3.74.7 \(\int \frac {25 x^2-6 e^x x^3+e^{2 x^2} (25-5 e^x x)+e^{x^2} (-50 x+e^x (x+11 x^2-2 x^3))}{25 e^{2 x^2} x-50 e^{x^2} x^2+25 x^3} \, dx\)

Optimal. Leaf size=28 \[ \frac {5}{2}-\frac {1}{25} e^x \left (5+\frac {x}{-e^{x^2}+x}\right )+\log (x) \]

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Rubi [F]  time = 1.71, 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 {25 x^2-6 e^x x^3+e^{2 x^2} \left (25-5 e^x x\right )+e^{x^2} \left (-50 x+e^x \left (x+11 x^2-2 x^3\right )\right )}{25 e^{2 x^2} x-50 e^{x^2} x^2+25 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25*x^2 - 6*E^x*x^3 + E^(2*x^2)*(25 - 5*E^x*x) + E^x^2*(-50*x + E^x*(x + 11*x^2 - 2*x^3)))/(25*E^(2*x^2)*x
- 50*E^x^2*x^2 + 25*x^3),x]

[Out]

-1/5*E^x + Log[x] + Defer[Int][E^x/(E^x^2 - x), x]/25 + Defer[Int][(E^x*x)/(E^x^2 - x)^2, x]/25 + Defer[Int][(
E^x*x)/(E^x^2 - x), x]/25 - (2*Defer[Int][(E^x*x^2)/(E^x^2 - x), x])/25 - (2*Defer[Int][(E^x*x^3)/(E^x^2 - x)^
2, x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 x^2-6 e^x x^3+e^{2 x^2} \left (25-5 e^x x\right )+e^{x^2} \left (-50 x+e^x \left (x+11 x^2-2 x^3\right )\right )}{25 \left (e^{x^2}-x\right )^2 x} \, dx\\ &=\frac {1}{25} \int \frac {25 x^2-6 e^x x^3+e^{2 x^2} \left (25-5 e^x x\right )+e^{x^2} \left (-50 x+e^x \left (x+11 x^2-2 x^3\right )\right )}{\left (e^{x^2}-x\right )^2 x} \, dx\\ &=\frac {1}{25} \int \left (-\frac {5 \left (-5+e^x x\right )}{x}-\frac {e^x x \left (-1+2 x^2\right )}{\left (e^{x^2}-x\right )^2}-\frac {e^x \left (-1-x+2 x^2\right )}{e^{x^2}-x}\right ) \, dx\\ &=-\left (\frac {1}{25} \int \frac {e^x x \left (-1+2 x^2\right )}{\left (e^{x^2}-x\right )^2} \, dx\right )-\frac {1}{25} \int \frac {e^x \left (-1-x+2 x^2\right )}{e^{x^2}-x} \, dx-\frac {1}{5} \int \frac {-5+e^x x}{x} \, dx\\ &=-\left (\frac {1}{25} \int \left (-\frac {e^x}{e^{x^2}-x}-\frac {e^x x}{e^{x^2}-x}+\frac {2 e^x x^2}{e^{x^2}-x}\right ) \, dx\right )-\frac {1}{25} \int \left (-\frac {e^x x}{\left (e^{x^2}-x\right )^2}+\frac {2 e^x x^3}{\left (e^{x^2}-x\right )^2}\right ) \, dx-\frac {1}{5} \int \left (e^x-\frac {5}{x}\right ) \, dx\\ &=\log (x)+\frac {1}{25} \int \frac {e^x}{e^{x^2}-x} \, dx+\frac {1}{25} \int \frac {e^x x}{\left (e^{x^2}-x\right )^2} \, dx+\frac {1}{25} \int \frac {e^x x}{e^{x^2}-x} \, dx-\frac {2}{25} \int \frac {e^x x^2}{e^{x^2}-x} \, dx-\frac {2}{25} \int \frac {e^x x^3}{\left (e^{x^2}-x\right )^2} \, dx-\frac {\int e^x \, dx}{5}\\ &=-\frac {e^x}{5}+\log (x)+\frac {1}{25} \int \frac {e^x}{e^{x^2}-x} \, dx+\frac {1}{25} \int \frac {e^x x}{\left (e^{x^2}-x\right )^2} \, dx+\frac {1}{25} \int \frac {e^x x}{e^{x^2}-x} \, dx-\frac {2}{25} \int \frac {e^x x^2}{e^{x^2}-x} \, dx-\frac {2}{25} \int \frac {e^x x^3}{\left (e^{x^2}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 35, normalized size = 1.25 \begin {gather*} \frac {1}{25} \left (\frac {e^x \left (-5 e^{x^2}+6 x\right )}{e^{x^2}-x}+25 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25*x^2 - 6*E^x*x^3 + E^(2*x^2)*(25 - 5*E^x*x) + E^x^2*(-50*x + E^x*(x + 11*x^2 - 2*x^3)))/(25*E^(2*
x^2)*x - 50*E^x^2*x^2 + 25*x^3),x]

