3.40.31 \(\int \frac {50 x+e^{\frac {1}{625} e^2 (9-150 x+625 x^2)} (-50 x+e^2 (-6 x^2+50 x^3))}{25-50 e^{\frac {1}{625} e^2 (9-150 x+625 x^2)}+25 e^{\frac {2}{625} e^2 (9-150 x+625 x^2)}} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^2}{1-e^{e^2 \left (-\frac {3}{25}+x\right )^2}} \]

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Rubi [F]  time = 1.41, 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 {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25-50 e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )}+25 e^{\frac {2}{625} e^2 \left (9-150 x+625 x^2\right )}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50*x + E^((E^2*(9 - 150*x + 625*x^2))/625)*(-50*x + E^2*(-6*x^2 + 50*x^3)))/(25 - 50*E^((E^2*(9 - 150*x +
625*x^2))/625) + 25*E^((2*E^2*(9 - 150*x + 625*x^2))/625)),x]

[Out]

-2*Defer[Int][x/(-1 + E^((E^2*(3 - 25*x)^2)/625)), x] - (6*E^2*Defer[Int][x^2/(-1 + E^((E^2*(3 - 25*x)^2)/625)
)^2, x])/25 - (6*E^2*Defer[Int][x^2/(-1 + E^((E^2*(3 - 25*x)^2)/625)), x])/25 + 2*E^2*Defer[Int][x^3/(-1 + E^(
(E^2*(3 - 25*x)^2)/625))^2, x] + 2*E^2*Defer[Int][x^3/(-1 + E^((E^2*(3 - 25*x)^2)/625)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{25 \left (1-e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2} \, dx\\ &=\frac {1}{25} \int \frac {50 x+e^{\frac {1}{625} e^2 \left (9-150 x+625 x^2\right )} \left (-50 x+e^2 \left (-6 x^2+50 x^3\right )\right )}{\left (1-e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2} \, dx\\ &=\frac {1}{25} \int \left (\frac {2 e^2 x^2 (-3+25 x)}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}+\frac {2 x \left (-25-3 e^2 x+25 e^2 x^2\right )}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}\right ) \, dx\\ &=\frac {2}{25} \int \frac {x \left (-25-3 e^2 x+25 e^2 x^2\right )}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \, dx+\frac {1}{25} \left (2 e^2\right ) \int \frac {x^2 (-3+25 x)}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2} \, dx\\ &=\frac {2}{25} \int \left (-\frac {25 x}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}-\frac {3 e^2 x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}+\frac {25 e^2 x^3}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}}\right ) \, dx+\frac {1}{25} \left (2 e^2\right ) \int \left (-\frac {3 x^2}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}+\frac {25 x^3}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \, dx\right )-\frac {1}{25} \left (6 e^2\right ) \int \frac {x^2}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2} \, dx-\frac {1}{25} \left (6 e^2\right ) \int \frac {x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \, dx+\left (2 e^2\right ) \int \frac {x^3}{\left (-1+e^{\frac {1}{625} e^2 (3-25 x)^2}\right )^2} \, dx+\left (2 e^2\right ) \int \frac {x^3}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 25, normalized size = 1.09 \begin {gather*} -\frac {x^2}{-1+e^{\frac {1}{625} e^2 (3-25 x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50*x + E^((E^2*(9 - 150*x + 625*x^2))/625)*(-50*x + E^2*(-6*x^2 + 50*x^3)))/(25 - 50*E^((E^2*(9 - 1
50*x + 625*x^2))/625) + 25*E^((2*E^2*(9 - 150*x + 625*x^2))/625)),x]

[Out]

-(x^2/(-1 + E^((E^2*(3 - 25*x)^2)/625)))

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fricas [A]  time = 0.67, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x^{2}}{e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+50*x)/(25*exp(1/625*(625*x^2-1
50*x+9)*exp(1)^2)^2-50*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+25),x, algorithm="fricas")

[Out]

