^x \left (80 e^{-3+x} x+(-4-8 x) \log (6)\right )}{100 e^{-3+x}-20 e^{-3+2 x} x+e^{-3+3 x} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-400*E^(-3 + x) - 4*E^(-3 + 3*x)*x^2 + 40*Log[6] + E^x*(80*E^(-3 + x)*x + (-4 - 8*x)*Log[6]))/(100*E^(-3
+ x) - 20*E^(-3 + 2*x)*x + E^(-3 + 3*x)*x^2),x]

[Out]

-4*x - 40*Log[6]*Defer[Int][E^(3 - x)/(-10 + E^x*x)^2, x] - 40*Log[6]*Defer[Int][E^(3 - x)/(x*(-10 + E^x*x)^2)
, x] - 8*Log[6]*Defer[Int][E^(3 - x)/(-10 + E^x*x), x] - 4*Log[6]*Defer[Int][E^(3 - x)/(x*(-10 + E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{3-x} \left (-400 e^{-3+x}-4 e^{-3+3 x} x^2+40 \log (6)+e^x \left (80 e^{-3+x} x+(-4-8 x) \log (6)\right )\right )}{\left (10-e^x x\right )^2} \, dx\\ &=\int \left (-4-\frac {40 e^{3-x} (1+x) \log (6)}{x \left (-10+e^x x\right )^2}-\frac {4 e^{3-x} (1+2 x) \log (6)}{x \left (-10+e^x x\right )}\right ) \, dx\\ &=-4 x-(4 \log (6)) \int \frac {e^{3-x} (1+2 x)}{x \left (-10+e^x x\right )} \, dx-(40 \log (6)) \int \frac {e^{3-x} (1+x)}{x \left (-10+e^x x\right )^2} \, dx\\ &=-4 x-(4 \log (6)) \int \left (\frac {2 e^{3-x}}{-10+e^x x}+\frac {e^{3-x}}{x \left (-10+e^x x\right )}\right ) \, dx-(40 \log (6)) \int \left (\frac {e^{3-x}}{\left (-10+e^x x\right )^2}+\frac {e^{3-x}}{x \left (-10+e^x x\right )^2}\right ) \, dx\\ &=-4 x-(4 \log (6)) \int \frac {e^{3-x}}{x \left (-10+e^x x\right )} \, dx-(8 \log (6)) \int \frac {e^{3-x}}{-10+e^x x} \, dx-(40 \log (6)) \int \frac {e^{3-x}}{\left (-10+e^x x\right )^2} \, dx-(40 \log (6)) \int \frac {e^{3-x}}{x \left (-10+e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.99, size = 34, normalized size = 1.17 \begin {gather*} -\frac {4 \left (10 x-e^x x^2+e^{3-x} \log (6)\right )}{10-e^x x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-400*E^(-3 + x) - 4*E^(-3 + 3*x)*x^2 + 40*Log[6] + E^x*(80*E^(-3 + x)*x + (-4 - 8*x)*Log[6]))/(100*
E^(-3 + x) - 20*E^(-3 + 2*x)*x + E^(-3 + 3*x)*x^2),x]

[Out]

(-4*(10*x - E^x*x^2 + E^(3 - x)*Log[6]))/(10 - E^x*x)

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fricas [A]  time = 1.66, size = 35, normalized size = 1.21 \begin {gather*} -\frac {4 \, {\left (x^{2} e^{\left (2 \, x\right )} - 10 \, x e^{x} - e^{3} \log \relax (6)\right )}}{x e^{\left (2 \, x\right )} - 10 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*exp(x-3)*exp(x)^2+(80*x*exp(x-3)+(-8*x-4)*log(6))*exp(x)-400*exp(x-3)+40*log(6))/(x^2*exp(x-
3)*exp(x)^2-20*x*exp(x-3)*exp(x)+100*exp(x-3)),x, algorithm="fricas")

[Out]

-4*(x^2*e^(2*x) - 10*x*e^x - e^3*log(6))/(x*e^(2*x) - 10*e^x)