[Out]

((E^x*(-5*E^x^2 + 6*x))/(E^x^2 - x) + 25*Log[x])/25

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fricas [A]  time = 1.27, size = 38, normalized size = 1.36 \begin {gather*} -\frac {6 \, x e^{x} - 25 \, {\left (x - e^{\left (x^{2}\right )}\right )} \log \relax (x) - 5 \, e^{\left (x^{2} + x\right )}}{25 \, {\left (x - e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*exp(x)*x+25)*exp(x^2)^2+((-2*x^3+11*x^2+x)*exp(x)-50*x)*exp(x^2)-6*exp(x)*x^3+25*x^2)/(25*x*exp
(x^2)^2-50*x^2*exp(x^2)+25*x^3),x, algorithm="fricas")

[Out]

-1/25*(6*x*e^x - 25*(x - e^(x^2))*log(x) - 5*e^(x^2 + x))/(x - e^(x^2))

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giac [A]  time = 0.20, size = 39, normalized size = 1.39 \begin {gather*} -\frac {6 \, x e^{x} - 25 \, x \log \relax (x) + 25 \, e^{\left (x^{2}\right )} \log \relax (x) - 5 \, e^{\left (x^{2} + x\right )}}{25 \, {\left (x - e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*exp(x)*x+25)*exp(x^2)^2+((-2*x^3+11*x^2+x)*exp(x)-50*x)*exp(x^2)-6*exp(x)*x^3+25*x^2)/(25*x*exp
(x^2)^2-50*x^2*exp(x^2)+25*x^3),x, algorithm="giac")

[Out]

-1/25*(6*x*e^x - 25*x*log(x) + 25*e^(x^2)*log(x) - 5*e^(x^2 + x))/(x - e^(x^2))

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maple [A]  time = 0.06, size = 23, normalized size = 0.82

method	result	size
risch	\(\ln \relax (x )-\frac {{\mathrm e}^{x}}{5}-\frac {{\mathrm e}^{x} x}{25 \left (-{\mathrm e}^{x^{2}}+x \right )}\)	\(23\)
norman	\(\frac {-\frac {6 \,{\mathrm e}^{x} x}{25}+\frac {{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{5}}{-{\mathrm e}^{x^{2}}+x}+\ln \relax (x )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*exp(x)*x+25)*exp(x^2)^2+((-2*x^3+11*x^2+x)*exp(x)-50*x)*exp(x^2)-6*exp(x)*x^3+25*x^2)/(25*x*exp(x^2)^
2-50*x^2*exp(x^2)+25*x^3),x,method=_RETURNVERBOSE)

[Out]

ln(x)-1/5*exp(x)-1/25*exp(x)*x/(-exp(x^2)+x)

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maxima [A]  time = 0.39, size = 29, normalized size = 1.04 \begin {gather*} -\frac {6 \, x e^{x} - 5 \, e^{\left (x^{2} + x\right )}}{25 \, {\left (x - e^{\left (x^{2}\right )}\right )}} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*exp(x)*x+25)*exp(x^2)^2+((-2*x^3+11*x^2+x)*exp(x)-50*x)*exp(x^2)-6*exp(x)*x^3+25*x^2)/(25*x*exp
(x^2)^2-50*x^2*exp(x^2)+25*x^3),x, algorithm="maxima")

[Out]

-1/25*(6*x*e^x - 5*e^(x^2 + x))/(x - e^(x^2)) + log(x)

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mupad [B]  time = 4.59, size = 30, normalized size = 1.07 \begin {gather*} \ln \relax (x)+\frac {5\,{\mathrm {e}}^{x^2+x}-6\,x\,{\mathrm {e}}^x}{25\,x-25\,{\mathrm {e}}^{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x^2)*(5*x*exp(x) - 25) + 6*x^3*exp(x) + exp(x^2)*(50*x - exp(x)*(x + 11*x^2 - 2*x^3)) - 25*x^2)/(2
5*x*exp(2*x^2) - 50*x^2*exp(x^2) + 25*x^3),x)

[Out]

log(x) + (5*exp(x + x^2) - 6*x*exp(x))/(25*x - 25*exp(x^2))

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sympy [A]  time = 0.16, size = 22, normalized size = 0.79 \begin {gather*} \frac {x e^{x}}{- 25 x + 25 e^{x^{2}}} - \frac {e^{x}}{5} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*exp(x)*x+25)*exp(x**2)**2+((-2*x**3+11*x**2+x)*exp(x)-50*x)*exp(x**2)-6*exp(x)*x**3+25*x**2)/(2
5*x*exp(x**2)**2-50*x**2*exp(x**2)+25*x**3),x)

[Out]

x*exp(x)/(-25*x + 25*exp(x**2)) - exp(x)/5 + log(x)

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