-x^2/(e^(1/625*(625*x^2 - 150*x + 9)*e^2) - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left ({\left (25 \, x^{3} - 3 \, x^{2}\right )} e^{2} - 25 \, x\right )} e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} + 25 \, x\right )}}{25 \, {\left (e^{\left (\frac {2}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} - 2 \, e^{\left (\frac {1}{625} \, {\left (625 \, x^{2} - 150 \, x + 9\right )} e^{2}\right )} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+50*x)/(25*exp(1/625*(625*x^2-1
50*x+9)*exp(1)^2)^2-50*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+25),x, algorithm="giac")

[Out]

integrate(2/25*(((25*x^3 - 3*x^2)*e^2 - 25*x)*e^(1/625*(625*x^2 - 150*x + 9)*e^2) + 25*x)/(e^(2/625*(625*x^2 -
150*x + 9)*e^2) - 2*e^(1/625*(625*x^2 - 150*x + 9)*e^2) + 1), x)

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maple [A]  time = 0.19, size = 22, normalized size = 0.96

method	result	size
risch	\(-\frac {x^{2}}{{\mathrm e}^{\frac {\left (25 x -3\right )^{2} {\mathrm e}^{2}}{625}}-1}\)	\(22\)
norman	\(-\frac {x^{2}}{{\mathrm e}^{\frac {\left (625 x^{2}-150 x +9\right ) {\mathrm e}^{2}}{625}}-1}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+50*x)/(25*exp(1/625*(625*x^2-150*x+9
)*exp(1)^2)^2-50*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+25),x,method=_RETURNVERBOSE)

[Out]

-x^2/(exp(1/625*(25*x-3)^2*exp(2))-1)

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maxima [A]  time = 0.39, size = 34, normalized size = 1.48 \begin {gather*} -\frac {x^{2} e^{\left (\frac {6}{25} \, x e^{2}\right )}}{e^{\left (x^{2} e^{2} + \frac {9}{625} \, e^{2}\right )} - e^{\left (\frac {6}{25} \, x e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x^3-6*x^2)*exp(1)^2-50*x)*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+50*x)/(25*exp(1/625*(625*x^2-1
50*x+9)*exp(1)^2)^2-50*exp(1/625*(625*x^2-150*x+9)*exp(1)^2)+25),x, algorithm="maxima")

[Out]

-x^2*e^(6/25*x*e^2)/(e^(x^2*e^2 + 9/625*e^2) - e^(6/25*x*e^2))

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mupad [B]  time = 0.73, size = 28, normalized size = 1.22 \begin {gather*} -\frac {x^2}{{\mathrm {e}}^{x^2\,{\mathrm {e}}^2}\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^2}{625}}\,{\mathrm {e}}^{-\frac {6\,x\,{\mathrm {e}}^2}{25}}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x - exp((exp(2)*(625*x^2 - 150*x + 9))/625)*(50*x + exp(2)*(6*x^2 - 50*x^3)))/(25*exp((2*exp(2)*(625*x
^2 - 150*x + 9))/625) - 50*exp((exp(2)*(625*x^2 - 150*x + 9))/625) + 25),x)

[Out]

-x^2/(exp(x^2*exp(2))*exp((9*exp(2))/625)*exp(-(6*x*exp(2))/25) - 1)

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sympy [A]  time = 0.17, size = 22, normalized size = 0.96 \begin {gather*} - \frac {x^{2}}{e^{\left (x^{2} - \frac {6 x}{25} + \frac {9}{625}\right ) e^{2}} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((50*x**3-6*x**2)*exp(1)**2-50*x)*exp(1/625*(625*x**2-150*x+9)*exp(1)**2)+50*x)/(25*exp(1/625*(625*
x**2-150*x+9)*exp(1)**2)**2-50*exp(1/625*(625*x**2-150*x+9)*exp(1)**2)+25),x)

[Out]

-x**2/(exp((x**2 - 6*x/25 + 9/625)*exp(2)) - 1)

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