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giac [A]  time = 0.16, size = 35, normalized size = 1.21 \begin {gather*} -\frac {4 \, {\left (x^{2} e^{\left (2 \, x\right )} - 10 \, x e^{x} - e^{3} \log \relax (6)\right )}}{x e^{\left (2 \, x\right )} - 10 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*exp(x-3)*exp(x)^2+(80*x*exp(x-3)+(-8*x-4)*log(6))*exp(x)-400*exp(x-3)+40*log(6))/(x^2*exp(x-
3)*exp(x)^2-20*x*exp(x-3)*exp(x)+100*exp(x-3)),x, algorithm="giac")

[Out]

-4*(x^2*e^(2*x) - 10*x*e^x - e^3*log(6))/(x*e^(2*x) - 10*e^x)

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maple [A]  time = 0.13, size = 35, normalized size = 1.21

method	result	size
norman	\(\frac {\left (4 \,{\mathrm e}^{3} \ln \relax (6)+40 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-x}}{{\mathrm e}^{x} x -10}\)	\(35\)
risch	\(-4 x -\frac {2 \,{\mathrm e}^{3-x} \ln \relax (2)}{5}-\frac {2 \,{\mathrm e}^{3-x} \ln \relax (3)}{5}+\frac {2 x \left (\ln \relax (2)+\ln \relax (3)\right ) {\mathrm e}^{3}}{5 \left ({\mathrm e}^{x} x -10\right )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2*exp(x-3)*exp(x)^2+(80*x*exp(x-3)+(-8*x-4)*ln(6))*exp(x)-400*exp(x-3)+40*ln(6))/(x^2*exp(x-3)*exp(x
)^2-20*x*exp(x-3)*exp(x)+100*exp(x-3)),x,method=_RETURNVERBOSE)

[Out]

(4*exp(3)*ln(6)+40*exp(x)*x-4*exp(x)^2*x^2)/exp(x)/(exp(x)*x-10)

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maxima [A]  time = 0.49, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4 \, {\left (x^{2} e^{\left (2 \, x\right )} - {\left (\log \relax (3) + \log \relax (2)\right )} e^{3} - 10 \, x e^{x}\right )}}{x e^{\left (2 \, x\right )} - 10 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2*exp(x-3)*exp(x)^2+(80*x*exp(x-3)+(-8*x-4)*log(6))*exp(x)-400*exp(x-3)+40*log(6))/(x^2*exp(x-
3)*exp(x)^2-20*x*exp(x-3)*exp(x)+100*exp(x-3)),x, algorithm="maxima")

[Out]

-4*(x^2*e^(2*x) - (log(3) + log(2))*e^3 - 10*x*e^x)/(x*e^(2*x) - 10*e^x)

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mupad [B]  time = 3.54, size = 22, normalized size = 0.76 \begin {gather*} \frac {4\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3\,\ln \relax (6)}{x\,{\mathrm {e}}^x-10}-4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(400*exp(x - 3) - 40*log(6) + exp(x)*(log(6)*(8*x + 4) - 80*x*exp(x - 3)) + 4*x^2*exp(2*x)*exp(x - 3))/(1
00*exp(x - 3) + x^2*exp(2*x)*exp(x - 3) - 20*x*exp(x - 3)*exp(x)),x)

[Out]

(4*exp(-x)*exp(3)*log(6))/(x*exp(x) - 10) - 4*x

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sympy [A]  time = 0.18, size = 34, normalized size = 1.17 \begin {gather*} - 4 x + \frac {2 x e^{3} \log {\relax (6 )}}{5 x e^{x} - 50} - \frac {2 e^{3} e^{- x} \log {\relax (6 )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2*exp(x-3)*exp(x)**2+(80*x*exp(x-3)+(-8*x-4)*ln(6))*exp(x)-400*exp(x-3)+40*ln(6))/(x**2*exp(x
-3)*exp(x)**2-20*x*exp(x-3)*exp(x)+100*exp(x-3)),x)

[Out]

-4*x + 2*x*exp(3)*log(6)/(5*x*exp(x) - 50) - 2*exp(3)*exp(-x)*log(6)/5

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